Deutsche Forschungsgemeinschaft Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties Deutsche Forschungsgemeinschaft (DFG) Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
Deutsche Forschungsgemeinschaft Macromolecular Systems: Microscopic Interactions and Macroscopic Properties Final report of the collaborative research centre 213, ``Topospezifische Chemie und Toposelektive Spektroskopie von MakromolekÏlsystemen: Mikroskopische Wechselwirkung und Makroskopische Funktion'', 1984^1995 Edited by Heinz Hoffmann, Markus Schwoerer and Thomas Vogtmann Collaborative Research Centres
Deutsche Forschungsgemeinschaft Kennedyallee 40, D-53175 Bonn, Federal Republic of Germany Postal address: D-53175 Bonn Phone: ++49/228/885-1 Telefax: ++49/228/885-2777 E-Mail: (X.400): S = postmaster; P = dfg; A = d400; C = de E-Mail: (Internet RFC 822):
[email protected] Internet: http://www.dfg.de
This book was carefully produced. Nevertheless, editors, authors and publisher do not warrant the information contained therein to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.
Library of Congress Card No.: applied for
A catalogue record for this book is available from the British Library.
Die Deutsche Bibliothek – CIP Cataloguing-in-Publication Data A catalogue record for this publication is available from Die Deutsche Bibliothek ISBN 3-527-27726-9
WILEY-VCH Verlag GmbH, D-69469 Weinheim (Federal Republic of Germany), 2000 Printed on acid-free and chlorine-free paper All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Cover Design and Typography: Dieter Hüsken Composition: ProSatz Unger, D-69469 Weinheim Printing: betz-druck gmbh, D-64291 Darmstadt Bookbindung: J. Schäffer GmbH & Co. KG, D-67269 Grünstadt Printed in the Federal Republic of Germany
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
XV
List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XVII
I
Mainly Solids
1
Model Systems for Photoconductive Material . . . . . . . Harald Meyer and Dietrich Haarer Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental techniques . . . . . . . . . . . . . . . . . . . . . . . . Transport models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moleculary doped polymers . . . . . . . . . . . . . . . . . . . . . . Side-chain polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . Conjugated systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.............. ... ... ... ... ... ... ... ... ...
3 4 5 7 7 9 10 12 13
Novel Photoconductive Polymers . . . . . . . . . . . . . . . . . Jörg Bettenhausen and Peter Strohriegl Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid crystalline oxadiazoles and thiadiazoles . . . . . . . . The basic idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Monomer synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oligo- and polysiloxanes with pendant oxadiazole groups Photoconductivity measurements . . . . . . . . . . . . . . . . . . Starburst oxadiazole compounds . . . . . . . . . . . . . . . . . . . Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Synthesis of starburst oxadiazole compounds . . . . . . . . . Thermal properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..............
15
........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ...........
15 16 16 17 19 21 22 22 24 27 30
1.1 1.2 1.3 1.4 1.4.1 1.4.2 1.4.3 1.5
2 2.1 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.3 2.3.1 2.3.2 2.3.3
........... ........... ........... ........... ........... ........... ........... ........... ...........
... ... ... ... ... ... ... ... ... ... ...
3
V
Contents 3 3.1 3.2 3.3 3.4 3.5
4
4.1 4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.3 4.4 4.4.1 4.4.2 4.4.3 4.5
5 5.1 5.2 5.3 5.4 5.5
6
6.1 6.2 6.3 6.4
VI
Theoretical Aspects of Anomalous Diffusion in Complex Systems . . . Alexander Blumen General aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Photoconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Matheron-de-Marsily model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polymer chains in MdM flow fields . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
...
31
... ... ... ... ... ... ...
31 32 36 38 42 42 43
...
44
... ... ... ... ... ... ... ... ... ... ... ... ... ...
44 46 46 47 49 52 54 55 55 58 62 65 66 66
Spectral Diffusion due to Tunneling Processes at very low Temperatures Hans Maier, Karl-Peter Müller, Siegbert Jahn, and Dietrich Haarer Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The optical cryostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
Low-Temperature Heat Release, Sound Velocity and Attenuation, Specific Heat and Thermal Conductivity in Polymers . . . . . . . . . . . . Andreas Nittke, Michael Scherl, Pablo Esquinazi, Wolfgang Lorenz, Junyun Li, and Frank Pobell Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phenomenological theory for heat release . . . . . . . . . . . . . . . . . . . . . . . Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The standard tunneling model with infinite cooling rate . . . . . . . . . . . . Influence of higher-order tunneling processes and a finite cooling rate . The influence of a constant and thermally activated relaxation rate . . . . Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . Specific heat and thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . Internal friction and sound velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . Heat release . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optically Induced Spectral Diffusion in Polymers Containing Water Molecules: A TLS Model System . . . . . . . . . . . . . . . . . . . . . . . . . . . . Klaus Barth, Dietrich Haarer, and Wolfgang Richter Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental setup for burning and detecting spectral holes . . . . . . . . . Reversible line broadening phenomena . . . . . . . . . . . . . . . . . . . . . . . . . Induced spectral diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68 69 71 73 74 76
...
78
... ... ... ... ...
78 79 80 83 87
Contents 7 7.1 7.2 7.2.1 7.2.2 7.2.3 7.2.4 7.3 7.3.1 7.3.2 7.3.3 7.4
8
8.1 8.2 8.3 8.4 8.5
9
9.1 9.2 9.2.1 9.2.2 9.2.3 9.2.3.1 9.2.3.2 9.2.3.3 9.2.4 9.3 9.3.1 9.3.2 9.3.3 9.3.4 9.3.5 9.3.6 9.3.7
Slave-Boson Approach to Strongly Correlated Electron Systems . . . . Holger Fehske, Martin Deeg, and Helmut Büttner Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Slave-boson theory for the t-t'-J model . . . . . . . . . . . . . . . . . . . . . . . . . SU(2)-invariant slave-particle representation . . . . . . . . . . . . . . . . . . . . . Functional integral formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saddle-point approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetic phase diagram of the t-t'-J model . . . . . . . . . . . . . . . . . . . . . Comparison with experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Normal-state transport properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetic correlations and spin dynamics . . . . . . . . . . . . . . . . . . . . . . . Inelastic neutron scattering measurements . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
...
88
... ... ... ... ... ... ... ... ... ... ... ...
88 90 90 93 95 97 102 102 105 106 109 110
...
113
... ... ... ... ... ... ...
113 114 116 117 119 120 120
..............
122
........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ...........
122 129 129 131 131 131 136 139 141 144 145 147 149 150 152 152 154
Non-Linear Excitations and the Electronic Structure of Conjugated Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Klaus Fesser Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-linear excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diacetylene Single Crystals . . . . . . . . . . . . . . . . . . . . . Markus Schwoerer, Elmar Dormann, Thomas Vogtmann, and Andreas Feldner Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Photopolymerization . . . . . . . . . . . . . . . . . . . . . . . . . . . Carbenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intermediate photoproducts . . . . . . . . . . . . . . . . . . . . . . Electronic structure of dicarbenes . . . . . . . . . . . . . . . . . . Electron spin resonance of quintet states (5DCn) . . . . . . ENDOR of quintet states . . . . . . . . . . . . . . . . . . . . . . . . ESR and ENDOR of triplet dicarbenes 3DCn . . . . . . . . . Flash photolysis and reaction dynamics of diradicals . . . . Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . General characterization . . . . . . . . . . . . . . . . . . . . . . . . Angular selectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prepolymerized samples . . . . . . . . . . . . . . . . . . . . . . . . . Chain length, polymer profile, and grating profiles . . . . . Multrecording . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
VII
Contents 9.3.8 9.4 9.4.1 9.4.1.1 9.4.1.2 9.4.1.3 9.4.2 9.4.2.1 9.4.2.2 9.4.2.3 9.4.2.4 9.4.3 9.4.4 9.5 9.5.1 9.5.2 9.5.3 9.5.4 9.5.5 9.5.6 9.5.7 9.5.8 9.5.9 9.5.10
Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Di-, pyro-, and ferroelectricity . . . . . . . . . . . . . . . . . . . . Dielectric properties of diacetylenes . . . . . . . . . . . . . . . . Correlation of polymer content and electric permittivity . Application to topospecifically modified diacetylenes . . . Additional applications . . . . . . . . . . . . . . . . . . . . . . . . . Pyroelectric diacetylenes . . . . . . . . . . . . . . . . . . . . . . . . IPUDO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NP/4-MPU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DNP/MNP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spurious piezo and pyroelectricity of diacetylenes . . . . . . The ferroelectric diacetylene DNP . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-linear optical properties . . . . . . . . . . . . . . . . . . . . . Aims of investigation . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical approaches . . . . . . . . . . . . . . . . . . . . . . . . . Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . Value and phase of the third order susceptibility w(3) . . . . Relaxation of the singlet exciton . . . . . . . . . . . . . . . . . . The w(3) tensor components . . . . . . . . . . . . . . . . . . . . . . Signal saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectral dispersion, phase, and relaxation of w(5) . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ...........
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
154 155 156 156 158 159 159 159 160 161 161 162 166 167 167 167 170 170 170 171 172 173 174 176 177
10
...
181
10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8
Matrix-Molecule Interaction in Dye-Doped Rare Gas Solids . . . . . . . Thomas Giering, Peter Geißinger, Wolfgang Richter, and Dietrich Haarer Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stochastic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rare gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inhomogeneous absorption lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rare gas mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
... ... ... ... ... ... ... ... ... ...
181 183 186 187 188 190 191 194 195 195
II
Mainly Micelles, Polymers, and Liquid Crystals
11
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heinz Hoffmann General behaviour of surfactants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.1 VIII
199 199
Contents 11.2 11.3 11.3.1 11.3.2 11.3.3 11.4 11.4.1 11.4.2 11.4.3 11.4.4 11.5 11.5.1 11.5.2 11.5.3 11.5.4 11.6 11.6.1 11.6.2 11.6.3 11.6.4 11.7 11.7.1 11.7.2 11.8 11.9
From globular micelles towards bilayers . . . . . . . . . . . . . . . . . . . . . . . . Viscoelastic solutions with entangled rods . . . . . . . . . . . . . . . . . . . . . . General behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscoelastic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanisms for the different scaling behaviour . . . . . . . . . . . . . . . . . . Viscoelastic solutions with multilamellar vesicles . . . . . . . . . . . . . . . . . The conditions for the existence of vesicles . . . . . . . . . . . . . . . . . . . . . Freeze fracture electron microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . Rheological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model for the shear modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ringing gels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The aminoxide system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The bis-(2-ethylhexyl)sulfosuccinate system . . . . . . . . . . . . . . . . . . . . . PEO-PPO-PEO block copolymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lyotropic mesophases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nematic phases and their properties . . . . . . . . . . . . . . . . . . . . . . . . . . . Cholesteric phases and their properties . . . . . . . . . . . . . . . . . . . . . . . . . Vesicle phases and L3 phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shear induced phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Under what conditions do we find drag-reducing surfactants? . . . . . . . . SANS measurements on micellar systems . . . . . . . . . . . . . . . . . . . . . . . A new rheometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
200 202 202 205 209 211 211 212 213 217 220 220 221 224 226 227 227 228 232 233 236 236 236 239 243 247
12
Photophysics of J Aggregates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hermann Pschierer, Hauke Wendt, and Josef Friedrich Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic aspects of pressure and electric field phenomena in hole burning spectroscopy of J aggregates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electric field-induced phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
...
251
...
251
... ... ... ... ... ... ... ...
252 253 254 256 256 258 258 259
..............
260
........... ........... ........... ...........
260 264 264 266
12.1 12.2 12.3 12.4 12.5 12.5.1 12.5.2
13 13.1 13.2 13.2.1 13.2.2
Convection Instabilities in Nematic Liquid Crystals . . Lorenz Kramer and Werner Pesch Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic equations and instability mechanisms . . . . . . . . . . The director equation . . . . . . . . . . . . . . . . . . . . . . . . . . . The velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
... ... ... ...
IX
Contents 13.2.3 13.2.3.1 13.2.3.2 13.2.4 13.3 13.4 13.5 13.5.1 13.5.2 13.5.3 13.5.3.1 13.5.3.2 13.6
Electroconvection . . . . . . . . . . . . . . . . . . . The standard model . . . . . . . . . . . . . . . . . . The weak electrolyte model . . . . . . . . . . . . Rayleigh-Bénard convection . . . . . . . . . . . . Theoretical analysis . . . . . . . . . . . . . . . . . . Rayleigh-Bénard convection . . . . . . . . . . . . Electrohydrodynamic convection . . . . . . . . Linear theory and type of bifurcation . . . . . Results of Ginzburg-Landau equation . . . . . Beyond the Ginzburg-Landau equation . . . . Experimental results . . . . . . . . . . . . . . . . . Theoretical results and discussion . . . . . . . . Concluding remarks . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . Note added . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .
14
Preparation and Properties of Ionic and Surface Modified Micronetworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michael Mirke, Ralf Grottenmüller, and Manfred Schmidt Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polymerization in normal microemulsion . . . . . . . . . . . . . . . . . . . . . . . Mechanism and size control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Surface functionalization of microgels . . . . . . . . . . . . . . . . . . . . . . . . . Polymerization in inverse microemulsion . . . . . . . . . . . . . . . . . . . . . . . Preparation of ionic microgels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of ionic microgels and interparticle interaction . . . . . . . . . . . Conclusion and relevance to future work . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.1 14.2 14.2.1 14.2.2 14.3 14.3.1 14.3.2 14.4
15
15.1 15.2 15.2.1 15.2.2 15.2.3 15.2.3.1 15.2.3.2 15.2.4 15.3 15.3.1 15.3.2 15.3.2.1 15.3.2.2 X
.......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... ..........
........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ...........
Ferrocene-Containing Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oskar Nuyken, Volker Burkhardt, Thomas Pöhlmann, Max Herberhold, Fred Jochen Litterst, and Christian Hübsch Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Addition polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radical polymerization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radical copolymerization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anionic polymerization of VFc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Living polymerization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Block copolymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polymeranalogeous reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polymers with ferrocene units in the main chain . . . . . . . . . . . . . . . . . . Polycondensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polymers by addition of dithiols to diolefins . . . . . . . . . . . . . . . . . . . . . Radical reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Base catalyzed reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
267 267 268 269 269 275 278 278 279 281 281 282 286 288 289 290
...
295
... ... ... ... ... ... ... ... ...
295 295 295 298 299 299 299 303 304
...
305
... ... ... ... ... ... ... ... ... ... ... ... ...
305 306 306 308 308 309 312 314 315 315 316 316 316
Contents 15.3.2.3 15.3.3 15.3.4 15.4
Acid catalyzed reactions . . . . . . . . . . . . . . . . . . . . . . . . 1,1'-dimercapto-ferrocene as initiator . . . . . . . . . . . . . . . Reductive coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mößbauer studies of polymers containing ferrocene . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
........... ........... ........... ........... ...........
... ... ... ... ...
318 319 320 320 323
16
Transfer of Vibrational Energy in Dye-Doped Polymers . . . . . . . . . . Johannes Baier, Thomas Dahinten, and Alois Seilmeier Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
...
325
... ... ... ... ...
325 326 327 332 332
...
333
... ... ... ... ... ... ... ... ... ...
333 334 337 337 337 339 341 342 343 343
..............
344
........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ...........
344 344 344 345 345 346 346 346 347 348 348 348 349 350 350 352
16.1 16.2 16.3 16.4
17
17.1 17.2 17.3 17.4 17.4.1 17.4.2 17.4.3 17.4.4 17.5
18 18.1 18.1.1 18.1.2 18.1.3 18.1.4 18.1.5 18.2 18.2.1 18.2.2 18.2.3 18.3 18.3.1 18.3.2 18.4 18.4.1 18.4.2
Picosecond Laser Induced Photophysical Processes of Thiophene Oligomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dieter Grebner, Matthias Helbig, and Sabine Rentsch Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectroscopic properties of oligothiophenes . . . . . . . . . . . . . . . . . . . . . Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Picosecond-transient spectra of oligothiophenes in solution . . . . . . . . . . Time behaviour of transient spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . Size dependence of spectroscopic properties of oligothiophenes . . . . . . Size dependence of the kinetic behaviour of oligothiophenes . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Topospecific Chemistry at Surfaces . . . . . . . . . . . . . . . Hans Ludwig Krauss Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preparative and analytical methods . . . . . . . . . . . . . . . . . Industrial applications . . . . . . . . . . . . . . . . . . . . . . . . . . The standard procedures of the Phillips process . . . . . . . Earlier work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unmodified silica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modified silica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Others . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transition metal surface compounds . . . . . . . . . . . . . . . . The metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Impregnation and activation . . . . . . . . . . . . . . . . . . . . . . Coordinatively unsaturated sites . . . . . . . . . . . . . . . . . . . Reduction of saturated surface compounds . . . . . . . . . . . Elimination of ligands . . . . . . . . . . . . . . . . . . . . . . . . . .
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
XI
Contents 18.5 18.5.1 18.5.2 18.6 18.6.1 18.6.2 18.6.3 18.7 18.8
Physical properties of the coordinatively unsaturated sites . . . . . . . . . . . Topologically different sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optical and magnetic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chemical properties of the coordinatively unsaturated sites . . . . . . . . . . Survey of catalytic reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Olefin polymerization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other catalytic reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deactivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
... ... ... ... ... ... ... ... ... ...
353 353 353 355 355 356 360 361 362 363
III
Biopolymers
Site-Directed Spectroscopy and Site-Directed Chemistry of Biopolymers Stefan Limmer, Günther Ott, and Mathias Sprinzl 19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Site-specific NMR spectroscopy of chemically synthesized RNA duplexes . . 19.2.1 Stability of tRNA-derived acceptor stem duplexes . . . . . . . . . . . . . . . . . . . . 19.2.2 Manganese ion binding sites at RNA duplexes . . . . . . . . . . . . . . . . . . . . . . 19.2.3 Structural determination of short RNA duplexes by 2D NMR spectroscopy . 19.2.4 NMR derived model of the tRNAAla acceptor arm . . . . . . . . . . . . . . . . . . . 19.2.5 Chemical shifts and scalar coupling as an indicator of RNA structure in the vicinity of a G-U pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2.6 Structure of aminoacyl-tRNA and transacylation of the aminoacyl residue . . 19.3 Structure of elongation factor Tu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3.1 Sequence of Thermus thermophilus EF-Tu . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3.2 Crystallization, X-ray analysis, and the tertiary structure . . . . . . . . . . . . . . . 19.3.3 Nucleotide binding and GTPase reaction . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3.4 Mechanism of GTP induced conformational change of EF-Tu . . . . . . . . . . . 19.3.5 Aminoacyl-tRNA in complex with EF-Tu 7 GTP . . . . . . . . . . . . . . . . . . . . . 19.3.6 1H NMR of yeast Phe-tRNAPhe EF-Tu 7 GTP complex . . . . . . . . . . . . . . . . 19.3.7 13C NMR studies of the Val-tRNAVal EF-Tu 7 GTP ternary complex . . . . . . . 19.3.8 Role of EF-Tu in complex with aminoacyl-tRNA . . . . . . . . . . . . . . . . . . . . . 19.3.9 EF-Tu interaction with EF-Ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19. 3. 10 Site-directed mutagenesis of EF-Tu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
369
19
20 20.1 20.2 20.2.1
XII
Spectroscopic Probes of Surfactant Systems and Biopolymers . . . . . . Alexander Wokaun Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diffusion in surfactant systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structural characteristics of micellar solutions, cubic phases, and multilamellar vesicles from NMR self-diffusion measurements . . . . . . . . . . .
369 370 371 374 377 379 380 382 383 384 386 387 388 389 391 393 395 395 396 397 397 398
...
401
... ...
401 402
...
402
Contents 20.2.2 20.2.3 20.2.4 20.3 20.3.1 20.3.2 20.3.3 20.3.4 20.4 20.5
21 21.1 21.2 21.3 21.4 21.5 21.6 21.7
Probing of mobilities in multilamellar vesicles by forced Rayleigh scattering Dimensionality of diffusion in lyotropic mesophases from fluorescence quenching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vibrational spectroscopy and conformational analysis of oligonucleotides . . Spectroscopic characterization of right and left-helical forms of a hexadecanucleotide duplex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SERS spectra of deoxyribonucleotides . . . . . . . . . . . . . . . . . . . . . . . . . . . . Studies of chromophore-DNA interaction by vibrational spectroscopy . . . . . Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Related projects carried out within the framework of the Collaborative Research Centre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Remarks and acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
405
Energy Transport by Lattice Solitons in a-Helical Proteins . . . . . . . . Franz-Georg Mertens, Dieter Hochstrasser, and Helmut Büttner Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quasicontinuum approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity range for the quasicontinuum approach . . . . . . . . . . . . . . . . . . Solitary waves for realistic parameter values . . . . . . . . . . . . . . . . . . . . . Iterative method and stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
...
424
... ... ... ... ... ... ... ...
424 426 428 431 432 434 437 438
...
443
... ... ... ...
443 444 444 444
... ... ... ... ... ...
445 446 447 450 457 459
IV
Appendix
22
Documentation of the Collaborative Research Centre 213 . . . . . . . . . Markus Schwoerer and Heinz Hoffmann List of Members . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heads of Projects (Teilprojektleiter) . . . . . . . . . . . . . . . . . . . . . . . . . . . Projektbereich A: Gemeinsame Einrichtungen . . . . . . . . . . . . . . . . . . . Projektbereich B: Festkörper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Projektbereich C: Funktionale Systeme – Mizellen, Oberflächen und Polymere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Projektbereich D: Biopolymere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Guests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Co-workers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . International Cooperation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Funding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22.1 22.2 22.2.1 22.2.2 22.2.3 22.2.4 22.3 22.4 22.5 22.6
409 412 413 413 415 417 418 419 421 422
XIII
Preface
At the end of 1983, about eight years after the inauguration of the University of Bayreuth the Deutsche Forschungsgemeinschaft (DFG) agreed to establish the Collaborative Research Centre 213, TOPOMAC, for the promotion of basic research in chemistry and physics of macromolecular systems. The title of TOPOMAC in its full length, “Topospezifische Chemie und Toposelektive Spektroskopie von Makromolekülsystemen: Mikroskopische Wechselwirkung und Makroskopische Funktion” expessed the intention of the original applicants: a productive cooperation between physicists, chemists and biochemists across the mutual borders of their original research fields. Until the end of 1995 TOPOMAC was supported by the Deutsche Forschungsgemeinschaft with 28 MDM. The present book documents the achievements of TOPOMAC. It is not a minute addition of all the results which have been published in periodical journals but rather a survey of important research fields of members of TOPOMAC. The articles have been written towards or after the end of the support period as both, review and original publications. The book covers the fields of: . Electronic, photoelectric, thermal, dielectric, optical and magnetic properties of macromolecular solids (polymers and polymer crystals), . Micellar structures, J-aggregates, liquid crystals, µ-gels, ferrocene-containing polymers and topospecific chemistry at surfaces and in single crystals, and . Biopolymers and surfactant systems as studied by site directed spectroscopy and site directed chemistry and also by the theory of energy transport. During the period of its support TOPOMAC had 30 members (Teilprojektleiter). Only ten of them have been members for the entire period, mainly because twenty times a call from other universities or research institutions reached one of the members of the Collaborative Research Centre 213. Fourteen of them followed this call and left the Collaborative Research Centre 213. Less than two years after the end of the period of support some of the remaining former members of Collaborative Research Centre 213 together with young and new faculty members began to continue the formal cooperation between chemists and physicists in the field of macromolecular research. Their actual cooperation in the meantime never had been terminated. The editors would like to express the sincere thanks of the members of TOPOMAC to the foreign guests of TOPOMAC, to the Deutsche Forschungsgemeinschaft, to the University XV
Preface of Bayreuth and also to the Freistaat Bayern. Our foreign guests stayed for long or short periods between one year and one day. They have contributed in an essential and special manner to the success on our research fields. They also strongly intensified the national and international scientific relations of both, the members and the research students of TOPOMAC. The cooperation of the speakers of TOPOMAC with Dr. Funk from the Deutsche Forschungsgemeinschaft office throughout the entire period of support was excellent. The help of the former president of the university, Dr. K. D. Wolff and his chancellor, W. P. Hentschel as well as the continuous support by the late Ministerialrat G. Grote and by J. Großkreutz from the Bayerisches Staatsministerium für Unterricht, Kultus, Wissenschaft und Kunst has been an essential stimulus for the scientific members of TOPOMAC. Last but not least we thank Doris Buntkowski for her faithful and reliable work as our secretary. The Editors
XVI
List of Contributors
Alexander Blumen Theoretische Polymerphysik Universität Freiburg Herrmann-Herder-Straße 3 79104 Freiburg
Dietrich Haarer Experimentalphysik IV Universität Bayreuth Universitätsstraße 30 95447 Bayreuth
Elmar Dormann Physikalisches Institut Universität Karlsruhe Engesserstr. 7 67131 Karlsruhe
Heinz Hoffmann Physikalische Chemie I Universität Bayreuth Universitätsstraße 30 95447 Bayreuth
Pablo Esquinazi Institut für Experimentelle Physik II Universität Leipzig Linnestraße 5 04103 Leipzig
Lorenz Kramer Theoretische Physik II Universität Bayreuth Universitätsstraße 30 95447 Bayreuth
Klaus Fesser Fachbereich Physik Universität Greifswald Domstraße 10 a 17489 Greifswald
Hans Ludwig Krauss Heunischstraße 5 b 96049 Bamberg
Josef Friedrich Lehrstuhl für Physik Technische Universität München 85350 Freising-Weihenstephan
Franz G. Mertens Theoretische Physik Universität Bayreuth Universitätsstraße 30 95447 Bayreuth
Holger Fehske Theoretische Physik I Universität Bayreuth Universitätsstraße 30 95447 Bayreuth
Oskar Nuyken Lehrstuhl für Makromolekulare Stoffe Technische Universität München Lichtenbergstraße 4 85747 München XVII
List of Contributors Sabine Rentsch Institut für Optik und Quantenelektronik Friedrich Schiller Universität Jena Max-Wien-Platz 1 07743 Jena
Alois Seilmeier Physikalisches Institut Universität Bayreuth Universitätsstraße 30 95447 Bayreuth
Wolfgang Richter Experimentalphysik IV Universität Bayreuth Universitätsstraße 30 95447 Bayreuth
Mathias Sprinzl Biochemie Universität Bayreuth Universitätsstraße 30 95447 Bayreuth
Paul Rösch Biopolymere Universität Bayreuth Universitätsstraße 30 95447 Bayreuth Manfred Schmidt Institut für Physikalische Chemie Universität Mainz Welder-Weg 11 55099 Mainz Markus Schwoerer Experimentalphysik II Universität Bayreuth Universitätsstraße 30 95447 Bayreuth
XVIII
Peter Strohriegel Makromolekulare Chemie I Universität Bayreuth Universitätsstraße 30 95447 Bayreuth Thomas Vogtmann Experimentalphysik II Universität Bayreuth Universitätsstraße 30 95447 Bayreuth Alexander Wokaun Bereich F5 Paul Scherrer Institut CH-5232 Villingen
I
Mainly Solids
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties Deutsche Forschungsgemeinschaft (DFG) Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
1
Model Systems for Photoconductive Materials Harald Meyer and Dietrich Haarer
1.1
Introduction
The effect of photoconductivity, i. e. the increase of the electrical conductivity of a material upon illumination with light of suitable photon energy, was first discovered in selenium by W. Smith in 1873. A typical application for these materials is the xerographic process [2, 3], which was developed by C. F. Carlson in 1942 [1]. Here, a photoconducting film on top of a grounded electrode is homogeneously charged by a corona discharge. Typical field strengths are up to 106 V/m. In a second step the image of the original document is projected onto the film. At the areas where the film is illuminated charge carriers are generated in the photoconducting film. One species traverses the film and recombines at the grounded electrode whereas the oppositely charged species neutralizes the surface charges. With this step the original image is transferred into an electrostatic image on the photoconductor film. Subsequently, small toner particles are deposited on the photoconductor. They stick to the charged regions and thus generate a real image. In the next step, this image is transferred onto paper and fixed by thermally fusing the toner particles on the paper. The process for a laser printer is similar except for the fact that the image is written onto the photoconductor directly by a laser or a diode array. If the toner particles are fused directly onto the photoconductor instead of transferring them onto paper the photoconductor can be used as an offset printing master [4]. Potential systems for commercial use have to meet several requirements: a) sensitivity in the visible region of the spectrum; b) good charge transport properties, i. e. charge carrier mobility in excess of 10 –7 cm2/Vs [3] and minor trapping effects; c) good mechanical and dielectrical properties; d) excellent film forming properties and possibility of manufacturing defect free large area films; e) mechanical flexibility for the use in small sized desktop devices. The last two requirements cannot be met neither by organic nor inorganic crystalline materials. Therefore both, organic and inorganic amorphous photoconductors, have been deMacromolecular Systems: Microscopic Interactions and Macroscopic Properties Deutsche Forschungsgemeinschaft (DFG) Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
3
1 Model Systems for Photoconductive Materials veloped. Inorganic materials like e. g. a-Se, As2Se3 or alloys of Se and Te show good transport properties with mobilities in the range of 0.1 cm2/Vs at room temperature [7], and high sensitivity for visible light together with moderate mechanical and dielectric properties. One of their most important disadvantages compared to organic systems is that most of these materials are highly toxic. In the past 30 years, since H. Hoegl discovered photoconductivity in the organic polymer poly(N-vinylcarbazole) (PVK) (Fig. 1.2) [5, 6], organic materials have almost completely replaced their inorganic counterparts, although the effective mobility in these systems is typically 5 orders of magnitude lower as compared to a-Se. Organic photoconductors can be regarded as large or medium bandgap semiconductors. A typical value for commercially applied systems is 3.5 eV for carbazole derivatives (Section 1.4). This inherent lack of sensitivity in the visible region has been overcome by using charge transfer systems [5] or multilayer systems with additional charge generation layers. For a review see e. g. Ref. [3]. On the other hand, the large bandgap of organic materials virtually eliminates the influence of thermally generated charge carriers and thus improves the dielectric properties as compared to low bandgap materials. Therefore the main target of the works, which will be described in the following, was to identify the key factors which limit the charge carrier mobilities in organic systems and to develop new high mobility materials.
1.2
Experimental techniques
Besides the xerographic discharge method [3, 10], the time-of-flight (TOF) technique is generally used to study the charge carrier transport of thin organic films. Here, the photoconducting film with a typical thickness of 10 mm is sandwiched between two electrodes (Fig. 1.1). Electron-hole pairs are generated by the energy hn of a strongly absorbed laser pulse, which is irradiated through one of the semitransparent electrodes. For most organic materials the charge carrier generation process can be described by an Onsager model, either in its one or three dimensional form [11–13]. The wavelength of the laser is chosen to ensure that the penetration depth is considerably less than the sample thickness. Thus the charge carriers are generated close to the illuminated surface. Under the influence of an externally applied electrical field the electronhole pairs are separated and one species, depending on the polarity of the external field, immediately recombines at the illuminated electrode. The opposite charged species drifts through the sample, thus giving rise to a time dependent photocurrent Ip (t). For a quantitative analysis the knowledge about the electrical field inside the sample is necessary. This can be achieved by performing the measurements in the small signal limit, i. e. disturbations due to space charge effects are negligible and the electric field in the bulk is determined by the externally applied field. It turns out, that this condition is fulfilled as long as the generated photocharge Qtot is less than 10 % of the charge Q = CU, which is 4
1.3
Transport models
Figure 1.1: Principle setup for TOF experiments.
stored on the electrodes. With typical experimental parameters (sample thickness d = 10 mm, sample area A = 4 cm2, 1021 monomer units per cm3, voltage U = 300 V, sample capacitance C = 1 nF) the above criterion leads to an upper limit for the photocharges of Qtot < 30 nF, corresponding to more than 26107 monomer units per charge carrier. For TOF experiments blocking contacts, e. g. Al for hole transport, have to be used to avoid trap filling due to an unduly large dark current. The quasi-Fermi energy e fp can be estimated from [69] p0 Nc exp
"f p =kB T; V
1
where V is the sample volume, p0 the number of charge carriers due to the dark current, and Nc is the density of states at the mobility edge. With typical values Nc = 1019 cm –3, T = 300 K, p0 = 36104 cm –3, equivalent to a dark current Id = 1 nA [68], the quasi-Fermi energy can be estimated to be approximately 0.83 eV. As will be shown later in Section 1.4, typical trap depths are of the order of 0.50 eV or less and therefore trap filling effects due to the dark current are not expected for the system under consideration.
1.3
Transport models
Basically two models are used to interpret the charge carrier transport properties of amorphous solids in the high temperature regime. The Multiple Trapping (MT) model [46] is based on the assumption that there exists a well-defined energy ed which separates electronic states where charge carriers are highly mobile from localized traps. In the conventional band picture this energy can be identified with the mobility edge. In the MT model the transport occurs by a sequence of trapping and detrapping events. Direct transfer between two traps is 5
1 Model Systems for Photoconductive Materials neglected, or in other words, the detrapping always occurs to states at ed . The traps can be caused by static disorder as well as by self trapping due to polaronic effects or both. Mathematically equivalent with the MT model is the Continuous Time Random Walk (CTRW) model [47–49], where the exponential trap distribution density g (e) in the MT model, " g
" / exp ;
2 kB T 0 corresponds to the well-known algebraic distribution of hopping times in the CTRW model [45],
t / t
1
;
3
where the disorder parameter a in Eq. 3 corresponds to the dimensionless temperature (a = T/T0). A variety of experimental data [16, 25, 26, 33, 34] can be described by the empirical formula, p "0 E ef f 0 exp ;
4 kB Tef f first proposed by Gill [33], where e0 is the zero field activation energy, E the electrical field and b the Poole-Frenkel factor. This equation describes the thermal activated release from localized traps where the activation energy is lowered by an external field according to the Poole-Frenkel effect [37]. The effective temperature Teff is related to the physical temperature T by 1 1 Tef f T
1 : T0
5
Initially, the characteristic temperature T0 simply was an empirical parameter. In Section 1.4, however, we shall see that in certain cases this parameter can be interpreted microscopically. An alternative approach [28, 50–54] is based on the assumption that the density of states can be modelled by a Gaussian distribution. Charge carrier transport occurs via direct hopping between the localized sites. In general, the differences between the two models are too small to be detected experimentally. Since our data can be quantitatively explained within the framework of multiple trapping we will restrict the discussion to this model.
6
1.4
1.4
Results
Results
As stated in Section 1.1, organic photoconductors can be regarded as large or medium bandgap semiconductors. In contrast to inorganic crystalline semiconductors the electronic coupling between adjacent sites is weak. Therefore the electronic states in these materials are primarily of molecular character and the charge carrier transport has to be regarded as an incoherent hopping process between two neighbouring sites. For the following discussion organic photoconductors will be divided into three main groups namely molecularly doped polymers (Section 1.4.1), side-chain polymers (Section 1.4.2.), and conjugated systems (Section 1.4.3).
1.4.1 Molecularly doped polymers In molecularly doped polymers (MDPs) the transport molecules are molecularly dispersed in an inert matrix. Due to crystallization the maximum concentration of chromophores, which can be achieved, is typically in the range of 50 mol%. Typical chemical compounds include oxadiazole derivatives [14], pyrazolines [15– 18], hydrazones [19–22], carbazole derivatives [23–26], triphenylmethane (TPM) derivatives [27, 28], triphenylamine (TPA) derivatives [29, 30], and TAPC [31], which can be regarded as a dimer of TPA. The charge carrier mobilities at room temperature are typically in the range from 10 –6 cm2/Vs for N-isopropylcarbazole [25] to 10–4 cm2/Vs for p-diethylaminobenzaldehyde diphenyl hydrazone (DEH) [22]. In order to study the effect of chemical substitutions of the respective transport molecule N-isopropylcarbazole (NIPC) and derivatives thereof, with electron donor as well as acceptor substituents (Fig. 1.2), have been investigated in a polycarbonate host [26]. As compared by 3,6-dibromo-N-isopropylcarbazole (DBr-NIPC) the effective hole mobility of 3,6-dimethoxy-N-isopropylcarbazole (DMO-NIPC) is slightly decreased. The reason for this behaviour is the fact that the cation, which is relevant for the transport process, is stabilized by electron donating substituents like the methoxy group, as can be seen from the shift of the ionisation potentials [26]. This leads to stronger localization of the charge within the aromatic rings as compared to derivatives with moderate electron acceptors like bromine. Therefore the spatial overlap of the electronic states of adjacent molecules, which are involved in the transport process, is reduced. With strong acceptors like the nitro groups in 3,6-dinitro-N-isopropylcarbazole (DNNIPC) the ionisation potential is already larger than for the host matrix polycarbonate. In this case even the matrix acts as a trap for holes, thus preventing efficient charge carrier transport [26]. Since the effective mobility is determined by the spatial overlap of the transport states, the concentration dependence of the effective mobility can be described by ef f / r2 exp
2r ; r0
6
7
1 Model Systems for Photoconductive Materials
H3CO
N
R
N N
C2H5 N BTA
R
R:
H: Br: OCH3: NO 2:
N
NIPC DBr-NIPC DMO-NIPC DN-NIPC
Si CH3 CH
Si
CH2
Ca
O
n
PMPS
(CH2)n
n
PVK
Ca
R
n
Polysiloxan
R N Ca:
R:
n
O DPOP-PPV H
PPV
Figure 1.2: Chemical structure of photoconducting materials. Abbreviations see text.
where r is the mean distance between adjacent transport molecules and r0 the wave function decay parameter. For carbazole derivatives r0 is typically in the range 1.3–1.6 Å, depending on the substituted groups [26]. In recent years, also percolation models have been successfully applied to experimental data [55] as an alternative approach to model the transport properties of doped disordered systems. For polycarbonate doped with a derivative of benztriazole (BTA) (Fig. 1.2) it has been shown that the concentration dependence of the effective mobility can be described at low concentrations by [55] ef f 0
p
pc t
7
for p > pc , where p is the concentration of transport molecules, pc the percolation threshold, and t the critical exponent. The experimental values (pc = 0.095 and t = 2.46) are consistent with 3D-continuum percolation calculations [56]. Since this system shows good transport properties over a wide range of concentrations, the influence of extrinsic traps could be studied [57]. Here, a sample containing 25 wt% of BTA in a polycarbonate host was doped with small amounts of the organic dye astrazone orange (AO). It turned out that the effective mobility is reduced by a factor of 2 when the 8
1.4
Results
concentration of AO is increased to 0.5 wt%. This finding becomes also important when thinking of new applications like organic electroluminescent devices. One way to increase the efficiency in these devices is to molecularly disperse the luminescent moiety into a charge transport material. But there is a trade-off between efficient trapping of the charge carriers – which is improved by higher concentrations of the luminescent groups – and concentration quenching of the fluorescence, which can be avoided by keeping the concentration as low as possible. The above described measurements [57] show that organic dyes can be very efficient traps for charge carriers. Therefore the concentration of the luminescent moiety can be as low as 2 % or less [61], thus avoiding concentration quenching without loosing high trapping efficiency.
1.4.2 Side-chain polymers In side-chain polymers, where the chromophore is covalently bonded to a polymer main chain, the concentration of transport molecules can be increased as compared to MDPs without causing crystallization. Because the underlying physical mechanism of charge carrier transport (nearest neighbour hopping between weakly coupled, localized states) is the same for both MDPs and side-chain polymers, the transport properties are qualitatively similar. This finding offers the opportunity to tailor the thermal and mechanical properties of the photoconductor without seriously affecting the transport properties. This can be achieved either by variation of the spacer length between the chromophore and the backbone or by variation of the backbone itself. A typical example for this group is PVK [12, 32, 34, 35]. Here, the transition from dispersive to non-dispersive transport could be observed [34]. Due to the fact, that the TOF curves have been measured over 10 decades in time, it was possible by means of a numerical inverse Laplace transform [34] to calculate from the measured photocurrent the trapping rate distribution, which is a measure for the density of localized states. In polysiloxane derivatives with pendant carbazolyl groups [58, 59] (Fig. 1.2) the effect of a variation of the spacer length was investigated. Starting from a spacer length of 3 or 5 methylene units up to 6 or 11 groups, the glass transition temperature drops from its initial value of 51 8C or 7 8C down to –5 8C or – 45 8C for the C11 spacer. It turns out, that the zero field activation energy eact for the hole transport remains unchanged and is the same for NIPC or PVK, eact = 0.51 eV [58]. This indicates, that the activation energy in side-chain polymers with pendant carbazolyl groups is dominated by intrinsic properties of the carbazolyl moiety and is not much affected by other factors like the morphology of the polymer [58]. However, the absolute values for meff differ by roughly one order of magnitude, where the compound with the shortest spacer shows an effective mobility in the range of 10 –6 cm2/Vs (T = 300 K, E = 3 7 105 V/cm). For the materials with a spacer length of 5 and 6 methylene units the mobility is lower by one order of magnitude and therefore comparable to the mobility in NIPC or PVK. A striking feature, when comparing NIPC with PVK, is the fact that polycarbonate doped with 20 wt% of NIPC shows the same hole mobility at room temperature as compared to PVK where the carbazole concentration amounts to 86 wt%. Obviously, the covalent attachment of the chromophore to a stiff backbone, like PVK with its glass transition temperature of 227 8C, induces a mutual orientation of adjacent pendant groups, 9
1 Model Systems for Photoconductive Materials which reduces the electronic coupling between the two sites. Therefore it is desirable to induce a well-defined amount of flexibility in both, the spacer and the backbone, to allow for mutual reorientation of the chromophores. For a polysiloxane backbone with carbazolyl transport groups the optimum spacer length n is equal to 3.
1.4.3 Conjugated systems In contrast to the two main groups described above the charge carrier transport in conjugated systems occurs via the polymer backbone. Various substituents are used to tailor the mechanical properties and the processibility. For instance, the chain can be a skeleton with quasi-s-conjugation like in polysilanes [36, 38–42] or polygermylenes [39]. The second group of main chain polymers contains a p-conjugated backbone like polyacetylene, polythiophene, polypyrrole, polyphenylene, poly(phenylenevinylene) (PPV), and their derivatives. For a review see [43, 44]. Both types of conjugated systems have been investigated. In the case of poly(methylphenyl silane) (PMPS), a quasi-s-conjugated polymer (Fig. 1.2), non-dispersive transport has been found down to a temperature of 250 K [60]. For lower temperatures, the transport is dispersive, which indicates that the transport in these materials is controlled by traps. Compared to the materials described above, the depths of the relevant traps are much lower. The zero field activation energy turns out to be as low as 0.37 eV as compared to 0.5 eV for materials containing carbazole. This gives a hole mobility at room temperature of roughly 10 –3 cm2/Vs, which is 3 orders of magnitude higher than for the materials described above. The high mobility and the transparency in the visible region makes PMPS also useful for applications in multilayer electroluminescent devices [61–63]. Besides the s-bonded PMPS, we investigated oligomeric conjugated compounds based on carbazole [64]. First experiments with a trimer of a carbazole derivative showed that this material is soluble in most of the common organic solvents. It forms a low molecular weight glass and shows no tendency to crystallize even after several months. In this material the p-system has been extended over three monomer units, which results in remarkably high hole mobilities as compared to the monomer model compound NIPC. First TOF experiments show an increase in mobility by roughly two orders of magnitude as compared to unconjugated carbazole systems, which reaches 2 7 10 –4 cm2/Vs [64]. The second group of p-conjugated polymers contains one pz-orbital per carbon atom which stands perpendicular to the plane of the s-bonded skeleton of the main chain. The interaction between these orbitals leads to electronic states, which are delocalized – at least partly – along the conjugated chain. As compared to the intrachain coupling, the intrachain coupling is lower by typically one or two orders of magnitude [66] thus leading to a large anisotropy of the electronic and optical properties. These materials can be regarded as basically one-dimensional organic semiconductors with bandgaps in the visible region of the spectrum, e. g. 1.5 eV (trans-(CH)x) [43], 2.0 eV (polythiophene), 2.5 eV (polyphenylene vinylene), and 3.0 eV [poly(para-phenylene)] [67]. We investigated the charge transport properties of the p-conjugated polymer poly (para-phenylenevinylene) and its soluble substituted derivative DPOP-PPV (Fig. 1.2). In 10
1.4
Results
contrast to e. g. (CH)x , PPV can be prepared in undoped form. If oxygen is carefully heated out, PPV exhibits excellent film forming properties as well as thermal stability under ambient conditions. In recent years, PPV has drawn additional attention due to its use in organic electroluminescent devices [45]. In the following we will show that the charge carrier transport in these materials can be described by the conventional models mentioned before, which have been developed for the characterization of disordered molecular systems. Based on the MT model a method has been developed to calculate the distribution of capture rates from the measured photocurrent [68, 69]. In contrast to Ref. [34], the method is based on a Fourier transform technique with better numerical stability as compared to the inverse Laplace transform used in Ref. [34]. With this approach the capture rate density oc is given by 2 !c
"! kB T p
Qtot sin tmic I
!
! ! ;
8
where Qtot is the total photogenerated charge, tmic the microscopic transit time, and I (o) the Fourier transform of the measured photocurrent, and f the phase shift (Eq. 9). For DPOPPPV, tmic = 7.5610–10 s for the given experimental conditions. tan
1
Im I
! Re I
!
9
The trap depth eo is related to the frequency o by the expression ! r0 exp
"! ; kB T
10
where r0 is the attempt-to-escape frequency (r0 = 1010 s –1 for DPOP-PPV). In DPOP-PPV the effective mobility is field dependent and thermally activated according to Gill’s formula (Eq. 4) [33] with a characteristic temperature T0 = 465 K (Fig. 1.3). The calculated trap density g (e) is exponential down to a pronounced cutoff energy ed , g
" / exp
" : kB T 0
11
This cutoff energy corresponds exactly to the measured activation energy of the effective mobility. Furthermore, the initially simply empirical parameter T0 in Gill’s formula (Eq. 4) could be correlated with the decay constant kBT0 of the trap distribution. By calculating the total number of trapping events it can be seen that the typical distance which a charge carrier travels between two trapping events is of the order of 4 Å. This value is comparable to the intrachain distance and indicates that the transport in these materials can be best described by conventional hopping between closely neighbouring sites. No evidence for bandlike transport has been found. 11
1 Model Systems for Photoconductive Materials 350 K
-2
10
300 K
500 kV/cm 350 kV/cm 200 kV/cm 75 kV/cm 0 kV/cm
-3
10
-4
10 2
µeff [cm /Vs]
230 K
270 K
-5
10
2
10
0
T0 = 465K
10
-6
10
-2
10
-4
10
-6
10
-7
-8
10
10
-10
10
0
2
4
6
8
-8
10
2.5
3.0
3.5 1/T [1000/K]
4.0
4.5
Figure 1.3: large frame: effective mobility meff of DPOP-PPV as a function of temperature for some electrical fields, data points for zero field were extrapolated from field dependent measurements; inset: Arrhenius fits for different electrical fields. For details see text. Data from [68, 69].
For the case of PPV the dispersion of the photocurrent is too large for transit time determination. It can be concluded that even at room temperature the release times from the deepest traps, which control the transport properties, are in the range of 1 s or less. This corresponds to an effective mobility of less than 10–8 cm2/Vs [69]. Based on TSC measurements from different groups [70], we conclude that this behaviour is primarily caused by the existence of grain boundaries. It is reasonable to assume that in conjugated polymers with a rigid backbone the preferred orientation of the main chain will be parallel to the film surface. Therefore, the charge carrier transport through the film has to occur perpendicular to the backbone, i. e. perpendicular to the direction with high intrinsic mobility, which would mean a principle limit for this class of materials for this kind of applications.
1.5
Outlook
As described above, disordered organic materials have been developed with effective hole mobilities of up to 10–3 cm2/Vs together with good mechanical and dielectric properties. However it seems that for disordered systems a considerable improvement of this value will 12
References be difficult to achieve, because for all investigated material groups the charge carrier transport is limited by the localized nature of the electronic states and by the hopping mechanism of the transport process. Based on recent other works [8, 9], we think that for developing even higher mobility materials it is essential to improve the structural order of the material. This may open new application fields, e. g. the use of active organic materials in semiconductor devices. One promising way is the use of highly ordered liquid crystals. With these materials, however, charge carrier mobilities up to 0.1 cm2/Vs or even more are realistic, which would make them comparable to single crystalline systems [65].
References
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.
C. F. Carlson: US Pat. 2 297 691 (1942) D. Haarer: Angew. Makromol. Chem., 183, 197 (1990) D.M. Pai, B.E. Springett: Rev. Mod. Phys., 65(1), 163 (1993) K.-W. Klüpfel, M. Tomanck, F. Endermann: German Patent No. DE-B-11 45 184 (1963) H. Hoegl, O. Süs, W. Neugebauer, Kalle AG: German Patent No. DBP 1068115. Chem. Abstr., 55, 20742 a (1961) H. Hoegl: J. Phys. Chem., 69(3), 755 (1965) G. Pfister: Phys. Rev. Lett., 36(5), 271 (1976) D. Adam, F. Closs, T. Frey, D. Funhoff, D. Haarer, H. Ringsdorf, P. Schuhmacher, K. Siemensmeyer: Phys. Rev. Lett., 70(4), 457 (1993) D. Adam, P. Schuhmacher, J. Simmerer, L. Häussling, K. Siemensmeyer, K.H. Etzbach, H. Ringsdorf, D. Haarer: Nature, 371, 141 (1994) H. Tung Li, P.J. Regensburger: J. Appl. Phys., 34, 1730 (1963) L. Onsager: Phys. Rev. B, 54, 554 (1938) H. Kaul, D. Haarer: Ber. Bunsenges. Phys. Chem., 91, 845 (1987) M. Gailberger, H. Bässler: Phys. Rev. B, 44(16), 8643 (1991) Y. Kanemitsu, Y. Sugimoto: Phys. Rev. B, 16(21), 14 182 (1992) Y. Sano, K. Kato, M. Yokoyama, Y. Shirota: H. Mikawa, Mol. Cryst. Liq. Cryst. 36(1–2), 137 (1976) A. Peled, L.B. Schein, D. Glatz: Phys. Rev. B, 41(15), 10 835 (1990) Y. Kanemitsu, H. Funada, Y. Masumoto: J. Appl. Phys., 71(1), 300 (1992) H. Bässler, P.M. Borsenberger: Chem. Phys., 177(3), 763 (1993) P.N. Sanda, T. Takamori, D.B. Dove: J. Appl. Phys., 64(3), 1229 (1988) J.X. Mack, L.B. Schein, A. Peled: Phys. Rev. B, 39, 7500 (1989) Y. Kanemitsu, H. Funada, S. Imamura: J. Appl. Phys., 67(9), 4152 (1990) P.M. Borsenberger, L.T. Pautmeier, H. Bässler: Phys. Rev. B, 46(19), 12 145 (1992) J. Mort, G. Pfister, S. Grammatica: Solid State Commun., 18, 693 (1976) P.M. Borsenberger, L.E. Contois, A.I. Ateya: J. Appl. Phys., 50(2), 914 (1979) S.J. Santos Lemus, J. Hirsch: Philos. Mag. B, 53(1), 25 (1986) D. Haarer, H. Meyer, P. Strohriegl, D. Naegele: Makromol. Chem., 192(3), 617 (1991) D.M. Pai, J.F. Yanus, M. Stolka, D. Renfer, W.W. Limburg: Philos. Mag., 48(6), 505 (1983) H. Bässler: Philos. Mag., 50(3), 347 (1984) G. Pfister: Phys. Rev. B, 16(8), 3676 (1977)
13
1 Model Systems for Photoconductive Materials 30. P.M. Borsenberger: J. Appl. Phys., 68(12), 6263 (1990) 31. P.M. Borsenberger L. Pautmeier, R. Richert, H. Bässler: J. Chem. Phys., 94(12), 8276 (1991) 32. J.M. Pearson, M. Stolka: Polymer Monographs, Vol. 6: Poly(N-vinylcarbazole), Gordon and Breach Sci. Publ., New York (1981) 33. W.D. Gill: J. Appl. Phys., 43, 5033 (1972) 34. E. Müller-Horsche, D. Haarer, H. Scher: Phys. Rev. B, 35, 1273 (1987) 35. F.C. Bos, D.M. Burland: Phys. Rev. Lett., 58, 152 (1987) 36. R.G. Kepler, J.M. Zeigler, L.A. Harrah, S.R. Kurtz: Phys. Rev. B, 35(6), 2818 (1987) 37. J. Frenekl: Phys. Rev., 54, 647 (1938) 38. K. Shimakawa, T. Okada, O. Imagawa: J. Non-Cryst. Solids, 114(1), 345 (1989) 39. M. Abkowitz, M. Stolka: Solid State Commun., 78(4), 269 (1991) 40. H. Kaul: PhD thesis, Universität Bayreuth (1991) 41. E. Brynda, S. Nespurek, W. Schnabel: Chem. Phys., 175(2–3), 459 (1993) 42. V. Cimrova, S. Nespurek, R. Kuzel, W. Schnabel: Synth. Met., 67(1–3), 103 (1994) 43. A.J. Heeger, S. Kivelson, J.R. Schrieffer, W.-P. Su: Rev. Mod. Phys., 60(3), 781 (1988) 44. J. L. Brédas, R. Silbey (eds.): Conjugated Polymers, Kluwer Academic Publishers, Dordrecht (1991) 45. J.H. Burroughes, D.D.C. Bradley, A.R. Brown, R.N. Marks, K. Mackay, R.H. Friend, P.L. Burns, A.B. Holmes: Nature, 347(6293), 539 (1990) 46. F.W. Schmidlin: Phys. Rev. B, 16(6), 2362 (1977) 47. H. Scher, E.W. Montroll: Phys. Rev. B, 12(6), 2455 (1975) 48. J. Noolandi: Phys. Rev. B, 16(10), 4474 (1977) 49. G. Pfister, H. Scher, Adv. in Phys, 27(5), 747 (1978) 50. P.M. Borsenberger, L. Pautmeier, H. Bässler: J. Chem. Phys., 94(8), 5447 (1991) 51. G. Schönherr, H. Bässler, M. Silver: Philos. Mag. B, 44(3), 369 (1981) 52. H. Bässler, G. Schönherr, M. Abkowitz, D.M. Pai: Phys. Rev. B, 26(6), 3105 (1982) 53. L. Pautmeier, R. Richert, H. Bässler: Philos. Mag. Lett., 59(6), 325 (1989) 54. L. Pautmeier, R. Richert, H. Bässler: Synth. Met., 37, 271 (1990) 55. H. Domes, R. Leyrer, D. Haarer, A. Blumen: Phys. Rev. B, 36(8), 4522 (1987) 56. D. Stauffer: Introduction to Percolation Theory, Taylor and Francis, London (1985) 57. H. Domes: PhD thesis, Universität Bayreuth (1988) 58. H. Domes, R. Fischer, D. Haarer, R. Strohriegl: Makromol. Chem., 190, 165 (1989) 59. H. Schnörer, H. Domes, A. Blumen, D. Haarer: Philos. Mag. Lett., 58(2), 101 (1988) 60. H. Kaul: PhD thesis, Universität Bayreuth (1991) 61. H. Suzuki, H. Meyer, J. Simmerer, J. Yang, D. Haarer: Adv. Mater., 5, 743 (1993) 62. H. Suzuki, H. Meyer, S. Hoshino, D. Haarer: J. Appl. Phys. in press (1995) 63. J. Kido, K. Nagai, Y. Okamoto, T. Skotheim: Appl. Phys. Lett., 59, 2760 (1991) 64. C. Beginn, J.V. Grazulevicius, P. Strohriegl,J. Simmerer, D. Haarer: Macromol. Chem. Phys., 195, 2353 (1994) 65. N. Karl: in: K. Sumino (ed.): Defect Control in Semiconductors, Elsevier Science Publishers, North Holland (1990) 66. P. Gomes da Costa, R.G. Dandrea, E.M. Conwell: Phys. Rev. B, 47(4), 1800 (1993) 67. G. Leising, K. Pichler, F. Stelzer: in: H. Kuzmany, M. Mehring, S. Roth, (eds.): Springer Series in Solid-State Sciences, Vol. 91: Electronic Properties of Conjugated Polymers III, Springer, Berlin (1989) 68. H. Meyer: PhD thesis, Universität Bayreuth (1994) 69. H. Meyer, D. Haarer, H. Naarmann, H.H. Hörhold: Phys. Rev. B, in press (1995) 70. M. Onoda, D.H. Park, K. Yoshino: J. Phys. (London), Condens. Matter, 1(1), 113 (1989)
14
2 Novel Photoconductive Polymers Jörg Bettenhausen and Peter Strohriegl
2.1
Introduction
Electrophotography is the only area in which the conductivity of sophisticated organic materials and polymers is exploited in a large scale industrial process today. Photoconductors are characterized by an increase of electrical conductivity upon irradiation. According to this definition photoconductive materials are insulators in the dark and become semiconductors if illuminated. In contrast to electrically conductive compounds photoconductors do not contain free carriers of charge. In photoconductors these carriers are generated by the action of light. The discovery of photoconductivity dates back to 1873 when W. Smith found the effect in selenium. Based on this discovery C. F. Carlson developed the principles of the xerographic process already in 1938. Photoconductivity in organic polymers was first discovered in 1957 by H. Hoegl, who found that poly(N-vinylcarbazole) (PVK) and charge transfer complexes of PVK with electron acceptors like 2,4,7-trinitrofluorenone act as photoconductors [1]. Besides the application of photoconductive polymers in photocopiers these materials are also widely used in laser printers in the last years. The third area in which photoconductors are applied is the manufacturing of electrophotographic printing plates. The organic photoconductors used in practice are based on two types of systems. The first one are polymers in which the photoconductive moiety is part of the polymer, for example a pendant or in-chain group. The second group involves low molecular weight compounds imbedded in a polymer matrix. These so-called moleculary doped polymers are widely used today. One interesting class of photoconductive materials are oxadiazoles. It is known for a long time that derivatives of 1,3,4-oxadiazole are good photoconductors. Compound 1 for example is described in patents and was frequently applied in photocopiers [2].
1
2
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties Deutsche Forschungsgemeinschaft (DFG) Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
15
2
Novel Photoconductive Polymers
The oxadiazole 1 is a hole transport material since the electron withdrawing effect of the oxadiazole group with three electronegative heteroatoms is overcompensated by two electron donating amino groups. Within the last years, oxadiazoles like 2-(biphenyl)-5-(4-tert.butylphenyl)-1,3,4oxadiazole (PBD) 2 have been frequently applied in organic light emitting diodes [3]. Here the electron withdrawing oxadiazole unit dominates the electronic properties and the oxadiazole compounds act as electron injection and transport layers. Furthermore, 2,5-diphenyloxadiazoles have been used as building blocks in thermostable polymers and they are highly fluorescent as well. In this paper the synthesis and characterization of a number of novel low molecular weight oxadiazole derivatives and polymers is described. The compounds show different molecular shapes, e. g. rod-like and star-shaped structures have been realized. In addition to the compounds described here, a series of different main chain and side group polymers with oxadiazole moieties are presently synthesized in our research group [4].
2.2
Liquid crystalline oxadiazoles and thiadiazoles
2.2.1 The basic idea The aim of this work is the preparation of photoconductive compounds with high carrier mobilities. There are a number of indications that the carrier mobility, i. e. the speed of the charge particles within the sample, depends on the geometric arrangement of the photoconductive moieties. At room temperature for instance, the mobility in single crystals of aromatic compounds like anthracene or perylene is very high, i. e. in the range of 10–1 cm2/Vs [5]. In amorphous polymers the carrier mobilities are orders of magnitude smaller and typical values are in the range of 10–6 and 10 –8 cm2/Vs [6]. So we were interested to investigate if the higher order in the liquid crystalline state leads to higher mobilities in comparison to less ordered amorphous solids. In liquid crystalline polymers a macroscopic orientation of the photoconductive groups in the mesophase can be achieved by means of electric or magnetic fields. The orientation can be frozen in by lowering the temperature below the glass transition. By this, materials with a degree of order between a perfect single crystal and totally disordered amorphous polymers are obtained. With this in mind we started the synthesis of a series of liquid crystalline model compounds and polymers in which the side groups possess both mesogenic and photoconductive properties.
16
2.2
Liquid crystalline oxadiazoles and thiadiazoles
2.2.2 Monomer synthesis In the past years we have synthesized a variety of compounds with oxadiazole and thiadiazole moieties [7]:
3X=S
4X=O
The rod-like thiadiazole derivative 3 is liquid crystalline and shows a smectic as well as a nematic LC phase with the following phase behaviour: c 103 sa 180 n 203 i. An acrylate monomer was prepared by hydroboration of the terminal double bond and subsequent esterification with acryloyl chloride and then polymerized. The resulting polymer 5 exhibits a mesophase which has not yet been identified.
5
Unfortunately the clearing point of the polymer is at 246 8C, where the material starts to decompose. Therefore orientation in an electric field was not possible. In contrast to the thiadiazole the oxadiazole 4 is neither liquid crystalline as monomer nor as polymer. The reason for the lack of mesogenic properties is that the substitution of the sulphur by oxygen introduces a bend into the molecule which prevents the formation of a LC phase. Recently liquid crystalline oxadiazoles (Scheme 2.1) have been described [8]. Here the oxadiazole is coupled to at least two benzene or cyclohexane rings. Thereby an increase of the mesophase stability is achieved.
Scheme 2.1: Liquid crystalline oxadiazole compounds [8].
17
2
Novel Photoconductive Polymers
6
7
8
9
10-13
10
R = Ph-OC6H13
12
R = Ph-N(CH3)2
11
R = C7H15
13
R = Ph-N(CH3)C6H13
Scheme 2.2: Monomer synthesis.
Our aim was to synthesize a series of different oxadiazole monomers which are functionalized for the preparation of polymers. Scheme 2.2 shows the synthesis of the monomeric 1,3,4-oxadiazoles. The oxadiazole moiety is formed by a cyclisation reaction of the diacylhydrazine derivatives 8 with phosphorous oxychloride. The unsymmetrical bishydrazides 8 are prepared by treatment of benzoyl chlorides 6 with the appropriate monohydrazides 7. The last reaction step is an esterification with 4-(5-hexenyloxy)benzoyl chloride. The esterification reaction is essential in this case because of two reasons. First the rod-like 18
2.2
Liquid crystalline oxadiazoles and thiadiazoles
mesogen is formed, and second a functional group for the preparation of LC polymers is introduced. Additionally the oxadiazole monomer 14 was synthesized, in which the ester group is replaced by a biphenyl unit. By this it is possible to investigate the influence of the ester function on the photoconductive properties of the liquid crystalline compounds.
14
The liquid crystalline behaviour and the phase transitions of the monomeric oxadiazole derivatives 10–14 were determined by DSC and polarizing microscopy. In Tab. 2.1 the phase transition temperatures are summarized: Table 2.1: Phase behaviour of the monomeric oxadiazoles 10–14 Transition temperature in 8C
Compound 10 11 12 13 14
k k k k k
141 90 162 127 72
n sa n (sa sa
167 116 179 111) 77
i i i i i
Except compound 13 all monomers are enantiotropic liquid crystalline and show nematic or smectic mesophases. Only derivative 13 shows a monotropic sa phase.
2.2.3 Oligo and polysiloxanes with pendant oxadiazole groups The next step was the synthesis of liquid crystalline polysiloxanes with oxadiazole groups in the mesogenic unit. The polymers were prepared by a polymeranalogous reaction of the monomers 10 and 11 with poly(hydrogenmethylsiloxane) 15. Both polymers are liquid crystalline. The transition into the isotropic phase takes place at about 200 8C and therefore at a much higher temperature than in the monomers. The high clearing temperature makes the orientation of the polymers difficult which is preferably carried out near the clearing point. Up to now it was not possible to prepare well-defined polymers with the monomers 12–14. A possible reason is the inactivation of the Pt catalyst by the amino groups. If the cyclic tetrasiloxane 18, which is commercially available in high purity, is used instead of 15, well-defined monodisperse model compounds are obtained [9]. These com19
2
Novel Photoconductive Polymers
10, 11
16
R = Ph-OC6H13
15
17
R = C7H15
Scheme 2.3: Synthesis of polysiloxanes with pendant oxadiazole groups.
pounds can be highly purified, for example by preparative gel permeation chromatography. This is very important for photoconductors, because it is well-known that even traces of impurities may reduce the carrier mobility.
18
So the monomeric compounds 10 and 12–14 have been reacted with 18 to yield the following tetrameric derivatives:
19
R = Ph-OC6H13 , n = 1
21
R = Ph-N(CH3)C6H13 , n =
20
R = Ph-N(CH3)2 , n = 1
22
R = Ph-N(CH3)C6H13 , n =
Scheme 2.4: Tetrasiloxanes with pendant oxadiazole groups.
20
2.2
Liquid crystalline oxadiazoles and thiadiazoles
All cyclosiloxanes are liquid crystalline and show several advantages compared to the polymers. So the cyclic siloxanes form stable glasses but their clearing points are much lower. For example compounds 21 and 22 possess clearing temperatures at only 102 8C. The phase behaviour of the tetramers are listed in Tab. 2.2. Table 2.2: Phase behaviour of the tetrameric siloxanes 19–22. Transition temperature in 8C
Comp. 19 20 21 22
g g g g
54 67 34 38
m1 a) sa k sa
108 138 66 82
m2 a) n n n
117 181 102 102
sc i i i
155
n
185
i
a) Mesophase not identified
2.2.4 Photoconductivity measurements For the characterization of the novel photoconductive compounds two different experimental techniques have been used. Some measurements were made by BASF AG using a steady-state method. These experiments allow a quick characterization, but in contrast to the time-of-flight (TOF) method no statements about transient photocurrents and carrier mobilities are possible. The TOF measurements were carried out in the group of D. Haarer, Universität Bayreuth. The tetrameric cyclosiloxanes are very suitable compounds for physical investigations. Because of the low viscosity the preparation of samples is much easier than in the case of the polymers. Additionally the dark currents of the tetramers are very low. All the compounds are photoconductive, as shown by the steady-state method. For example, the measurements of the thiadiazole 3 showed a distinct rise of the photocurrent at the transition from the crystalline to the mesophase, as shown in Fig. 2.1. The decrease of
Figure 2.1: Temperature dependence of the photocurrent of the thiadiazole 3.
21
2
Novel Photoconductive Polymers
the current at the transition from the smectic to the nematic phase is directly correlated with the loss of order. These results show that the basic idea of this work is correct. Nevertheless, we were not able to carry out TOF measurements with calamitic monomers, because the dark currents in the liquid crystalline phases were too high. As mentioned before, such measurements are possible with the cyclic tetramers. The transient photocurrents of the tetrasiloxanes 19, 21, 22 illustrate that the carrier transport is totally dispersive, i. e. dominated by deep traps in which the charge carriers are captured. This is a typical behaviour of amorphous polymers. No transit time could be detected and no statements about the carrier mobility can be made for the rod-like mesogens. In contrast it was shown in the past years that discotic liquid crystals like hexapentyloxytriphenylene (HPT) 23 have carrier mobilities up to 10 –3 cm2/Vs in the liquid crystalline Dho phase [10]. Here, the disc-shaped molecules are ideally stacked above each other and therewith allow a fast carrier transport. The discotic hexathioether 24 with a highly ordered helical columnar (H) phase exhibits mobilities up to 10–1 cm2/Vs, which are almost as high as in organic single crystals [11].
6
6
23
24
One important observation during the photoconductivity measurements of the tetramer 19 was, that it showed almost the same photocurrent for holes and electrons. This was interesting because of the lack of electron transporting materials and led us synthesize a variety of starburst oxadiazole compounds, which are described in the next Chapter.
2.3
Starburst oxadiazole compounds
2.3.1 Motivation The synthesis of novel materials with high carrier mobilities is one of the major goals in the field of photoconductive polymers. Within the last years different approaches have been pursued to reach this goal. First, the photoconductive properties of conjugated polymers like poly(phenylenevinylene) and poly(methyl phenylsilane) have been investigated [12]. Another approach are liquid crystals which are the topic of the first Chapter. The third way to realize 22
2.3
Starburst oxadiazole compounds
the goal are glasses of large extended aromatic amines. The best investigated representatives of this class of compounds are N,N'-diphenyl-N,N'-bis(3-methylphenyl)-[1,1'-biphenyl]-4,4'diamine (TPD) 25 from Xerox [13] and 1,1-bis(di-4-tolyl-aminophenyl)cyclohexane (TAPC) 26 from Kodak [14]. Both compounds are derivatives of triphenylamine, a well-known photoconductor. If thin films of TPD and TAPC are prepared by vacuum evaporation both compounds form metastable glasses. In such glasses carrier mobilities up to 10–3 cm2/Vs for TPD [13] and 10 –2 cm2/Vs for TAPC [14] have been reported. But both TAPC and TPD glasses are metastable and have a strong tendency to crystallize. If the molecules are imbedded in a polymer matrix, e. g. polycarbonate or polystyrene, morphologically stable materials are formed, but the mobilities decrease drastically [13].
25
26
Several attempts have been made to overcome the problems with metastable TPD and TAPC glasses. Recently we described the synthesis of 3,6-bis[(9-hexyl-3-carbazolyl)ethynyl]-9-hexylcarbazole 27, a trimeric model compound of poly(carbazolylene ethynylene) [15]. This material shows a glass transition temperature of 41 8C and forms glasses which are stable for more than a year. Mobilities up to 10–4 cm2/Vs (E = 6 7 105 V/cm, T = 30 8C) have been measured by the time-of-flight technique.
27
Another interesting approach are starburst compounds with high glass transition temperatures. So 4,4',4@-tris(N-carbazolyl)triphenylamine, which has been published recently [16], shows a glass transition at 151 8C and forms a stable glass. Beginning with the work of Thomalia [17] in 1986, starburst molecules (dendrimers) have achieved enormous interest within the last years. Dendrimers are highly branched regular molecules, which are usually prepared by stepwise reactions. In many cases the behaviour of dendritic macromolecules are different from that of linear polymers, e. g. the former 23
2
Novel Photoconductive Polymers
usually show enhanced solubility. The reasons for these differences are the unique three-dimensional structure and the large number of chain ends in dendrimers. Two different methods have been developed for the stepwise synthesis of starburst dendrimers: a) the divergent approach, where the synthesis starts from a core molecule with two or more reactive groups; b) the convergent approach, in which the synthesis starts at the outer sphere of the dendrimer. In the divergent approach, the reaction of the core molecule with two or more reagents containing at least two protected branching sites is followed by removal of the protecting groups and subsequent reaction of the liberated reactive groups which leads to the starburst molecule of the 1st generation. The process is repeated until the desired size is reached. In the convergent approach the synthesis starts at what will become the outer surface of the dendrimer. Step by step large dendrimer arms are prepared and finally the completed arms are coupled to the core. Both methods produce well-defined large dendritic molecules whose structures are manifolds of the building blocks. They allow structural as well as functional group variation. On the other hand, hyperbranched polymers can be synthesized in a one-step reaction using highly functionalized monomers of the type AxB, where x is 2 or larger. This method does not yield polymers of such a well-defined structure, but has the advantage to provide rapidly large quantities of material. The control of the degree of branching is difficult and mainly depends on statistics, steric effects, and the reactivity of the functional groups. The aim of our work is the synthesis of starburst compounds with oxadiazole moieties. In contrast to the triphenylamine and carbazole derivatives oxadiazoles are strong electron acceptors. Therefore their transport characteristics can be switched from hole transport materials like 2,5-(4-diethylaminophenyl)-oxadiazole 1, in which the electron withdrawing effect of the oxadiazole ring is overcompensated by the two electron donating amino groups, to electron transport materials like 2-(biphenyl)-5-(4-tert.butylphenyl)-oxadiazole (PBD) 2. Recently, Saito demonstrated that oxadiazoles in a polycarbonate matrix show electron transport with mobilities up to 10 –5 cm2/Vs [18]. Such electron transport materials are attractive for the use in copiers and in organic light-emitting diodes.
2.3.2 Synthesis of starburst oxadiazole compounds Starburst oxadiazole compounds have been mentioned for the first time in a thermodynamic study of the structure-property-relationship in low molecular weight organic glasses [19, 20]. Upon rapid cooling these compounds form glasses with Tgs between 77 and 169 8C. We have used three different approaches for the preparation of starburst oxadiazole compounds, which are schematically shown in Fig. 2.2. The methods A and B can be compared to the divergent synthesis of dendrimers. Method A starts from a central core and involves the reaction of the acid chloride groups and a subsequent cyclisation with phosphorous oxychloride to the oxadiazole ring (Chap24
Cl
O
O
C
C
Cl
C O
Cl
Method B
Method A O H 2N
O C
NH
NH
O
O
C
C
O
NH
C
O NH
NH
C
C C
NH
N
NH O
N N
NH
C
N
N
N
O
N O
O
N N
= core
= shell
Method C
X
Z
+ X
X =
Y
X
OH ,
C
Z
CH
= core
Y =
Br ,
F
Z =
O
, C
Z
C
= shell
Figure 2.2: The different approaches to starburst oxadiazoles.
25
2
Novel Photoconductive Polymers
ter 1). The main difficulty in this case is that three functional groups must react in one step. Partially incomplete cyclisation causes problems, because the products contain one or two bishydrazides and are difficult to separate from the target compounds, which have three oxadiazole groups. Therefore we recently developed a totally different route (method B). The compounds 31 a–f were prepared by the reaction of benzene tricarbonyl chloride 30 with the tetrazoles 29. The latter were synthesized from benzonitriles 28 with potassium azide (Scheme 2.5). With the loss of nitrogen the tetrazole ring is transferred to the oxadiazoles [21]. This gives an easy access to unsymmetrically substituted oxadiazoles which we used for the first time in the synthesis of starburst oxadiazole compounds. In the third approach (method C) which is similar to the convergent synthesis a preformed oxadiazole precursor is prepared by stepwise synthesis and finally coupled to the core molecule. R1
R1
R2
CN
+
NH4Cl
NaN3
R3
N
R2
C N
R3
28
Cl
N
29
O
O
C
C
R1
Cl +
C O
NH
3
R2
N C N
R3
NH N
31a-f
pyridine
31a-f 2h, reflux
Cl
30
29
Scheme 2.5: Synthesis of starburst oxadiazoles via tetrazole intermediates.
For the preparation of the novel starburst molecules, different core molecules have been used: 1,3,5-benzene tricarboxylic acid chloride, 1,3,5-tris(4-benzoyl)benzene, 1,3,5triethynylbenzene, and 4,4',4@trihydroxytriphenylamine. Except benzenetricarboxylic acid which is commercially available, all core molecules were synthesized following well-known literature procedures [22, 23]. The structures of the starburst oxadiazole compounds are summarized in Tab. 2.3. 31 c and 31 g have been described before [19, 20], but no synthetisis procedure was given. The oxadiazoles 31 a–h have been prepared according to method B (Fig. 2.3), starting from benzene tricarboxylic acid chloride. Compound 31 g with tert.-butyl substituents was synthesized by method A, too. So a comparison of methods A and B is possible. The tetrazole route (method B) exhibits several advantages compared to the ring closure with dehydrating agents (method A). One advantage is the short reaction time. The reaction is finished within 2 hours whereas heating for 1–3 days is necessary if the ring closure is carried out with phosphorous oxychloride. Even more important is the facile work up procedure. In the case of the dehydration with POCl3 column chromatography and subsequent recrystallization is necessary to purify the product which is finally obtained in 33 % yield. In contrast only one filtration on a short silica gel 26
2.3
Starburst oxadiazole compounds
Table 2.3: Structures of the starburst oxadiazole compounds 31–34. Core a)
Comp.
Shell a)
N
31 a 31 b 31 c 31 d 31 e 31 f 31 g 31 h
R1
N
31
R2
O
R3
N
32
N
C
C
R = C(CH3)3
R
O
C
R1 = R3 = H, R2 = CH3 R2 = R3 = CH3, R1 = H R1 = R3 = CF3, R2 = H R1 = R3 = H, R2 = C2H5 R1 = R3 = H, R2 = OC2H5 R1 = R3 = H, R2 = CH(CH3)2 R1 = R3 = H, R2 = C(CH3)3 R1 = R3 = H, R2 = N(C2H5)2
C N
33
N R
O
R = C(CH3)3
C C
N
34
N
O
N O
R
R = C(CH3)3
a) cf. Figure 2.2.
column is sufficient for the purification of 31 g, prepared by the tetrazole route. In this case the yield of the pure product is 69 %. The starburst oxadiazole 32 has been prepared by method B too, whereas for 33 and 34 method C (Fig. 2.2) was used. The identity of all compounds was confirmed by NMR and FTIR spectroscopy.
2.3.3 Thermal properties The thermal behaviour of the starburst molecules has been investigated by differential scanning calorimetry and thermogravimetric measurements. 27
2
Novel Photoconductive Polymers
Table 2.4: Thermal properties of the starburst compounds 31–34. Compound
Molecular mass [g/mol]
Tg a) [8C]
Tm a) [8C]
Tdec b) [8C]
31 a 31 b 31 c 31 d 31 e 31 f 31 g 31 h 32 33 34
553 595 919 595 643 637 679 724 907 979 1122
– – – – 108 97 142 128 165 – 137
334 333 250 276 259 225 257 299 297 320 –
348 358 336 360 337 349 366 329 409 356 391
a) 3rd heating, determined by DSC with 20 K/min b) onset of decomposition in nitrogen, thermogravimetric measurement with 10 K/min
Among the compounds 31 a–h with a small benzene core both crystalline and glass forming materials exist. So 31 a–d with small methyl, ethyl, or trifluoromethyl substituents show only melting points in the DSC experiment with cooling rates of 20 K/min. Naito and Miura reported that it is possible to obtain glasses even with small methyl or trifluoromethyl substituents if the compounds are rapidly cooled with liquid nitrogen [19]. In contrast, the compounds 31 e with ethoxy, 31 f with iso-propyl, 31 g with tert-butyl substituents, and 31 h with diethylamino groups form glasses upon cooling with 20 K/min in the DSC equipment. When the amorphous samples are heated again the glass transition appears first but on further heating the samples start to recrystallize and consequently show a melting point at higher temperatures. Compound 32 with the triphenylbenzene core behaves similar like 31 e–h. The most stable glasses are formed by 34 with the triphenylamine core. The DSC diagram of the novel glass forming oxadiazole compound is shown below. Upon both, heating and cooling, only a glass transition is observed at 137 8C (Fig. 2.3). In our DSC experi-
Figure 2.3: DSC scan of the starburst oxadiazole compound 34, 3rd heating and cooling, with 20 K/min.
28
2.3
Starburst oxadiazole compounds
ments we never found any evidence for crystallization of 34. Even in the first heating no melting point is observed up to 350 8C. Consequently, transparent amorphous films are obtained from the starburst oxadiazole compound 34 by solvent casting. The thermal behavior of compound 33 is somewhat different. No reproducible DSC scans are obtained in subsequent heating-cooling cycles. We attribute this to thermal crosslinking of the triple bonds [24]. The thermal stability has been monitored by thermogravimetric measurements. In most cases the onset of decomposition is in the range from 330–370 8C. The oxadiazole compound 32 with a triphenyl benzene core shows a somewhat higher thermal stability up to 410 8C. (CH3)3C
N
N
N
N
C(CH3)3
O
O O
O N
O
N
O
N
C(CH3)3
34
The starburst oxadiazole compounds are now being tested as electron injection and transport layer in organic LEDs and as photoconductors. First tests of two layer LEDs with PPV show that the novel materials possess properties comparable to 2 but have the great advantage to show no recrystallization if thin films were made by spin-coating. We will report on these measurements in the near future.
29
2
Novel Photoconductive Polymers
References
1. H. Hoegl, O. Süs, W. Neugebauer, Kalle AG: DBP 1 068 115, Chem. Abstr., 55, 20742 a (1961), H. Hoegl: J. Phys. Chem., 69, 755 (1965) 2. W. Wiedemann: Chem.-Ztg., 106, 275 (1982) and references therein 3. A.R. Brown, D.D.C. Bradley, J.H. Burroughes, R.H. Friend, N.C. Green-ham, P.L. Burn, A.B. Holmes, A. Kraft: Appl. Phys. Lett., 61, 2793 (1992) 4. E. Buchwald, M. Meier, S. Karg, W. Rieß, M. Schwoerer, P. Pösch, H.-W. Schmidt, P. Strohriegl: Adv. Mater., 7, 839 (1995), M. Greczmiel, P. Pösch, H.-W. Schmidt, P. Strohriegl, E. Buchwald, M. Meier, W. Rieß, M. Schwoerer: Makromol. Symposia, 102, 371 (1996) 5. W. Warta, R. Stehle, N. Karl: Appl. Phys., A 36, 163 (1985) 6. P.M. Borsenberger, D.S. Weiss: in: Organic Photoreceptors For Imaging Systems, M. Dekker, New York, (1993) 7. R.G. Müller: PhD thesis, Bayreuth, (1992) 8. D. Girdziunaite, C. Tschierske, E. Novotna, H. Kresse, A. Hetzheim: Liq. Cryst., 10, 397 (1991) 9. F.H. Kreuzer, D. Andrejewski, W. Haas, N. Häberle, G. Riepl, P. Spes: Mol. Cryst. Liq. Cryst., 199, 345 (1991) 10. D. Adam, F. Closs, T. Frey, D. Funhoff, D. Haarer, H. Ringsdorf, P. Schuhmacher, K. Siemensmeyer: Phys. Rev. Lett., 70, 457 (1993) 11. D. Adam, P. Schuhmacher, J. Simmerer, L. Häussling, K. Siemensmeyer, K.H. Etzbach, H. Ringsdorf, D. Haarer: Nature, 371, 141 (1994) 12. M. Gailberger, H. Bässler: Phys. Rev. B, 44, 8643 (1991), M.A. Abkowitz, M.J. Rice, M. Stolka: Phil. Mag. B, 61, 25 (1990) 13. M. Stolka, J.F. Yanus, D.M. Pai: J. Phys. Chem., 88, 4707 (1984) 14. P.M. Borsenberger, L. Pautmeier, R. Richert, H. Bässler: J. Phys. Chem., 94, 8276 (1991) 15. C. Beginn, J.V. Grazulevicius, P. Strohriegl, J. Simmerer, D. Haarer: Macromol. Chem. Phys., 195, 2353 (1994) 16. Y. Kuwabara, H. Ogawa, H. Inoda, N. Noma, Y. Shirota: Adv. Mater., 6, 677 (1994) 17. D.A. Tomalia, H. Baker, J. Dewald, M. Hall, G. Kallos, S. Martin, J. Roeck, J. Ryder, P. Smith: Macromolecules, 19, 2466 (1986) 18. H. Tokuhisa, M. Era, T. Tsutsui, S. Saito: Appl. Phys. Lett., 66, 3433 (1995) 19. K. Naito, A. Miura: J. Phys. Chem., 97, 6240,(1993) 20. S. Egusa, Y. Watanabe: EP 553 950 A2 to Toshiba Corp. 21. R. Huisgen, J. Sauer, H.J. Sturm, J.H. Markgraf: Chem. Ber., 93, 2106 (1960) 22. E. Weber, M. Hecker, E. Koepp, W. Orlia, M. Czugler, I. Csöregh: J. Chem. Soc., Perkin Trans. II, 1251 (1988) 23. S.J.G. Linkletter, G.A. Pearson, R.I. Walter: J. Am. Chem. Soc., 99, 5269 (1977) 24. S.-C. Lin, E.M. Pearce: in: High-Performance Thermosets – Chemistry, Properties, Applications, Carl Hanser Verlag, Munich, 137 (1994)
30
3
Theoretical Aspects of Anomalous Diffusion in Complex Systems Alexander Blumen
3.1
General aspects
Many relaxation phenomena in the solid phase depend on diffusion, which is usually treated in the framework of statistical methods. Regular diffusion, known as Brownian motion, is characterized by a linear increase of the mean-squared displacement with time. On the other hand, for a whole series of phenomena this simple relation does not hold; their temporal evolution of the mean-squared displacement is non-linear and thus obeys at long times:
2 r
t t
1
with g 0 1. Relation 1 is referred to as anomalous diffusion. In the case that g < 1 one denotes the behaviour as subdiffusive. A subdiffusive pattern of motion often results from disorder [1–4]. One has to note, however, that the asymptotic law, Eq. 1, emerges only when the disorder influences the motion on all scales. In the case that g > 1 the motion is termed superdiffusive. A classical example for superdiffusive behaviour is furnished by the motion of particles in a turbulent flow. In this paper we focus on several models for anomalous diffusion which involve polymeric systems. Now, the mean-squared displacement is a basic characteristic feature for the motion but, as an averaged quantity, it can provide only restricted information about the basic microscopic mechanisms involved. In several works we have also studied the propagator P(r,t), the probability to be at r at time t having started at the origin at t = 0. Space limitations prevent us from going into details. Here we focus on Ar2 (t)S and refer to Klafter et al. and Zumofen et al. [4, 5] for in-depth analyses of the propagator. From the beginning, the role of time-dependent aspects in disordered media has to be emphasized. In usual random walk problems it is often assumed that the disorder is quenched, so that the dynamics evolves over a static substrate, i. e. the geometrical or the energetical disorder is frozen in. However, it is also possible that the dynamical processes are directly under the influence of temporal (possibly medium induced) fluctuations. First, in Section 3.2 we concentrate on photoconductivity, whose canonical description involves the continuous time random walk (CTRW) approach [1, 4, 6–8]. Basic ingredients Macromolecular Systems: Microscopic Interactions and Macroscopic Properties Deutsche Forschungsgemeinschaft (DFG) Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
31
3
Theoretical Aspects of Anomalous Diffusion in Complex Systems
in CTRWs are waiting time distributions with long time-tails. For photoconductivity one usually has g < 1. Second, in Section 3.3 we display the Matheron-de-Marsily (MdM) model, in which particles move under the influence of external, randomly quenched flow fields [9–11]. Here, depending on the correlations between distinct field lines, superdiffusive behaviour with g > 1 shows up. An interesting dynamical feature is observed when the object to move has itself an internal structure; that is the case, for instance, for a polymer. In this case, because of the connections which exist between different segments of the polymer, the motion gets a very rich structure. The details of this situation will be discussed in Section 3.4. Finally, we close in Section 3.5 with some conclusions.
3.2
Photoconductivity
A major field of application of scaling ideas in the time domain is photoconductivity [1–3, 7, 12]; here the use of polymeric photoconductors in printing and copying represents one of the most sophisticated applications for organic materials. Such materials are nowadays superior to their inorganic counterparts. Now, the issue at stake in amorphous photoconductors is the appearance of dispersive transport, as contrasted to the familiar diffusive Gaussian behaviour. One observes experimentally that the motion of the charge carriers becomes slower and slower with the passage of time, a situation mirrored by Eq. 1 with g < 1. Introducing the diffusion coefficient D as the first derivative of Ar2 (t)S with respect to time one has from Eq. 1 an algebraic decay (since g < 1) of the form D
t t 1 :
2
Also the current I (t) which flows under the influence of an external field often depends algebraically on time. Here charge carriers, generated by a short light pulse, move through the sample and give rise to a transient photocurrent in the external circuit. Under this time-of-flight technique two regimes can be distinguished. For times shorter than the transit time tT (the time needed by the fastest carriers to cross the sample) the current obeys I t
1
;
3
whereas for t > tT one finds I t
1
:
4
For instance measurements on polysiloxanes with pendant carbazole groups [13] follow Eqs. 3 and 4 with g = 0.58 very well over four decades in time, see below. In fact this is 32
3.2
Photoconductivity
typical for g significantly lower than one. However, one should note that the behaviour of I (t) is complex, since for g > 1 in the long-time limit the transport behaviour is non-dispersive. This is related to the existence of a finite mean waiting-time t = g/(g – 1). It follows that around g = 1 there is a crossover from dispersive to non-dispersive behaviour [13], as is also confirmed experimentally. From the preceding discussion it is obvious that the transport of charge carriers is much influenced by the waiting-time distributions (WTD) between hops. Physically the WTD arise from the local disorder in the sample. For large disorder the WTD are not exponential but decay much more slowly. In line with the preceding arguments one takes for the WTD expressions c (t) which behave at long times algebraically [7, 12] 1
t t
;
5
where 0 < g < 1. This choice for c (t) reproduces Eqs. 2–4, see below. Equation 5 is not valid near the time-origin. Therefore in calculations one prefers to work with functions welldefined for t 6 0. A suitable choice is for instance the series
t
1 aX
1 a
an bn exp
bn t
6
n1
with a < 1 and b < a –1 [8]. This function is everywhere continuous and finite, and for purely imaginary t it turns into the Weierstrass function. One verifies readily that for large t Eq. 6 obeys Eq. 5 and that the corresponding g in Eq. 5 is g = ln a/ln b. Equation 6 is very useful, since it allows to vary g freely by a judicious choice of a and b. Another choice for c (t) which is continuous for t 6 0 is
t
1 t :
7
Here again the long-time behaviour follows Eq. 5. We now recall the basic ingredients of random walks in continuous time, the so-called CTRW [6–8]. Let c (r, t) be the probability distribution of making a step of length r in the time interval t to t + dt. The total transition probability in this time interval is
t
X
r; t:
8
r
Furthermore the survival probability at the initial site is
t 1
Rt 0
d ;
9
so that, switching to the Laplace space (t ? u), one has
u 1
u=u:
10 33
3
Theoretical Aspects of Anomalous Diffusion in Complex Systems
The probability density Z (r, t) of just arriving at r in the time interval t to t + dt obeys the iterative relation
r; t
P Rt r0 0
r0 ;
r
r0 ; t
d
t r;0 ;
11
in which the initial condition of starting at t = 0 from r = 0 is incorporated. One then has for the probability P (r,t), that the particle is at r at time t, P
r; t
Rt 0
r; t
d :
12
Now P (r, t) also obeys an iterative relation: P
r; t
X Rt r0
0
P
r0 ;
r
r0 ; t
d
t r;0
13
as may be seen either by inspection or, more formally, by using Eqs. 11 and 12. Clearly, a description of such convolutions, Eqs. 11–13, is more compact in Fourier-Laplace space. For P (k, u) one has from Eq. 13: P
k; u P
k; u
k; u
u
14
with the immediate solution P
k; u
u=1
k; u
1
u u
1
1 :
k; u
15
The last expression generalizes the usual diffusion relation to random media, in which spatial and temporal aspects are coupled through c (k, u). The analysis is much simplified if such aspects decouple, which is the case for instance when the disorder is mainly energetic and the random walker moves over a rather regular lattice. In the decoupled case:
r; t
r
t :
16
From this it follows immediately that also c (k, u) = l (k) c (u) is decoupled. In the decoupled scheme P (k, u) takes the form P
k; u
1
u u
1
1
k
u
;
17
P where l (k) = q p (q)e–ik7q is the structure function of the infinite lattice and p (q) denotes the probability that a step extends over the distance q. 34
3.2
Photoconductivity
Coupling is very important for superlinear behaviour. Thus for the c (r, t) given by Eq. 16 the mean-squared displacement is either divergent or increases sublinearly or at most linearly in time. In order to obtain finite Ar2 (t)S with a superlinear temporal behaviour, coupled c (r, t) forms have to be used [4]. An example is the WTD
r; t Ar
r
t ;
18
in which the d-function couples r and t. This WTD leads to Lévy walks. Let us now focus on the mean-squared displacement. Evidently, one has
R r2
t r2 P
r; t dr
rk2 P
k; t jk0
19
from which, in the decoupled scheme, using Eqs. 5 and 17 it follows that in the absence of any bias Eq. 2 is fulfilled. The procedure is more readily examplified by calculating the current from the mean displacement of the carrier Ar(t)S in a biasing field [8]. Setting L –1 for the inverse Laplace transform it follows:
P r
t rP
r; t irk P
k; t k0 L 1
irk P
k; ujk0 r
@ P
k; u
u 1 j q ; L 1 L ir
k k k0 @ u 1
u 1
20
P where in the last line we work in the decoupled scheme and AqS = q qp (q) is the mean displacement per hop. In the presence of a bias AqS 0 0. Thus, setting |AqS| = 1, we have for the current I (t) in an infinite lattice of any dimension, d r
t 1 L I
t 1 dt
u :
u
21
For a finite chain of N sites Eq. 21 takes in the Laplace space the form [3, 13]: I
u
N X n1
un
1
u 1
u
uN :
22
It is precisely this function which was used in Refs. [13, 14] to analyse, together with the WTD Eq. 6, the time-of-flight currents of photoconductive carriers in polysiloxanes with pendant carbazole groups. In this work one achieved with g = 0.58 a good agreement with the experimental findings. The short time behaviour indeed obeys Eq. 3, whereas at long times the form of Eq. 4 is reproduced fairly well.
35
3
3.3
Theoretical Aspects of Anomalous Diffusion in Complex Systems
The Matheron-de-Marsily model
As mentioned in the introduction, superdiffusive (enhanced) behaviour is often found in turbulent flows. In this Section we adopt a picture different from CTRW. We focus on the influence of randomly distributed biasing external fields and follow the description of Oshanin and Blumen [11]. We take a three-dimensional (3D) solvent for which the flow is parallel to the Y-axis. The direction and magnitude of the flow depend only on the X coordinate of the position vector [9] so that VY, the non-vanishing component of the flow field, obeys: VY
X; Y; Z V X :
23
Here furthermore V[X] is a random function of X. Geometrically the system consists of parallel layers perpendicular to the X-axis. In each layer the value of VY is constant but varies from layer to layer [9]. We assume the random function V[X] of Eq. 23 to be Gaussian with zero mean, AV[X]S = 0, and with the covariance
V X1 V X2
jX1
X2 j:
24
Here the brackets denote configurational averages, which are conveniently expressed through Fourier integrals,
j X1
X2 j
R1 1
dwQ
w exp iw
X1
X2 :
25
Now many possibilities for Q (w) can be envisaged. For simplicity we take here only a flat spectrum, Q (w) = W/2p, as in the original Matheron-de-Marsily (MdM) model [9] in which the flows are delta-correlated,
jX1
X2 j W
X1
X2 :
26
We start from the Langevin dynamics of a single spherical bead subject to the MdM flow. This allows us to display enhanced (superlinear) diffusion in a simple situation [9, 10, 22]. The study of the dynamics of Rouse polymers in such flows is deferred to the next Section. Let R (t) be the position of the center of mass of the bead at time t, and we assume that R (0) = 0. The components X(t), Y(t) and Z(t) of R(t) obey the following Langevin equations
36
m
d2 X dt2
dX fX
t ; dt
27
m
d2 Z dt2
dZ fZ
t ; dt
28
3.3
d2 Y m 2 dt
dY dt
The Matheron-de-Marsily model
V X fY
t :
29
Here m denotes the mass of the bead and z the friction constant. The terms fX (t), fY (t), and fZ (t) give the random (thermal-noise) forces exerted on the bead by the solvent molecules. These forces are Gaussian, with the moments f i
t 0
30
and fi
t fj
t0 2 T i;j
t
t0 ;
31
where i,j B {X,Y,Z}. The dash stands for thermal averaging, di,j is the Kronecker-delta and the temperature T is measured in units of the Boltzmann constant kB . Conventionally the acceleration terms in Eqs. 27–29 are neglected, since they are small relative to the other terms [15]. This leads to
dX fX
t ; dt
32
dZ fZ
t ; dt
33
dY VX
t fY
t : dt
34
Note that in Eq. 34 the X and Y coordinates are coupled. Equations 32 and 33 are readily solved, X
t
1
Z
t
1
Rt
d fX
;
35
d fZ
:
36
0
Rt 0
Thus the bead undergoes a conventional diffusive motion between the layers (along the X-axis) and in the Z direction. One sees it readily by evaluating, say:
X 2
t
2
Rt 0
Rt Rt Rt d 1 d 2 fX
1 fX
2 2 1 T d 1 d 2
1 0
0
0
2 2
T= t ;
37
so that X 2
t 2D1 t, where D1 = T/z is the diffusion coefficient of a single bead. 37
3
Theoretical Aspects of Anomalous Diffusion in Complex Systems
A similar procedure can be performed with respect to the solution of Eq. 34, Y
t
Rt 0
d V X
1
Rt 0
d fY
:
38
The averaging involves now both, the thermal noise and the configurational disorder: Y 2
t
2
Rt 0
Rt Rt Rt
d 1 d 2 fY
1 fY
2 d 1 d 2 VX
1 VX
2 0
0
2D1 t
W=2
Rt 0
0
Rt R1 d 1 d 2 dw exp iw
X
1 1
0
X
2 ;
39
where in the last line use was made of the representation Eqs. 25 and 26 with the flat spectrum of the delta-function. The remaining average on the rhs. of Eq. 39 is readily evaluated by remembering that it is the characteristic functional of the Brownian trajectory
1 ; 2 ; w exp iw
X
1
X
2 exp
w2 D1 j 1
2 j :
40
Inserting Eq. 40 into Eq. 39 one recovers the following result for the average squared displacement (ASD) in the direction of the flow field,
4W Y 2
t 2 D1 t 3
t3 p D1
1=2 :
41
One should remark that at times greater than tc = 9 pD31/(4W2) the superlinear growth t3=2 in Eq. 41 dominates and the first, diffusive term can be neglected. One has then a superdiffusive behaviour with an exponent of g = 3/2 in Eq. 1 [9]. We now turn to the analysis of the behaviour of polymers in MdM flow fields. hY 2
ti
3.4
Polymer chains in MdM flow fields
The conformational properties and the dynamics of polymers in solutions under various types of flows have been a subject of considerable interest within the last decades. Much progress has been gained in the explanation of experimental data for systems in which the flow velocities are given functions in space and time, see Refs. [16–19]. On the other hand, the behaviour of polymers in random flows is less understood. In recent works [11] we (Oshanin and Blumen) succeeded in establishing analytically the behaviour of Rouse polymers [20] in MdM flow fields. The presentation here follows closely Ref. [11]. 38
3.4
Polymer chains in MdM flow fields
In the Rouse model N monomers (beads) are coupled to each other via harmonic springs [16, 17, 20]. As is well-known, the forces are of entropic origin. It is customary to revert to a continuous picture in which n, the bead’s running number, takes real values. For a detailed discussion see Doi and Edwards [17]. The Langevin equations of motion for such a polymer in the MdM flow field are
@ Xn
t @2 Xn
t K fx
n; t ; @t @ n2
42
@ Zn
t @2 Zn
t fZ
n; t ; K @t @ n2
43
@ Yn
t @2 Yn
t K VXN
t fY
n; t ; @t @ n2
44
see Ref. [11]. Equations 42–44 are the generalization of Eqs. 27–29 to polymers. They are to be solved subject to the Rouse boundary conditions at the chain’s ends, n = 0 and n = N [17]: @ Xn
t @ Yn
t @ Zn
t 0: @n @n @n
45
As before, the fluctuating forces on the rhs. of Eqs. 42–44 are Gaussian and also delta-correlated with respect to the running index [11, 17]. For the X and Z components, which are not subject to the flow, the procedure is standard [17]. Say, for the averaged X component of the end-to-end vector one has P2X
t
X0
t
XN
t2
b2 N ; 3
46
where b is the so-called persistence length. Furthermore, the X component of the radius of gyration is Xg2
1 2N 2
ZN ZN dn dm
Xn
t 0
0
Xm
t2
b2 N : 18
47
For isotropic situations (in the absence of flow fields) the end-to-end vector PR and the radius of gyration Rg of the polymer are related to PX and Xg through P2R = 3P2X and R2g = 3X 2g . Because of the anisotropy one has to consider in the MdM model the different components separately. The dynamics of a flexible polymer chain is richer than that of a single bead. In the Rouse model the dynamics of Xn (t) depends essentially on the time of observation t and on the Rouse time tR, R
b2 N 2 : 3p2 T
48 39
3
Theoretical Aspects of Anomalous Diffusion in Complex Systems
Now tR is the largest internal relaxation time of the chain [17, 20]. Exemplarily for t P tR one finds for the mean-squared displacement of a bead of the chain, say the zeroth one,
X0
t
X0
02 2b
D1 t 3p
1=2
:
49
Equation 49 is subdiffusive with g = 1/2 in Eq. 1. This is due to the fact that the trajectory of a bead in a chain is spatially confined by its neighbours. In the limit t p tR the chain diffuses as one entity and the bead’s trajectory follows mainly the motion of the chain’s center of mass. The chain’s center of mass obeys X 2
t 2DR
N t
50
with DR (N) = D1/N { T/(zN). Let us now turn to the Y component of the center of mass for which Eq. 44 leads readily to 1 Y
t N
ZN 0
ZN
Zt
1 dnYn
t N
d 0
dn V Xn
1 fY
n; ;
51
0
so that we obtain
W Y 2
t 2DR
N t 2N
Zt
Zt d 1
0
Z1 dw g
w; 1 ; 2 ;
d 2
52
1
0
where g(w; t1, t2) denotes the dynamic structure factor of the chain, 1 g
w; 1 ; 2 N
ZN ZN dn dm exp iw
Xn
1 0
Xm
2 :
53
0
It turns out [11], that all these integrations can be performed analytically. In the longtime limit t p tR one finds
4W Y 2
t 2DR
N t 3
N pT
1=2
t3=2 1
O
t
1=2
;
54
whereas for t P tR the short-time behaviour is given by
Y 2
t 2DR
N t 40
p 3 W 2 t 1 bN 1=2
O
t
1=4
:
55
3.4
Polymer chains in MdM flow fields
The interpretation of Eq. 54 is that at long times the t3/2 dynamics dominates the picture; the Rouse chain behaves like a compact bead. At short times the term t2 may become important. This g = 2 case in Eq. 1 is called ballistic; at very short times the center of mass of the chain hardly moves and it practically does not change the flow pattern to which it is subjected. We remark that at short times the motion of the center of mass of the chain and the motion of a tagged bead are characterized by different dependences on time. The mean-squared displacement along the Y-axis of a tagged bead, say the zeroth one, grows in time as
2 W Y0
t e2DR
N t 1=2 D1 1=4 t7=4 ; b
56
see Ref. [11]. This g = 7/4 dependence is, of course, related to the fact that the segmental motion at short times is confined, the number of distinct flow layers visited by the bead growing as t1/4. Results of such fractal-type behaviour may be formulated exactly [21, 22]. We now turn to the question of the elongation of the Rouse chain in the MdM flow and sketch the evaluation of the end-to-end distance along the Y-axis [11]. The solution of Eq. 44 under the boundary conditions Eq. 45 has the form of a Fourier series, Yn
t Y
0; t 2
1 X
cos
ppn
p1
N
Y
p; t ;
57
where the Y (p,t), p = 0, 1, …, denote the normal coordinates [17]. At t = 0 the chain is assumed to be in thermal equilibrium, i. e. to have a Gaussian conformation. This can be accounted for automatically by stipulating it to be subject to the thermal fluctuations since t = –?. Furthermore, the MdM flow fields are switched on at t = 0. This leads to Y
p; t 1 N
Zt
Rt 1
dn cos 0
= R f~Y
p;
d exp
p2
t
ZN d
0
1
ppn N
VXn
t
exp
p2 = R ;
58
where the functions f~Y denote the Fourier components of the thermal fluctuations [17]. The Y component of the end-to-end vector follows now from: PY
t Y0
t
YN
t 2
1 P p1
1
1p Y
p; t :
59
After performing the averaging over both, the thermal fluctuations and the realizations of the flow fields, one obtains from Eqs. 58 and 59 in the long-time limit the equilibrium value of the end-to-end vector for delta-correlated flows [11], 41
3
Theoretical Aspects of Anomalous Diffusion in Complex Systems
2 b2 N CW2 bN 5=2 1 PY
1 ; 3 T2
60
where C is a constant. Therefore, a Rouse polymer stretches along the Y-axis under MdM flows, taking the form of a prolate ellipsoid. The correction term to the Gaussian behaviour grows as N7/2. It is now interesting to confront this finding with the situation in simple shear flows, for which the result is [18, 19]: P2Y
t
b2 N 3 _ 2 2 b4 N 4 1 : 3 6480 T2
61
In Eq. 61 _ stands for the (constant) shear rate. Because all flow lines point now in the same direction, the correction term shows a stronger N-dependence and obeys a N5 law.
3.5
Conclusions
In this review several situations were presented, which lead naturally to the appearance of anomalous diffusion. Besides the already well-discussed subdiffusive behaviour, which is often seen in disordered media, we also considered superdiffusive dynamics, such as encountered in layered random flows. Anomalous diffusion was analysed through several models and we focussed on the behaviour of polymeric materials under such conditions. The basic aspect underlying these phenomena is dynamical scaling, which is often encountered experimentally and theoretically. On the other hand not all systems scale with time, and care is required in applying the models presented here; one has to be aware of intrinsic limitations of the scaling range (e. g. size limitations, other temporal scales involved). Because of this, close cooperations between experimentalists and theoreticians are much needed for the analysis of systems similar to the ones described here. Furthermore, despite the success of the now-closing Collaborative Research Center 213, much work still remains to be done.
Acknowledgements
The research collaboration with Prof. D. Haarer, Prof. J. Klafter, Dr. G. Oshanin, Dr. H. Schnörer, and Dr. G. Zumofen in our joint work reviewed here was always very helpful and pleasant. The support of the Deutsche Forschungsgemeinschaft through the Sonder42
References forschungsbereich 213 and Sonderforschungsbereich 60 was fundamental for the whole project. Additional help was provided by the Fonds der Chemischen Industrie and – in the later stages – by the PROCOPE-Program of the DAAD.
References
1. A. Blumen, J. Klafter, G. Zumofen: in: I. Zschokke (ed.): Optical Spectroscopy of Glasses, Reidel, Dordrecht, p. 199 (1986) 2. D. Haarer, A. Blumen: Angew. Chem. Int. Ed. Engl., 27, 1210 (1988) 3. A. Blumen, H. Schnörer: Angew. Chem. Int. Ed. Engl., 29, 113 (1990) 4. G. Zumofen, J. Klafter, A. Blumen: in: R. Richert, A. Blumen. (eds.): Disorder Effects on Relaxational Processes: Glasses, Polymers, Proteins, Springer, Berlin, p. 251 (1994) 5. J. Klafter, G. Zumofen, A. Blumen: J. Phys., A24, 4835 (1991) 6. E.W. Montroll, G.H. Weiss: J. Math. Phys., 6, 167 (1965) 7. H. Scher, M. Lax: Phys. Rev. B, 7, 4491; 4502 (1973) 8. M.F. Shlesinger: J.Stat. Phys., 36, 639 (1984) 9. G. Matheron, G. de Marsily: Water Resour. Res., 16, 901 (1980) 10. G. Zumofen, J. Klafter, A. Blumen: Phys. Rev. A, 42, 4601 (1990) 11. G. Oshanin, A. Blumen: Macromol. Theory Simul. 4, 87 (1995); G. Oshanin, A. Blumen: Phys. Rev. E, 49, 4185 (1994) 12. H. Scher, E.W. Montroll: Phys. Rev. B, 12, 2455 (1975) 13. H. Schnörer, H. Domes, A. Blumen, D. Haarer: Philos. Mag. Lett., 58, 101 (1988) 14. D. Haarer, H. Schnörer, A. Blumen: Dynamical Processes in Condensed Molecular Systems, in: J. Klafter, J. Jortner, A. Blumen (eds.): World Scientific, Singapore, p. 107 (1989) 15. M. Fixman: J. Chem. Phys., 42, 3831 (1965) 16. P.G. de Gennes: in: Scaling Concepts in Polymer Physics, Cornell Univ. Press, Ithaca, N.Y., (1979) 17. M. Doi, S.F. Edwards: in: The Theory of Polymer Dynamics, Oxford Univ. Press, Oxford, (1986) 18. R.B. Bird, C.F. Curtiss, R.C. Armstrong, O. Hassager: in: Dynamics of Polymeric Liquids, Vol.2, 2nd Ed. Wiley, New York, (1987) 19. W. Carl, W. Bruns: Macromol. Theory Simul., 3, 295 (1994) 20. P.E. Rouse: J. Chem. Phys., 21, 1273 (1953) 21. J.-U. Sommer, A. Blumen: Croat. Chem. Acta, 69, 793 (1996) 22. G. Zumofen, J. Klafter, A. Blumen: J.Stat. Phys., 65, 991 (1991)
43
4
Low-Temperature Heat Release, Sound Velocity and Attenuation, Specific Heat and Thermal Conductivity in Polymers Andreas Nittke, Michael Scherl, Pablo Esquinazi, Wolfgang Lorenz, Junyun Li, and Frank Pobell
We have measured the long time (t = 5 h to 200 h) heat release of polymethylmethacrylate (PMMA) and polystyrene (PS) at 0.070 K ^ T ^ 0.300 K. After cooling from a temperature (the charging temperature) of 80 K the heat release in PMMA shows a t –1-dependence in the measured time and temperature ranges in agreement with the tunneling model. In contrast, for PS we observe strong deviations from a t –1-dependence and a heat release smaller than in PMMA in by a factor of ten, in apparent contradiction to specific heat and thermal conductivity data for PS. To compare the heat release with other low-temperature properties and to verify the consistency of the tunneling model we have measured also the acoustical properties (sound velocity and attenuation), the specific heat and the thermal conductivity of PMMA and PS in the temperature ranges 0.070 K ^ T ^ 100 K, 0.070 K ^ T ^ 0.200 K and 0.3 K ^ T ^ 4 K, respectively. We show that the anomalous time dependence of the heat release of PS is due to the thermally activated relaxation of energy states with excitation energies above 15 K.
4.1
Introduction
A disordered material releases heat after cooling it from an equilibrium or charging temperature T1 to a measuring temperature T0 [1]. This heat release Q_ (T1,T0, t) depends on the charging temperature T1 as well as on the temperature T0, at which the measurement is performed, and the elapsed time during and after cooling [2, 3]. The time-dependent heat release, observed in several disordered systems, is a consequence of the long-time relaxation of the low-energy excitations, identified as two-level tunneling systems (TS) [4–6]. As a consequence of the finite relaxation time of the TS through their interaction with thermal phonons [7] the specific heat depends also on the time scale of the experiment [5], as first measured by Zimmermann and Weber [1]. In recent publications [5, 6] it was shown that within the tunneling model we can quantitatively understand the observed temperature and time dependence of the specific heat 44
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties Deutsche Forschungsgemeinschaft (DFG) Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
4.1
Introduction
and heat release of vitreous silica (SiO2) over ten orders of magnitude in time. It has also been pointed out that the measurements of the heat release at workable long times is probably the only feasible method to obtain at least approximately the low-energy limit of the tunnel splitting D0 of the distribution function of TS. This low-energy limit is considered in the literature by the cut-off parameter umin = (D0 /E)min of the distribution function (E is the energy splitting of the TS) and is introduced to keep the number of TS finite, avoiding divergences in the calculated properties. The interpretation of the heat release data in terms of the tunneling model is a difficult task due to the not well-known: a) temperature and time dependence of the specific heat due to TS at T > 3 K; b) influence of relaxation processes of the TS at T > 3 K other than one-phonon tunneling relaxation, i. e. high-order phonon tunneling and thermally activated relaxation; c) influence of the cooling procedure. In a recently published paper Parshin and Sahling [8] showed the complexity of the interpretation of the heat release data when thermally activated relaxation is taken into account within the framework of the soft potential model. Further theoretical work on the residual properties of two-level systems and its dependence on the cooling procedure has been published by Brey and Prados [9]. In this paper we present a further example of the complexity in the interpretation of the heat release data and an experimental proof of the influence of the relaxation, probably by thermally activated processes, of excited states that contribute to the heat release of the TS at low temperatures. We have studied the long-time heat release of two amorphous polymers with similar low-temperature specific heats and thermal conductivities, cooled under similar conditions. In spite of those similarities we have found a large difference in the absolute value and in the time dependence of the heat release between the two polymers when cooled from temperatures above 15 K. The similarities and differences in the low-temperature properties, their interpretation within the tunneling model, and the influence of thermally activated relaxation are the main scope of this work. Preliminary results were published in Ref. [10]. The paper is organized in five sections. In Section 4.2 we describe the phenomenological theory for the heat release, based on the tunneling model and the influence of different relaxation rates, and the cooling process. In Section 4.3 we briefly describe the experimental procedures and samples. In Section 4.4 we show and discuss the experimental results. Conclusions are drawn in Section 4.5.
45
4
4.2
Low-Temperature Heat Release, Sound Velocity and Attenuation, …
Phenomenological theory for the heat release
4.2.1 Generalities New theoretical work for the calculation of the heat release within the soft potential model has been published recently [8]. In order to simplify the calculations, to minimize the number of free parameters, and to assure a more transparent interpretation of the results we decided, however, to interpret our results in terms of the standard tunneling model. According to Ref. [8] and to our numerical calculations the differences between the soft potential and standard tunneling model are not significant at the low temperatures of our measurements. The standard tunneling model, including a thermally activated relaxation of the twolevel systems, was used successfully to interpret the acoustical properties of vitreous silica in eight orders of magnitude in phonon frequency from approximately 0.1 K up to room temperature [11]. For a system of N two-level systems with energy difference E, the difference in the population of the two levels at a given temperature T and at thermal equilibrium is given by n0 N tanh
E=2kB T :
1
If the thermodynamic equilibrium of the system is slightly perturbed, e. g. the system is rapidly cooled to a temperature T, the dynamical behaviour of the population difference can be calculated according to the relaxation time approximation formula: n0
T
d
n
t dt
n
t n0
T ;
E; T
2
where t (E,T) is the relaxation time of a tunneling system with energy E at a temperature T. The heat released by the N two-level systems after the temperature change is given by _ Q_ nE=2 :
3
The problem of calculating Q_ simplifies to calculating n from Eq. 2. However, the cooling process has to be taken into account. In this case Eq. 2 must be rewritten as n_
@n0 dT @T dt
n
n0
:
4
Equation 4 describes the response of two-level systems during a temperature change given by the function T (t). In the general case Eq. 4 has to be solved numerically since n0 and t are temperature-dependent variables.
46
4.2
Phenomenological theory for the heat release
4.2.2 The standard tunneling model with infinite cooling rate If N two-level systems are in thermal equilibrium at a temperature T1 and they are cooled to T0 with infinite cooling rate, i. e. the time to cool the sample from T1 to T0 is zero, from Eq. 2 we obtain n
t
n0
T1
n0
T0 exp
t=
E; T0 n0
T0 :
5
The standard tunneling model assumes that at low temperatures (T < 2 K) the one-phonon process is the dominating mechanism. The one-phonon relaxation rate is given by p 1 pm1 u2
6
pm1 AE 3 coth
E=2KB T ;
7
with
where A = (g 2l /n 5l + 2 g 2t n 5t )/2 pr–4. The indices l, t refer to the longitudinal and transversal phonon branches, r is the mass density, gl and gt are the coupling constants between phonons and TS, and u = D0 /E (D0 is the tunneling splitting). For symmetrical TS, i. e. D0 = E (u = 1), tp reaches its minimum value tpm . Furthermore, the standard tunneling model assumes that the distribution function of TS is constant in terms of two independent variables, namely the asymmetry D and the tunneling parameter l, i. e. P
; dd Pdd :
8
According Ref. [7] the distribution function can be written in terms of the variables E and u as
1 P
E; u P=u
u2 1=2 :
9
It is also convenient to have the distribution function in terms of the asymmetry and the barrier height V between the potential wells. Following the work of Tielbürger et al. [11] and assuming two well-defined harmonic potentials, it can be shown that in a first approximation l = V/E0 where E0 represents the zero-point energy. In this case the distribution function is P Pdd ddV : E0
10
Replacing the total number N of TS with the integrals in E and u, the heat release is given by 47
4
Low-Temperature Heat Release, Sound Velocity and Attenuation, …
1 Z PV E Q_ dEE tanh E0 2kB T0 0
E tanh 2kB T1
Z1 umin
du p 1
T0 exp
t=
T0 ; u 1 u2 (11)
where V is the sample volume and umin is the cut-off in the distribution function at u ? 0. Figure 4.1 shows the heat release at T0 = 0.1 K as a function of time for different charging temperatures T1 following Eq. 11 and taking into account one-phonon relaxation rate, Eqs. 6 and 7. The calculation was performed with parameters appropriated for vitreous silica, Ak 3B = 46106 s–1 K–3, P = 1.661038 J–1 g–1 and umin = 5610–8 [5]. We observe that at short times and small T1 the heat release Q_ ! t–1 T 21. This dependence follows from the logarithmic time dependence of the specific heat and holds for t P tm/u2min . In this limit the heat release follows the often used approximation from Eq. 11 [1]: p2 2 Q_ k PV
T12 24 B
T02
1 ; t
12
where Q_ and P are measured in W and (Jg)–1. We recognize in Fig. 4.1, however, that at longer times the theory deviates from the t –1-dependence. This deviation comes from the exponential term in Eq. 11 that decreases strongly at long times and is determined by the product Au2min. The smaller this product the larger is the time-range where the approximation given by Eq. 12 holds. The influence of the cut-off umin can be recognized comparing the calculated heat release in Fig. 4.1 with that in Fig. 4.2 (dashed lines) calculated with a smaller umin. In Fig. 4.1 we note also that at large times the heat release becomes independent of T1. This feature was recognized experimentally in Refs. [2, 3] and was discussed in Ref. [6]. This T1-independence within the assumptions described above is a direct consequence of the finite number of TS given by the cut-off umin. We should note, however, that in SiO2 a saturation of the heat release for large charging temperatures (T1 > 20 K) is observed and still a t –1-dependence was measured [2]. This result cannot be explained within the standard tun-
Figure 4.1: Heat release as a function of time t (in s) according to the standard tunneling model and infinite cooling rate with Ak 3B = 46106 s–1 K–3, P = 1.661038 J–1 g–1 and umin = 5610–8 at a measuring temperature of 0.1 K and at different charging temperatures T1, bottom: 5 K, top: 80 K.
48
4.2
Phenomenological theory for the heat release
Figure 4.2: Heat release as a function of time t (in s) within the standard tunneling model and infinite cooling rate. The parameters are the same as in the previous figure but with umin = 5610–9 (dashed lines) and an additional constant (energy-independent) relaxation time tTA = 103 s (continuous lines).
neling model. The reason for this discrepancy is the relaxation rate, which is different from the one-phonon process at higher temperatures and the influence of a finite cooling rate (see below).
4.2.3 Influence of higher-order tunneling processes and a finite cooling rate At temperatures above 3 K tunneling processes of higher-order are, in principle, possible. One particular high-order process was studied theoretically in Ref. [12] and is similar to the optical Raman process; the relaxation time of the TS is given by an interaction that involves two phonons. Following Ref. [12] we added this Raman relaxation rate to the one-phonon relaxation rate (Eq. 6). Following Ref. [12] a new free parameter, i. e. the coupling constant R for the Raman process, is assumed. Due to its strong temperature dependence (T 7), the Raman process can influence the population difference of the energy states of the TS mainly at high temperatures (T > 5 K) during the cooling process. We should note that for infinite cooling rate and for the temperature range of our measurements (T0 ^ 1 K) the contribution of the Raman process to the time and temperature dependence of the heat release is negligible. This follows from Eq. 5 where only the relaxation of the TS at the measurement temperature T0 enters. In order to take into account the cooling process we used the algorithm explained below. The zero-time point is chosen as the time when the cooling process is started. The function n (E,u,t) represents the difference in population of the TS energy states at any time t and n0 (E,T) means this difference in equilibrium (t ? ?) at a temperature T. Analog to Eq. 11 the heat release can be written as: 1 Q_ PV 2
Z1 Z1 dEE 0
umin
du p 1
T0 e u 1 u2
1
T0 t
n0
E; T0
n
E; u; t 0
13
49
4
Low-Temperature Heat Release, Sound Velocity and Attenuation, …
Note that the population difference n depends on E and u because the relaxation rates, that influence the relaxation of the TS, depend on these variables. As already described above, to obtain the function n we need to solve the differential equation given by Eq. 4. In general this is only possible by numerical methods. The time dependence of the temperature during the cooling process performed in this work can be well approximated by a linear function given by T
t T1
T0
T1 tA
:
14
We assume that the sample is at t = 0 (tA) at T1 (T0); tA is the time needed to cool the sample from T1 to T0. The calculations have been done splitting the function T (t) in N steps of time Dt (Fig. 4.3). During the time Dt the temperature remains constant and n (E,u,t) shows an exponential behaviour given by Eq. 5; this is qualitatively shown in Fig. 4.3. To obtain the population difference at the time Dt cooling from T1 at t = 0 for a given E and u we calculate iteratively the value n
t 0 t n0
0 t
n0
0 t
n
0 e
1
T
0tt
:
15
The calculations given in this work have been made taking into account 10 to 200 time-steps or iterations depending on the convergence of the numerical results. Figure 4.4 shows the heat release as a function of time taking into account one-phonon (dashed lines) and the Raman process (Rk7B = 100 s–1 K–7) with a cooling time tA = 3000 s from T1 to T0 = 0.1 K. Taking into account our experimental time scale the calculations were performed for the time interval 104 s ^ t ^ 106 s only. For one-phonon
Figure 4.3: Time dependence of the sample temperature during a cooling process and its splitting in N steps (dashed lines) assumed for numerical calculations. Bottom: qualitative behaviour of the population difference of the tunneling systems n0 (T) as a function of time.
50
4.2
Phenomenological theory for the heat release
Figure 4.4: Heat release as a function of time for a finite cooling time TA = 3000 s and with (continuous lines) or without (dashed lines) Raman processes.
process only, our calculations indicate that a finite cooling rate with tA = 3000 s to 5000 s has a negligible influence (< 3 %) on the heat release in our experimental time scale (compare the dashed lines in Fig. 4.4 with the dashed lines in Fig. 4.2 obtained with an infinite cooling rate). The influence of the Raman process at T0 = 0.1 K can be well observed if we take into account a finite cooling rate. In comparison with the results using the one-phonon process only and for T1 > 20 K, Q_ is smaller and shows a slightly different time dependence. Note that the results for T1 6 20 K resemble those obtained taking into account the one-phonon process only but with a smaller charging temperature T1. We note also that at T1 ^ 20 K the heat release still shows a T 21-dependence but reaches its saturation at a smaller T1 in comparison with the results with the one-phonon process only (Fig. 4.4). We conclude that taking into account Raman processes, that influence the relaxation rate of TS at temperatures larger than 2 K, it is possible to understand qualitatively the results of, for example, SiO2 where Q_ reached a saturation at T & 10 K but still shows a t –1dependence at t ^ 36105 s [2]. This result is only valid if the product Au2min in Eq. 7 remains small enough. Because of the uncertainty in the coupling constant R between TS and phonons a quantitative comparison, however, of the heat release results with the theory taking into account the contribution of higher-order processes is not useful. Phenomenologically and in order to simplify the calculations, one can take into account the influence of higher-order processes by choosing an appropriate smaller charging temperature than the real one. In that sense charging temperatures T1 6 10 K can be considered for comparison with theory as free parameters. From our calculation we learn that it is practically impossible to obtain reliable information on the density of states of TS from heat release experiments for charging _ temperatures in the region of the T1-saturation of Q.
51
4
Low-Temperature Heat Release, Sound Velocity and Attenuation, …
4.2.4 The influence of a constant and thermally activated relaxation rate There are different attempts to link the high-temperature (T > 10 K) with the low-temperature properties of disordered solids [13, 14]. In particular, it has been proposed that the maximum in the attenuation of phonons observed in amorphous materials at T > 10 K can be interpreted assuming a thermally activated relaxation of the two-level systems. In this Section we discuss the influence of thermally activated relaxation rate on the heat release for a finite cooling rate. For a single activation barrier V0 between the two potential wells and at a measuring temperature T0 the thermally activated relaxation rate is given by TA1 0 1 e
V0 =kB T0
:
16
Taking into account this relaxation rate in addition to quantum tunneling (Eq. 6) and assuming that both processes are independent, the total relaxation rate can be written as
1
P 1 TA1 :
17
Obviously, at a given temperature T0 and for a single activation barrier we add a constant relaxation rate to the tunneling process. Figure 4.2 shows the time dependence of the heat release at different temperatures T1 and using Eq. 17 with tTA = 103 s. As expected, at t P tTA the introduction of an additional constant relaxation rate does not influence the heat release. At t & tTA the heat release is larger than taking only into account the one-phonon process (Fig. 4.2). At longer times Q_ decreases exponentially with time. This decrease can be easily understood: the energy levels with long (tunneling) relaxation time have been depopulated at t & tTA at a larger rate. Since the total amount of heat released by the TS should be finite a decrease below the standard result (dashed lines in Fig. 4.2) is expected. To take into account the distribution of potential barriers V and a thermally activated rate we will follow the approach used by Tielbürger et al. [11] and transform Eq. 11 in a double integral in the variables D and V using the approximation l = V/E0 and D0 & 2E0 e–l/p: V 1 Zmax Z PV E E _ Q d dV tanh 2 E0 2kB T0 0
0
tanh
E 1
T0 exp
t=
T0 : 2kB T1
18
For the following discussion it is not relevant to introduce in Eq. 18 a specific distribution of potential barriers V [11]. This is done below for comparison with experimental results and for computing the acoustic properties. After finding the range of potential barriers relevant for the heat release we divided the calculations in two regions: T1 ^ 1 K and T1 > 1 K as described below. In order to find the range of potential barriers relevant for the heat release in our measuring time and temperature ranges we have calculated it following Eq. 18, splitting the Vlimits of the inner integral from Vmin = Vmax – 10 K to Vmax, using only the one-phonon process, and measuring temperature T0 = 0.200 K. We have recognized that for V < 130 K and 52
4.2
Phenomenological theory for the heat release
at t & 103 s the TS are mostly relaxed and do not contribute appreciably to the heat release in our measured time range (104 s ^ t ^ 106 s) any more. The TS that relax through the one-phonon process and are relevant to the heat release are found to be those with potential barriers 130 K < V < 200 K. The contribution of tunneling systems with V > 200 K to the heat release is negligible. At T1 < 1 K the contribution of a thermally activated rate for TS with potential barriers 130 K < V < 200 K is irrelevant, i. e. using t & 10–13 s (from acoustic measurements, see below) we obtain a tTA(V = 130 K kB) of several years. The potential barriers which are relevant in our time and temperature range through a thermally activated rate are V/kB < 10 K, e. g. V (102 s (106 s))/kB & 7 (9) K. We note, however, that according to our calculations TS with those potential barriers already relax at the beginning of the measurement through the one-phonon process. Therefore, at T1 < 1 K we do not expect that thermally activated relaxation influences appreciably the heat release. Our calculations show that thermally activated processes can indeed influence the heat release by depopulating some of the relevant states (100 K ^ V/kB ^ 200 K) during the cooling process if T1 > 1 K. Figure 4.5 shows the calculated heat release as a function of T1 at a given time t = 1.56105 s taking into account the cooling process and one-phonon process as well as thermally activated relaxation rate. The numerical results (Fig. 4.5) show clearly a saturation of Q_ for T1 > 5 K, i. e. at lower temperatures in comparison with the results without thermally activated relaxation, in qualitative agreement with the results assuming Raman processes and experimental results [2]. It is also interesting to note that a thermally activated relaxation rate decreases slightly the exponent in the time dependence of the heat release, i. e. Q_ ! t –a with a < 1. This result is shown in Section 4.4.
Figure 4.5: Heat release at t = 1.56105 s as a function of charging temperature T1 calculated with and without thermally activated processes and a finite cooling rate.
53
4
4.3
Low-Temperature Heat Release, Sound Velocity and Attenuation, …
Experimental details
The experimental setup for measuring the heat release and the specific heat consists of a calorimeter on which the sample is mounted. The samples were cooled from 80 K to 0.050 K in about (5–8)6103 s. The calorimeter is attached to a holder through a thermal resistance or a superconducting heat switch made of an Al strip. The holder is screwed to the mixing chamber of a top loading dilution refrigerator. Two sample holders were used, one from plastic and a second from Teflon. The first one showed a heat release approximately proportional to t –1 whereas for the second one no heat release was measured in agreement with previous measurements [2]. We used two different methods to measure the heat release. Firstly, after cooling to low temperatures (T & 0.060 K) the heat switch was opened and the time dependence of the sample temperature (warmup rate) was measured till about 0.120 K. Then the sample was cooled again to about 0.050 K before starting another warmup run. The heat release is obtained from the slope of the temperature in the warmup run dT (t)/dt with the knowledge of the specific heat of the sample plus calorimeter C (T), i. e. Q_ = C (T) dT (t)/dt. The measurements were completely automated with a personal computer. The background heat leak was about (0.03–0.50) nW at 0.090 K and at t & 105 s depending on the sample holder and the vibration of the connecting cables. To test the measuring setup and procedure we have produced well-known heat leaks with a electrical heater, fixed to the sample. The second method for measuring the heat release is based as before on the measurement of the time dependence of the sample temperature at constant bath (mixing chamber) temperature. The difference lies on the selection of a fixed thermal resistance Rth between sample and holder that enables the semiadiabatic measurement of the heat release in a time scale larger than the intrinsic thermal relaxation time of the arrangement. Approximately 30 minutes after reaching a constant bath temperature Tb the decrease of the sample temperature Ts with time provides directly the heat release, i. e. Q_ = (Ts – Tb)/Rth(Ts ,Tb). The measurement of the thermal resistance together with the background contribution to the heat release is made in situ after reaching the time-independent minimum temperature. Within experimental error (& 10 %) both methods show the same results. The specific heat of both polymers was measured with the heat pulse technique in semiadiabatic fashion. The thermal conductivity was measured with a top loading 3He refrigerator using the standard procedure. The acoustic properties, sound velocity and attenuation, were measured with the vibrating reed technique [16] in the frequency range (0.2–3) kHz. Both polymers were prepared following standard procedures. The PMMA sample had additionally 10 –2 mol% of tetra-4-tert.butyl-phthalocyamin (dye molecule) because it was used in an early optical hole burning experiment [15]. For the specific heat and heat release measurements the mass and density of the PMMA sample were determined as 11.97 g and 1.15 g/cm3, respectively. We have measured two PS samples prepared from different batches. The densities of these samples were 1.05 g/cm3, and the masses 11.4 g (sample PS1) and 38.0 g (sample PS2). For the thermal conductivity measurements two slices were cut from the bulk samples with length l = 1 cm, width w = 2 mm, and thickness d = 300 mm. For the acoustic measurements the reeds had the geometry l = (0.7–1) cm, w = (0.1–0.3) cm and d = (100–300) mm. 54
4.4
4.4
Experimental results and discussion
Experimental results and discussion
4.4.1 Specific heat and thermal conductivity Figure 4.6 shows for both polymers the specific heat devided by the temperature as a square function of temperature. These measurements were performed only when the heat release is negligible (t & 106 s). For t ? ? the specific heat can be written in terms of two contributions: c (T) = c1T + c3T 3 with coefficients for the tunneling system c1 and phonon contribution c3. From Fig. 4.6 we obtain the values c1 = (3.0 + 0.3) mJ/gK2 and c3 = (93 + 18) mJ/gK4 for PMMA, and c1 = (4.6 + 0.5) mJ/gK2 and c3 = (77 + 23) mJ/gK4 for PS. The values of the linear term are in fair agreement with those published by Stephens [17, 18] c1 = 4.6 mJ/gK2 for PMMA and c1 = 5.1 mJ/gK2 for [PS], but we obtained larger phonon contribution (Stephens: c3 = 29 mJ/gK4 for PMMA and c3 = 45 mJ/gK4 for PS). In Figs. 4.7 and 4.8 we compare our specific heat results with data from Ref. [18]. Within the scatter of the data it is difficult to conclude whether the c values are really different.
Figure 4.6: Specific heat divided by the temperature as function of the squared temperature for PMMA (a) and PS (b).
Figure 4.7: Comparison of our specific heat for PMMA with data from Ref. [18].
55
4
Low-Temperature Heat Release, Sound Velocity and Attenuation, …
Figure 4.8: Comparison of our specific heat for PS with data from Ref. [18].
According to the tunneling model and for t ? ? the specific heat due to the TS is given by [5]: c
T
p2 2 2 k PV ln
T c1 T : umin 6 B
19
If we assume (from heat release data, see below) umin = 10 –10 for PMMA and umin = 6610 –9 for PS and from the measured c1 we obtain a density of states P = 4.061038 (7.561038) 1/Jg for PMMA (PS) in reasonable agreement with the values obtained in earlier specific heat measurements [17] and also from acoustic measurements for PMMA [24]. Recent optical hole burning experiments on PMMA and PS [15] indicate that the density of states of TS for PS is about two times larger than for PMMA in reasonable agreement with our specific heat measurements.
Figure 4.9: Thermal conductivity as a function of temperature for PMMA and PS.
56
4.4
Experimental results and discussion
Figure 4.9 shows the thermal conductivity for PMMA and PS as a function of temperature. For PMMA and at T < 0.7 K we obtain k = 28T 1.84610–3 W/mK2.84 and for PS k = 19T 1.93610 –3 W/mK2.93, in agreement with earlier measurements [17, 19–22], (Figs. 4.10 and 4.11), and with measurements in epoxies [23]. According to the tunneling model and for T < 1 K the thermal conductivity is given by:
T
k3B v 2 1 2
P T ; 6p2
20
where v is the sound velocity and r the mass density. From Eq. 20 and with the sound velo 2 = 9.26105 J/m3 (11.56105 J/m3) for PMMA (PS). If we city from Ref. [17] we obtain Pg replace the density of states P from the specific heat results we obtain similar values for the coupling constant between TS and phonons in both materials g = 0.28 eV (0.27 eV) for PMMA (PS). The results for the thermal conductivity indicate that the density of states of TS for the PS sample is about a factor of two larger than for the PMMA sample.
Figure 4.10: Comparison of our thermal conductivity data for PMMA with earlier publications. The dashed line has been calculated within the tunneling model with parameters taken from our internal friction data.
Figure 4.11: Comparison of our thermal conductivity data for PS with earlier publications. The dashed line has been calculated within the tunneling model with parameters taken from our internal friction data.
57
4
Low-Temperature Heat Release, Sound Velocity and Attenuation, …
4.4.2 Internal friction and sound velocity Figures 4.12 and 4.13 show the internal friction of PMMA and PS. Below 1 K the internal friction of PMMA and PS is approximately temperature-independent. According to the tunneling model the sound absorption should be temperature-independent in the region otp P 1 (fulfilled in our temperature range) where o = 2 pv is the phonon frequency and tp the relaxation time of the TS (Eq. 6). In this temperature range a simple relationship is found for the internal friction [11, 16] Q
1
p C 2
21
Figure 4.12: Squares: PMMA internal friction as a function of temperature at a frequency of 535 Hz. Full circles: indicate the attenuation data taken from the ultrasonic measurements at 15 MHz from Ref. [24]. Solid and dotted lines: the calculated values for v = 535 Hz and v = 15 MHz following the modified tunneling model considering a thermally activated relaxation rate. For more details see text.
Figure 4.13: Squares and Circles: PS internal friction as a function of temperature at two different frequencies. Solid line: the calculated values for v = 0.24 kHz following the modified tunneling model described in the text.
58
4.4
Experimental results and discussion
2/ru2 and the sound velocity u. Equation 21 should hold even if with the constant C = Pg higher-order processes dominate the relaxation of the TS (if the relaxation rate is proportional to u2 ! D20, Eq. 6). From the temperature-independent value of the internal friction and assuming no background contribution (e. g. due to the clampling) we obtain for PMMA (PS) C % 2.6610–4 (8.3610 –4). Taking into account the mass density, sound velocity, and the coupling constant obtained from the thermal conductivity and specific heat results described above, the values for the parameters C from the internal friction indicate that the density of states of TS for PS is larger than for PMMA by a factor & 2.5. This is slightly larger than that obtained from the specific heat (about 1.5). This might be attributed to the unknown clamping contribution to the measured attenuation (which is always present) or to different umin values (Eq. 19). In Figs. 4.10 and 4.11 we show the calculated thermal conduc 2 values from our internal friction at the plateau. Excellent agreetivity using Eq. 20 with Pg ment is obtained for PMMA (Fig. 4.10) but for PS the thermal conductivity is by a factor of two smaller. This difference might be attributed partially to the background contribution which is not subtracted from the measurement before computing the parameter C. The behaviour of the internal friction above about 3 K is qualitatively different for both polymers. PMMA shows a decrease in the internal friction reaching a minimum at about 30 K and increasing monotonously to the highest temperature of our experiment, 120 K (not shown in Fig. 4.12) in very good agreement with previous work at similar frequencies [25]. On the contrary, the internal friction of PS shows the typical temperature dependence measured for other amorphous materials, for example SiO2 [16]. It increases, reaching a frequency-dependent maximum at T & 37 K (;40 K), (Fig. 4.13), and increases again at T > 60 K (70 K) at the frequency 0.24 kHz (3.2 kHz). This internal friction increase at T > 60 K for PS together with the increase at T > 30 K for PMMA will not be discussed here. Instead, we will discuss the behaviour of the internal friction at lower temperatures in terms of an extension of the tunneling model following the procedure described in Ref. [11]. We assume the simplified relation for a thermally activated relaxation rate given by Eq. 16 and add it to the tunneling rate (Eq. 17). For two well-defined harmonic potentials and within the approximations given by Eq. 10 and with l = V/E0, it can be shown that the internal friction increases linear with temperature just above the temperature-independent region (plateau) [11]: Q
1
pCkB T : E0
22
This increase is observed for PS, (Fig. 4.13); applying Eq. 22 to the results below 20 K we obtain a zero point energy E0 = 13 + 2 K. In the same temperature region and due to the influence of the thermally activated relaxation the relative change of the sound velocity can be written as [11]: uu CkB T ln
! 0 : u E0
23
A nearly linear temperature dependence of uu/u is observed in both polymers at temperatures about below 50 K (Figs. 4.14 and 4.15). At T ~ 1 K a crossover, due to tunneling and due to thermally activated relaxation, from the linear T-dependence to the logarithmic 59
4
Low-Temperature Heat Release, Sound Velocity and Attenuation, …
Figure 4.14: Relative change of sound velocity at v = 535 Hz for PMMA. The solid line represents the linear temperature dependence of the sound velocity at temperatures below 20 K used for the numerical calculations. Inset: the same data but below 10 K in a semilogarithmic scale.
Figure 4.15: Relative change of sound velocity at two different frequencies for PS. The solid line was calculated according to the modified tunneling model described in the text. Inset: the same data but below 10 K in a semilogarithmic scale.
T-dependence can be observed for both polymers (insets in Figs. 4.14 and 4.15). Although the tunneling model with the assumption of thermally activated relaxation of the TS provides a reasonable fit for the linear temperature dependence, its origin is still controversial. We note that a linear temperature dependence of the sound velocity above a few Kelvin is a rather general behaviour observed in several amorphous [26, 27], disordered [28], and polycrystalline metals [29]. Nava [28] argued recently against an interpretation in terms of thermally activated relaxation of the TS for the linear T-dependence of the sound velocity. However, new acoustical results in polycrystalline materials indicate a linear T-dependence of the sound velocity comparable with those found in amorphous materials [29]. Applying Eq. 23 to the measurements at the two phonon frequencies for PS and with the value of E0 from the internal friction we obtain t0 % (1 + 0.5)610 –17 s. As discussed in Ref. [11] the meaning of the prefactor t0 is still unknown. The very small value of t0 found for PS in comparison with the one for SiO2 (t0 % 10 –13 s) should be taken as an ef60
4.4
Experimental results and discussion
fective value until a systematic comparison of the applied model to other experimental data is available. To obtain a maximum in the internal friction an upper limit of the potential height must be assumed. Instead of a cut-off in the distribution P (D, V) and following Ref. [11], a Gaussian distribution with a width s0 will be assumed: P
; V
P
e E0
V 2 =202
:
24
Figures 4.13 and 4.15 show the result of numerical calculations with only the width of the distribution s as free parameter. The fits were obtained assuming s0 = 1200 K and with the values C = 8.3610 –4, v = 0.24 kHz, t0 = 10–17 s and E0 = 13 K using the equations given in Ref. [11]. A reasonable agreement with the experimental data is achieved for both acoustic properties in the expected temperature region. The minimum obtained numerically at T ~ 3 K (Fig. 4.13) has been discussed in Ref. [11] and is attributed to the difference in the number of TS contributing to the relaxation process in the tunneling or thermally activated regime. As pointed out above, the internal friction of PMMA behaves differently from PS above the plateau. It decreases at T > 3 K and reaches a minimum at T ~ 30 K. The predicted linear temperature dependence of the internal friction (Eq. 22), at the measured frequency (v = 535 Hz) is not observed for PMMA. It is tempting to interpret the decrease of the internal friction results above 3 K assuming an upper bound of the TS density of states around 15 K. This assumption has been indeed used to interpret the irreversible line broadening of the optical hole burning experiments [30]. However, ultrasonic attenuation measurements at 15 MHz for PMMA [24] show clearly a thermally activated maximum at 12 K. Therefore we have decided to search for a set of parameters that might explain the low and high frequency results under the assumptions described above and as done for PS. From the sound velocity data below 20 K (Fig. 4.14) and assuming the validity of Eq. 23 (ot P 1) we obtain (E0 /kB) / ln (ot0) % C/(u ln (u)/uT) ~ 0.9 + 0.2. If we assume t0 ~ 10 –17 s like PS, we obtain E0 /kB ~ 30 K. With this value we are not able to obtain a reasonable set of parameters that explain the data. If we take the value for SiO2 from Ref. [11], t0 % 10–13 s, we obtain E0 /kB % (20 + 5) K, a value comparable to that for SiO2. After several trials, eventually we have found a set of parameters that fits reasonably the low and high frequency measurements without the assumption of an energy cut-off. In Fig. 4.12 we compare the experimental results of the internal friction with the numerical calculations for frequencies at 535 Hz and at 15 MHz using E0/kB = 10 K, t0 = 10 –13 s, and a rather small distribution width s0 = 150 K. The model reproduces fairly well the position of the maximum in the attenuation at 15 MHz as well as its relative value in comparison with the plateau [24]. It is worth pointing out, that for PMMA, since the changes produced by the free parameters E0, t0 and s0 compete with each other, it is not possible to find another set of values which can explain the results as well as we can. It is important to note that the internal friction data for both polymers indicate a much smaller thermally activated contribution to the phonon attenuation for PMMA than for PS, in spite of the fact that in the quantum tunneling regime both samples show similar results. This difference may influence the low-temperature heat release because the number of excited states that are available after the cooling process will be different. 61
4
Low-Temperature Heat Release, Sound Velocity and Attenuation, …
4.4.3 Heat release Figure 4.16 shows the heat release as a function of time for the 12 g PMMA sample at T = 0.090 K and after cooling from 80 K. The heat release follows very well the predicted t –1 law from the tunneling model (Eq. 12). The same result was obtained in different runs with slightly different cooling rates and measuring temperatures 0.070 K ^ T0 ^ 0.110 K. For finite cooling rates and for charging temperatures T1 p 1 K this independence cooling rate is to be expected (Section 4.2.3). As discussed in Section 4.2 the unknown higher-order relaxation processes are taken into account through an effective charging temperature T1. With the tunneling systems density of states from the specific heat and using Eq. 12 we obtain T1 % 26 K. As stated in Section 4.2 for charging temperatures T > 10 K it is not possible to obtain reliable values of the density of states for TS from heat release measurements. From the observed t –1-dependence and using the value for the coupling constant between TS and phonons from Ref. [24], Ak3B = 1.66109 K–3 s –1, we obtain an upper limit for the parameter umin < 10 –10. According to theory, described above, larger values of umin should influence the t –1-dependence in our measuring time scale. From the theoretical results, discussed above, and for a finite cooling rate, thermally activated relaxation (Eq. 16) should influence the absolute value of the heat release, but only slightly its time dependence in our time and temperature ranges. In Figure 4.16 we show the results of the numerical calculations with and without thermally activated processes for a charging temperature T1 = 80 K and using the parameters obtained from the acoustical data (umin = 10 –10). We note that the experimental data lie between the two computed curves (2)
Figure 4.16: Heat release as a function of time. (.): PMMA at measuring temperature T0 = 0.090 K and cooling from T1 = 80 K; (_): Polystyrene (sample PS1) with T0 = 0.090 K and T1 = 80 K, (y): (sample PS2) with T0 = 0.300 K and T1 = 80 K. For PS the curves (1), (3), and (4) are calculated according to the tunneling model with finite cooling rate using the parameters from the fits to the acoustic data. For PMMA the curves (2) and (5) are calculated. In curves (1) and (2) no thermally activated processes are taken into account but for (3), (4) and (5). The parameter umin = 10 –10 with exception of curve (4) where umin = 6610 –9 has been chosen.
62
4.4
Experimental results and discussion
and (5). Even worse, the computed curve (5) with thermal activation shows much smaller values for the heat release by almost two orders of magnitude. This result indicates that too many energy states were depopulated during the cooling process through thermal activation. This disagreement might be ascribed to the assumed potential distribution function (Eq. 24) or the approximation l = V/E0. Curve (2), which is calculated without thermally activated relaxation, lies only a factor of two higher than the measured one (Fig. 4.16) and is in qualitative agreement with the acoustical data where no influence from thermal activation were taken into account. For PS, according to the specific heat and acoustical data (at T < 3 K) and due to a larger density of states of tunneling systems, we expected a larger heat release than for PMMA. Surprisingly, the opposite is observed. After cooling from 80 K with the same cooling rate, the heat release for PS measured at the same temperature as before was at least of a factor of ten smaller than for PMMA (Figure 4.16), and it does not follow the t –1-dependence. In order to understand the observed deviations we have measured the heat release of PS at lower charging temperatures. Figure 4.17 shows the heat release as a function of time at charging temperatures 0.5 K ^ T1 ^ 1 K. The heat release follows very well the t –1-dependence as well as an increase with T21 (Eq. 12, Fig. 4.18). It is interesting to note that a fair quantitative agreement with the prediction of the tunneling model is obtained. If we calculate the density of states of tunneling systems P from these data with Eq. 12 (straight line in Fig. 4.18) we obtain P = 9.0610–38 J–1 g –1 which is similar to the one obtained from the specific heat assuming umin = 6610–9.
Figure 4.17: Heat release as a function of time (in hours) for Polystyrene at a measuring temperature T0 = 0.200 K cooling from different T1. The straight lines have a t –1-dependence.
At higher temperatures T1 we observe the expected saturation of the heat release (Fig. 4.18), however, no deviation from the t –1-dependence has been measured within experimental error. Deviations are observed if we charge the sample at temperatures T1 > 15 K (Fig. 4.16). In this case we have cooled the sample from 80 K and measured the heat release at 0.090 K (sample PS1) and 0.300 K (sample PS2). According to theoretical estimates the observed difference in the heat release between the two PS samples is not attributed to the difference in measuring temperatures. The rather abrupt saturation of the heat release at T1 6 7 K is attributed to the depopulation of the excited states through thermally activated relaxation in the cooling process (Figs. 4.5 and 4.18), as it will become clear below. 63
4
Low-Temperature Heat Release, Sound Velocity and Attenuation, …
Figure 4.18: Heat release multiplied by the time t as a function of the difference between the squares of charging temperature T1 and measuring temperature T0 for PS. The solid line follows the theoretical prediction with a density of states of tunneling systems 1.45 larger than the one obtained from the specific heat results.
The following experiment provides further verification that the non-simple time dependence of the heat release for PS is due to energy states excited above 15 K and depopulation during the cooling process, very likely through thermally activated relaxation. We have charged the sample at 80 K for several hours and cooled it then to 1 K. We have left the sample for 22 h at 1 K and later continued to 0.3 K, the temperature at which the heat release was measured (Curve (1) in Fig. 4.19). We observe now a clear deviation from the t –1 law. That means that even after 22 h at 1 K the energy states excited above 15 K have not relaxed completely. After 60 h at 0.3 K we warmed the sample to 1 K for 17 h, cooled it to 0.3 K and measured the heat release (Curve (2) in Fig. 4.19). The heat release follows the theoretical t –1-dependence very well. We have calculated the heat release taking one-phonon,thermally activated processes, and the experimental cooling process into account. We used the same parameters obtained from the fit to the acoustical data (Figs. 4.13 and 4.15). With only one free parameter, umin,
Figure 4.19: Heat release as a function of time for PS. (1): cooled from 80 K to 1 K and leaving the sample 17 h at this temperature, T0 = 0.300 K. (2): the sample was warmed to a charging temperature T1 = 1 K for 22 h and cooled again to T0 = 0.300 K. The solid line in (1) is only a guide and the solid line (2) has a t –1-dependence.
64
4.5 Conclusions we can reproduce the measured heat release of PS reasonably well (Fig. 4.16). In this figure we present the two curves (3) and (4) using the parameter umin = 10–10 and 6610 –9. This fit indicates that we can observe the influence of a finite umin through the steeper decrease of the heat release at long times. Curve (1) in Fig. 4.16 was calculated with parameters for PS but without thermally activated relaxation.
4.5
Conclusions
We conclude that thermal conductivity, specific heat and acoustical properties at low temperatures (T < 1 K) can be quantitatively interpreted within the tunneling model for PMMA and PS. At higher temperatures we have measured very different temperature dependence of the internal friction for the two polymers. At the used frequencies and at 0.1 K ^ T ^ 120 K PMMA shows no thermally activated dissipation peak. PMMA shows a minimum in the internal friction at T ~ 30 K where PS shows a maximum. Nevertheless, within the tunneling model and introducing a thermally activated relaxation rate as well as a Gaussian distributon density of states of tunneling systems, it is possible to understand the acoustical properties. From the fits we may conclude that the density of states of tunneling systems for PMMA is restricted to smaller energies, i. e. a smaller width s0, than for PS. Although the thermal conductivity, specific heat, and sound velocity temperature dependence for both samples are similar, the heat release shows different time dependences as well as different values in apparent contradiction with the other measured low-temperature properties. Taking into account the internal friction results, it is tempting to correlate the differences in the heat release between both samples to the difference in the contribution of thermally activated processes in the relaxation of the tunneling states during the cooling process. We have demonstrated that the deviations of the time dependence of the heat release from the t –1-dependence predicted by the tunneling model is due to the relaxation of energy states excited above 15 K. The smaller values of the measured heat release of PS in comparison with the standard tunneling model can be explained by the depopulation of energy states relaxing mainly by thermally activated processes during the cooling process. Without detailed knowledge of the short relaxation times at T1 > 1 K, a quantitative fit or even a qualitative understanding of the heat release Q_ (T1,T0, t) data for high charging temperatures seems to be impossible. Therefore, the heat release with high charging temperatures can be hardly used to obtain information on the density of states of tunneling systems. Heat release experiments at high measuring temperatures T0 > 1 K are necessary to obtain information on the dynamics of the tunneling systems with a relaxation rate not given by the tunneling one-phonon process. These experiments may clarify the influence of the two different relaxation mechanisms discussed nowadays in the literature to interpret the low-temperature properties of amorphous solids above 1 K, namely incoherent tunneling 65
4
Low-Temperature Heat Release, Sound Velocity and Attenuation, …
[31] and thermal activation within the soft potential model [8]. Both approaches have been used with impressive success in the last years. It is, however, not yet clear if incoherent tunneling, a mixture of the two processes (incoherent and thermal activation), or only thermal activation is the main relaxation mechanism of the TS above 1 K.
Acknowledgements
We wish to acknowledge W. Joy for the sample preparation, and S. Hunklinger, D. Haarer and J. Friedrich for valuable discussions. This work was supported by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 213. A. Nittke was supported by the “Graduiertenkolleg Po 88/13” of the Deutsche Forschungsgemeinschaft.
References
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
11. 12. 13. 14. 15. 16. 17. 18.
66
J. Zimmerman, G. Weber: Phys. Rev. Lett., 46, 661 (1981) M. Schwark, F. Pobell, M. Kubota, R.M. Mueller: J. Low Temp. Phys., 58, 171 (1985) S. Sahling, A. Sahling, M. Kolac: Solid State Commun., 65, 1031 (1988) P.W. Anderson, B.I. Halperin, C.M. Varma: Phil. Mag., 25, 1 (1972); W.A. Phillips, J. Low Temp. Physics 7, 351 (1972) M. Deye, P. Esquinazi: Z. Phys. B – Condensed Matter 76, 283 (1989) M. Deye, P. Esquinazi: in: S. Hunklinger, W. Ludwig, G. Weiss (eds.): Phonon 89, World Scientific, Singapore, p. 468, (1990) J. Jäckle: Z. Phys., 257, 212 (1972) D. Parshin, S. Sahling: Phys. Rev. B, 47, 5677 (1993) J.J. Brey, A. Prados: Phys. Rev. B, 43, 8350 (1991) P. Esquinazi, M. Scherl, Li Junyun, F. Pobell: in: M. Meissner, R. Pohl (eds): 4th International Conference on Phonon Scattering in Condensed Matter, Springer Series in Solid State Sciences, Vol. 112, p. 287 (1993) D. Tielbürger, R. Merz, R. Ehrenfels, S. Hunklinger: Phys. Rev. B, 45, 2750 (1992) P. Doussineau, C. Frenois, R. G. Leisure, A. Levelut, J. Y. Prieur: J. Phys. (Paris), 41, 1193 (1980) S. Hunklinger, W. Arnold: in: W. P. Mason, R. N. Thurston (eds.): Physical Acoustics, Vol XII, Academic Press, New York, 1976 W.A. Phillips: in: S. Hunklinger, W. Ludwig, G. Weiss (eds.): Phonon 89, World Scientific, Singapore, p. 367 (1990) K.-P. Müller, D. Haarer: Phys. Rev. Lett., 66, 2344 (1991) A.K. Raychaudhuri, S. Hunklinger: Z. Phys. B – Condensed Matter, 57, 113 (1984) R.B. Stephens: Phys. Rev. B, 8, 2896 (1973) R.B. Stephens, G.S. Cieloszyk, G.L. Salinger: Physics Letters, 28A, 215 (1972)
References 19. 20. 21. 22. 23. 24.
25. 26. 27. 28. 29. 30. 31.
D. Cahill, R. Pohel: Phys. Rev. B, 35, 4067 (1987) C. Choy, G. Salinger, Y. Chiang: J. of Applied Physics, 41, 597 (1970) J. Freeman, A. C. Anderson: Phys. Rev. B, 34, 5684 (1986) J. Mack, J. Freeman, A.C. Anderson: J. of Non Crystalline Solids, 91, 391 (1987) G. Hartwig: Progr. Colloid and Polymer Sci., 64, 56 (1978) G. Federle, S. Hunklinger: J. de Physique, C9(12), Tome 43, p. C9–505 (1982); G. Federle: PhD Thesis, M. Planck Stuttgart, 1983, unpublished; A.K. Raychaudhuri: in: T.V. Ramakrishnan, M. Raj Lakshmi (eds.): Non-Debye Relaxation in Condensed Matter, World Scientific, p. 193 (1987) J. Crissman, J. Sauer, E. Woodward: J. of Polymer Science, A2, 5075 (1964) G. Bellesa: Phys. Rev. Lett., 40, 1456 (1978) J.-Y. Duquesne, G. Bellesa: J. de Physique Lettres, 40, L-193 (1979) R. Nava, R. Oentrich: J. of Alloys and Compounds, in press; R. Nava, Phys. Rev. B, 49, 4295 (1994) E. Gagemidze, P. Esquinazi, R. König: Europhys. Letters, 31, 13 (1995) W. Köhler, J. Zollfrank, J. Friedrich: Phys. Rev. B, 39, 5414 (1989) A. Würger: in: From coherent tunneling to relaxation, Springer Tracts in Modern Physics, Vol. 135, (1997)
67
5
Spectral Diffusion due to Tunneling Processes at very low Temperatures Hans Maier, Karl-Peter Müller, Siegbert Jahn, and Dietrich Haarer
5.1
Introduction
Pioneered by the work of Zeller and Pohl [1] it was discovered in the early 1970s that amorphous solids show low-temperature thermal and acoustic properties which are very different from those observed in crystals. For reviews see for example Refs. [2, 3]. The most wellknown of these anomalous properties is the specific heat, which is in general considerably larger than would be expected from the Debye model and varies linear with temperature in contrast to the Debye T3-dependence. Other anomalies are the temperature dependence of the thermal conductivity, the properties of phonon echoes, and ultrasonic absorption. These features seem to be quite universal for all kinds of amorphous solids, irrespective of their chemical composition and structure, i. e. inorganic as well as organic or polymeric. Very soon after the first experimental evidence for these anomalous low-temperature excitations Anderson et al. [4] and independently Phillips [5] developed a theoretical description, called the tunneling model [4, 5]. It is based on the assumption that the low energy degrees of freedom in amorphous solids arise from tunneling motions of atoms or groups of atoms between local energetic minima which should exist in any disordered solid. If only the lowest energy level of each minimum is considered, this leads to two eigenstates. For this reason, these degrees of freedom are often referred to as two-level system (TLS). An important consequence of this model is the formal analogy to a system of particles with spin 1/2, which for example immediately explains the existence of phonon echoes. Another very striking feature of the tunneling model was the prediction of a time-dependent specific heat [4]. This was confirmed experimentally several years later [6]. Furthermore, the model can explain the heat release [7] of amorphous solids, which is a time-dependent non-equilibrium phenomenon. From these kinds of experiments, it could be concluded that TLS dynamics occur on time scales extending over many orders of magnitude. The sensitivity of optical experiments on amorphous solids was hindered, in many instances, by the large inhomogeneous broadening which arises from the distribution of local environments of the involved optical transitions. Therefore a very important step was the discovery of persistent spectral hole burning in 1974 [8, 9]. This method of high-resolution laser spectroscopy eliminates the inhomogeneous effects induced by the static disorder in 68
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties Deutsche Forschungsgemeinschaft (DFG) Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
5.2
The optical cryostat
amorphous hosts. For this purpose dye molecules are embedded in the solid which change their absorption spectra when irradiated with resonant narrow band laser light. This produces a dip in the inhomogeneous absorption band, called a spectral hole. A narrow spectral hole can be regarded as a highly sensitive spectral probe for any kind of distortion of the matrix, which induces small spectral changes [10]. In this way variations of strain fields caused by pressure changes of almost some 10 hPa can be detected [11]. Spectral holes are also sensitive to small changes of other external parameters including electric fields [12, 13]. After observing quite a few anomalous properties of optical transitions in glasses and attributing them to the dynamics of TLS [14], the tunneling model was adopted by Reinecke [15] to explain the low-temperature line widths of optical transitions in amorphous solids using the concept of spectral diffusion. This concept had originally been developed for the description of spin resonance experiments [16] and had already been applied to the theoretical treatment of the above mentioned ultrasonic properties of glasses [17]. Soon after this step, the possibility of a connection between thermal and optical properties of amorphous solids was supported by the observation of time dependence of spectral hole widths [18]. The application of the tunneling model to the description of spectroscopic properties proposed an important link suggesting a correlation between the specific heat and the optical line width. This implies that, besides the mentioned calorimetric and acoustic methods, optical spectroscopy yields information about the dynamics of the same low energy excitations of amorphous solids, which dominates the calorimetric experiments. A crucial point in establishing this connection is the temperature dependence of these physical solid state parameters. The tunneling model in its original form predicts linear temperature dependence for the specific heat as well as for the optical line width. This prediction, however, is only valid at temperatures where the contribution of phonons is of minor importance. Earlier hole burning measurements on the temperature dependence of optical line widths [19] indicated the necessity of expanding the temperature range accessible to optical absorption spectroscopy to values far below 1 Kelvin in order to exclude any other mechanisms except TLS dynamics. For this reason we have constructed a 3He/ 4He dilution refrigerator with optical windows allowing transmission spectroscopy at temperatures down to 0.025 K, a temperature regime in which only very few data from optical spectroscopy existed before [20, 21]. In addition, cooling with a 3He/ 4He mixture can be performed continuously for very long times (in principle unlimited) if the necessary care in cryostat design and operation is taken. In our experiments we have reached operation times up to three months.
5.2
The optical cryostat
In the design of a cryostat for optical spectroscopy two main problems have to be overcome. In an absorption experiment the sample has to dissipate the absorbed energy in order to avoid overheat during the measurement. Therefore the best possible thermal contact to the cold stage is of extreme importance. The second problem is how to guide the laser light to the sample without creating an intolerably large additional heat leak, for example by infrared 69
5
Spectral Diffusion due to Tunneling Processes at very low Temperatures
irradiation from the windows. Since various construction methods for optical cryostats in the temperature range below 0.500 K did not yield the desired results, special care had to be taken in the design of our dilution refrigerator. For example it had been reported [22] that for samples located outside the mixing chamber temperatures below 0.300 K could not be reached, even if the best thermal contact is realized by pressing the samples to the wall of the mixing chamber with indium metal as contact material. In another kind of experiment a glass fibre winding around the cold finger of a dilution refrigerator shows a significant temperature gradient with respect to the mixing chamber below 0.100 K [20]. For these reasons, we decided to place the sample directly into the dilute phase of the mixing chamber. This way optimal thermal contact and a constant temperature of the sample was achieved via the superfluid 4He. For performing optical experiments the mixing chamber was designed as a glass cylinder. The optical path, used in our cryostat, is shown in Figure 5.1 [23]. Three sets of windows are mounted on cold copper shields to eliminate most of the room temperature radiation in three steps: at liquid nitrogen temperature (77 K), liquid helium temperature (4.2 K), and at approximately 1 K, the latter being achieved by cooling with the pumped 3He of the distillation chamber. In contrast to other constructions, the vessel containing the dilution refrigerator part of the cryostat is in our design not surrounded by the helium tank which has the advantage that the laser beam is not scattered by boiling liquid helium. We estimate that the total heat leak caused by room temperature radiation is below 50 nW. The minimum temperature reached is 0.024 K. The cooling power is about 6 mW at 0.100 K. Although this is a very low value, it is sufficient since the power of the absorbed laser light was always far below 1 mW. The light powers used for hole burning in our experiments were on the order of several nanowatts to several tens of nanowatts depending on the temperature. For hole detection the power is reduced to the picowatt range. The cooling power of our cryostat and the thermal conductivity of the involved polymers are high enough to guarantee a uniform temperature distribution with no significant sample heating during the hole burning process. mixing - chamber He - shield 1-K - shield LN2 - shield sample 4
Figure 5.1: Alignment of optical windows and sample in the cryostat. The sample is placed inside the mixing chamber of the dilution refrigerator. The shields are connected to the distillation chamber, the helium tank, and the nitrogen tank, respectively. The liquid N2 windows possess an infrared reflective coating.
70
5.3
Theoretical considerations
The temperature is measured with a RuO2 thick film resistor (TFR) supplied by Phillips (type RC-01). The heat capacity and thermal relaxation time of the TFR are equivalent to those of a carbon resistor but its thermal reproducibility during temperature cycles is better [24]. The resistor is calibrated against several primary and secondary thermometers including a NBS fixed point device, CMN and Ge, thermometers. Its accuracy is better than 2 % in the temperature range between 4.2 K and 0.025 K. The thermometer is placed into the dilute phase of the mixing chamber in direct thermal contact to the sample. The resistance is measured using a four wire ac resistance bridge (AVS-46 RV Elektroniikka, Finland). At the minimum temperature, the energy dissipated in the resistor is less than 0.5 pW.
5.3
Theoretical considerations
In the tunneling model [4] the TLS was described by two parameters (Fig. 5.2), the asymmetry parameter D and the overlap parameter l, which contains the barrier height, the distance of the two minima, and the mass of the tunneling particle. They are related with the total energy splitting E of the two levels and the tunneling matrix element D0 by the relations E
q 2 20
and
0 k exp
;
1
where kO is a typical zero point energy. One of the most important ingredients of the TLS model is the distribution function characterizing the two parameters. The originally used assumption for this function, which has also been applied to explain spectral diffusion [15, 17], is a flat distribution in both parameters, namely P
; dd Pdd :
2
Figure 5.2: Double minimum potential as a model for a TLS. All relevant parameters are shown in the figure.
71
5
Spectral Diffusion due to Tunneling Processes at very low Temperatures
It is convenient, however, to use experimentally accessible quantities as variables, i. e. the energy E and a dimensionless relaxation rate R, which can be defined as: R = r/rmax = D20 / E2. The transformation to the new variables yields for the distribution function [25]: P
E; R P
1 p : 2R 1 R
3
A very important feature of this distribution is the fact that P (E, R) is independent of E. In order to keep the total number of TLS finite, some cut-off value for R ? 0 has to be introduced [3], which is usually denoted by Rmin . It was shown in Ref. [15] that for optical transitions in glasses the TLS dynamics results in spectral diffusion, which shows up in the experiment as a time and temperature-dependent Lorentzian line broadening. The width of this Lorentzian line must be calculated by averaging over the distribution of energies and relaxation rates P(E, R). It can be written as: 2p2
t; T Cij 3k
E Zmax
Emin
Z1 dEn
T
dR
Rmin
P
E; R
1 E
exp trmax R :
4
Here ACijS represents an average coupling constant between the TLS and the optical transition, n (T) is a thermal occupation factor of the TLS states. For the system which is investigated in this work, the parameter rmax can be estimated from experimental data on ultrasonic attenuation [26] to be on the order of 107 s –1 at a temperature of 0.100 K and even larger at higher temperatures. Therefore, the condition t 7 rmax p 1 is well fulfilled on all time scales exceeding several microseconds. In this limit the rate integration in Eq. 4 can be performed analytically [27], which leads to the well-known logarithmic time dependence of spectral diffusion: !
t; T
p2 Cij P kB T ln
rmax t : 3k
5
Equation 5 represents the theoretical prediction of the tunneling model for the time and temperature-dependent broadening of spectral holes. This result, however, is the result of a particular form of the density of tunneling states P (E, R), which is based on the a priori assumption of Eq. 2. A uniform density of states in D is a physically reasonable choice; the independence of l, however, is difficult to justify, since l consists of several parameters [28]. The temperature dependence of spectral diffusion is dominated by D the time evolution stems mainly from l. Therefore, these two predictions from Eq. 5 do not have the same validity. In our experiments we have investigated time and temperature dependence separately.
72
5.4
5.4
Temperature dependence
Temperature dependence
In these experiments, we investigated polystyrene (PS) doped with phthalocyanine and polymethylmethacrylate (PMMA) doped with tetra-4-tert-butyl phthalocyanine. Both sample materials have also been investigated by heat release and specific heat measurements [29]. The samples had optical densities of about 0.4 at a typical thickness of 3 mm. The samples were prepared by bulk polymerisation of the solution of the dye in the monomer. The observed hole shapes are Lorentzian with no detectable deviations in all cases. To eliminate saturation effects at least 10 holes with different energies were burned at each temperature and the homogeneous line width was determined by extrapolation to zero burning fluence [30]. Due to their low relative depths the extrapolation was done assuming a linear dependence of the hole area on energy [10]. A weak variation of the line width with light power was observed at our minimum temperature of 0.025 K for PMMA. We attribute this variation to sample heating. The temperature dependence of the hole width is shown in Fig. 5.3 [23]. The results are: a) both systems display a linear temperature dependence over the whole temperature range from 0.500 K down to 0.025 K and no crossover to a constant line width is seen. The linearity of the data plot shows that the density of states P (E, R) is indeed independent of the TLS energy; b) the relative magnitude of the line width in the two systems correlates with the respective specific heat [31]; c) the extrapolated line width for T ? 0 is nearly the same for both systems, confirming that in an amorphous system the line width reaches the lifetime limited value Gh = 1/2 pT1 (Heisenberg limit) of the electronic transition only when T ? 0. This limit is reached in our experiment within 10 %!
Figure 5.3: Temperature dependence of the hole burning line width for PS and PMMA between 0.025 K and 0.500 K.
73
5
Spectral Diffusion due to Tunneling Processes at very low Temperatures
In the case of PS, where unsubstituted phthalocyanine was used as dye, the result for the extrapolated line width G (T ? 0) is in very good agreement with the fluorescence lifetime measurements of this molecule [32]. We are not aware of measurements of the fluorescence lifetime of substituted phthalocyanine, but our results show that there is no major difference between both dyes as far as their excited state lifetimes are concerned. These results imply that in amorphous solids there are dynamical processes of twolevel systems with an energy spectrum extending down to values as low as kB 7 0.025 K. These low-energy excitations dominate the line widths of optical transitions even at low temperatures and the lifetime limited value can only be reached by extrapolating to zero temperature.
5.5
Time dependence
After the system has been cooled down spectral holes are burned immediately and after some delay times, while the sample was at constant temperature, we observe a hole broadening which depends on this delay time. This behaviour is demonstrated in Fig. 5.4. Dataset A is the broadening of a hole which is burned immediately after the sample (phthalocyanine in PMMA) has been cooled from liquid nitrogen temperature to 0.300 K. The curves show the evolution of holes burned about one day, three days, and one week after cooling down. The amount of spectral diffusion in a given time interval decreases continuously until a small residual effect becomes observable, which is independent of the time delay after reaching the final temperature. This is represented by dataset B in Fig. 5.4.
A
B
Figure 5.4: Time evolution of spectral holes burned at different times after sample cooling. Datasets corresponding to A and B are investigated in the following.
74
5.5
Time dependence
Investigating the temperature dependence of the time evolution of the hole width [33], we performed experiments on a PMMA sample doped with phthalocyanine at different temperatures T, varying from 0.100 K to 1 K. For each of the temperatures T = 0.100, 0.300, 0.500, and 0.700 K, the respective experiments were carried out in a separate run. It took about one hour to cool the system from 77 K to 4 K. The time for cooling down from 4 K to the final temperature T depends slightly on T, but is in the range of about (6 + 1) hours. Immediately after reaching T, holes are burned and their subsequent broadening is observed for about one week. These experiments yield data corresponding to set A in Fig. 5.4. The results of these measurements are shown in Fig. 5.5 [33]. In this plot the time origin for each run is identical, namely the time at which the cooling down procedure from 77 K was started. For better visibility, however, the data corresponding to T = 0.500, 0.300, and 0.100 K are shifted by 0.5, 1.0, and 1.5 orders of magnitude along the logarithmic time axis, respectively. Except for the data taken at 0.500 K the total observation time was 6–10 days.
Figure 5.5: Broadening of holes burned immediately after reaching T. Data shifted for better visibility (see text).
Keeping the samples at constant temperature new holes are burned after 1–2 weeks and their broadening is observed for several hours, corresponding to set B in Fig. 5.4. These data are shown in Fig. 5.6 [33]. Here the observation time is 2–16 hours. Note that the range of the DG-axis is about 10 times less as compared with Fig. 5.5. Comparing our data of Figs. 5.5 and 5.6, it is obvious that the measurements which are started immediately after cooling down show no variation of the observed broadening with the phonon bath temperature, while the residual behaviour observed many days later exhibits a rather pronounced dependence on the experimental temperature. It has to be emphasized that the data shown in Fig. 5.5 and the corresponding data in Fig. 5.6 for the same values of T were measured without any change of temperature in between the two subsequent experiments. In spite of the variation of the final temperature over almost one order of magnitude, all four datasets in Fig. 5.5 show the same hole broadening behaviour. We attribute this to the existence of a non-equilibrium state of the TLS ensemble due to the fast cooling procedure: The large amount of spectral diffusion observed after fast 75
5
Spectral Diffusion due to Tunneling Processes at very low Temperatures
Figure 5.6: Broadening of holes burnt after 1–2 weeks at low temperature.
cooling is due to the relaxation of TLS to their thermal equilibrium [34]. This relaxation process is known to produce the thermal phenomenon of heat release [7]. Thus, it can be concluded that we have established an important connection between optical and calorimetric phenomena. The data of Fig. 5.6 exhibit a pronounced dependence on temperature, as can be expected from theoretical considerations outlined in Section 5.3 and from the non-time resolved experiments of Section 5.4. The time evolution, however, shows a distinct non-logarithmic behaviour in contradiction to the prediction of the tunneling model. For short times the Fig. 5.6 shows clearly that the hole broadening starts fairly logarithmic. We have performed transient hole burning experiments on the millisecond time scale, which also yielded a logarithmic behaviour [35]. At longer times, however, the diffusional broadening increases faster than logarithmic. This increase occurs later at lower temperatures, which is to be expected from the temperature dependence of the relaxation rates. These new features are in qualitative agreement with theoretical work that invokes strong coupling of TLS with phonons [28, 36]. We believe that our experimental work shows that the method of spectral hole burning spectroscopy is suitable for studying the low-temperature properties of amorphous solids. Especially for the investigation of the long-time behaviour of TLS equilibrium dynamics, where other methods cease to function, it is a very powerful experimental technique.
References
1. R.C. Zeller, R.O. Pohl: Phys. Rev. B, 4, 2029 (1971) 2. Amorphous Solids. Low Temperature Properties, in:W.A. Phillips (ed.): Topics in current Physics, Vol. 24, Springer, Berlin, (1981)
76
References 3. S. Hunklinger, A. K. Raychaudhuri: in: D. F. Brewer (ed.), Progress in Low Temp. Physics, Vol. IX, Elsevier Science, Amsterdam, p. 265 (1986) 4. P.W. Anderson, B.I. Halperin, C.M. Varma: Philos. Mag., 25, 1 (1972) 5. W.A. Phillips: J. Low Temp. Phys., 7, 351 (1972) 6. M.T. Loponen, R.C. Dynes, V. Narayanamurti, J.P. Garno: Phys. Rev. Lett., 45, 457 (1980) 7. J. Zimmermann, G. Weber: Phys. Rev. Lett., 46, 661 (1981) 8. B.M. Kharlamov, R.I. Personov, L.A. Bykovskaya: Opt. Commun., 12, 191 (1974) 9. A.A. Gorokhovskii, R.K. Kaarli, L.A. Rebane: JETP Lett., 20, 216 (1974) 10. J. Friedrich, D. Haarer: Angew. Chem., 96, 96 (1984); Angew. Chem. Int. Ed. Engl., 23, 113 (1984) 11. Th. Sesselmann, W. Richter, D. Haarer, H. Morawitz: Phys. Rev. B, 36(14), 7601 (1987) 12. A.P. Marchetti, M. Scozzafara, R.H. Young: Chem. Phys. Lett., 51(3), 424 (1977) 13. V.D. Samoilenko, N.V. Rasumova, R.I. Personov: Opt. Spectr., 52(4), 580 (in Russian) (1982) 14. P.M. Selzer, D.L. Huber, D.S. Hamilton, W.M. Yen, M.J. Weber: Phys. Rev. Lett., 36, 813 (1976) 15. T.L. Reinecke: Solid State Commun., 32, 1103 (1979) 16. J.R. Klauder, P.W. Anderson: Phys. Rev., 125(3), 912 (1962) 17. J.L. Black, B. I. Halperin: Phys. Rev. B, 16, 2879 (1977) 18. W. Breinl, J. Friedrich, D. Haarer: J. Chem. Phys., 81(9), 3915 (1984) 19. G. Schulte, W. Grond, D. Haarer, R. Silbey: J. Chem. Phys., 88(1), 679 (1988) 20. M.M. Broer, B. Golding, W.H. Haemmerle, J.R. Simpson, D.L. Huber: Phys. Rev. B, 33, 4160 (1986) 21. A. Gorokhovskii, V. Korrovits, V. Palm, M. Trummal: Chem. Phys. Lett., 125, 355 (1986) 22. Korrovits, M. Trummal: Proceedings of the Academy of Sciences of the Estonian SSR, 35(2), 198 (1986) 23. K.-P. Müller, D. Haarer: Phys. Rev. Lett., 66, 2344 (1991) 24. W.A. Bosch, F. Mathu, H.C. Meijer, R.W. Willekers: Cryogenics, 26, 3 (1985) 25. J. Jäckle: Z. Phys., 257, 212 (1972) 26. G. Federle: PhD thesis, Max-Planck-Institut für Festkörperforschung Stuttgart (1983) 27. S. Hunklinger, M. Schmidt: Z. Phys. B, 54, 93 (1984) 28. K. Kassner: Z. Phys. B, 81, 245 (1990) 29. A. Nittke, M. Scherl, P. Esquinazi, W. Lorenz, Junyun Li, F. Pobell: J. Low Temp. Phys., 98(5/6), 517 (1995) 30. L. Kador, G. Schulte, D. Haarer: J. Phys. Chem., 90, 1264 (1986) 31. K.P. Müller: PhD thesis, Universität Bayreuth (1991) 32. W.H. Chen, K.E. Rieckhoff, E.M. Voigt, L.W. Thewalt: Mol. Phys., 67(6), 1439 (1989) 33. H. Maier, D. Haarer: J. Lum., 64, 87 (1995) 34. S. Jahn, K.-P. Müller, D. Haarer: J. Opt. Soc. Am. B, 9, 925 (1992) 35. S. Jahn, D. Haarer, B.M. Kharlamov: Chem. Phys. Lett., 181, 31 (1991) 36. K. Kassner, R. Silbey: J. Phys. Condens. Matter, 1, 4599 (1989)
77
6
Optically Induced Spectral Diffusion in Polymers Containing Water Molecules: A TLS Model System Klaus Barth, Dietrich Haarer, and Wolfgang Richter
6.1
Introduction
In the past decade much experimental and theoretical work was done on spectral diffusion in amorphous solids. It turned out that at low temperatures, optical dephasing phenomena [1] as well as the numerous caloric data [2] can be well described by assuming the presence of low-energy excitations, the so-called two-level systems (TLS). Photochemical hole burning [3] and optical echo experiments [4] have provided experimental evidence for different dynamics of optical transitions in glassy systems as compared to crystalline matrices. For glassy organic solids the TLS concept was first proposed by Small and co-workers [5, 6] and was later used by Reinecke [7] to explain low-temperature optical line widths. Especially the photochemical hole burning data over long observation periods (observation times longer than three months) at temperatures down to 0.050 K allowed a critical test of the theoretical approach within the tunneling model [8]. Theoretical and experimental results to this topic are also given in this book by H. Maier et al. At temperatures above 1 K tunneling transitions and localized vibrations influence the physical properties [9]. Both types of mechanisms have recently been incorporated in the soft potential model [10] which contains the well-known tunneling model as a special case. A microscopic interpretation of optical line broadening phenomena in terms of e. g. local excitations of the matrix or switching of optically addressed chemical groups suffered from the lack of experimental data. In the past, several experimental techniques have been developed to shed light on the microscopic mechanisms of the phenomena. These techniques are all based on the generation of phonons in the matrix material, especially in the neighbourhood of the spectral probe. Heat pulses from external heaters [11] produce a broad distribution of phonon frequencies inside the chromophore host system and permit the investigation of the dynamics of barrier crossings and the coupling between the TLS and the dye molecules [12]. Light-induced phonon generation in hole burning systems can in principle be achieved either by exciting non-radiative decay processes in appropriate chromophores [13] or by direct absorption of IR light in the matrix material [13, 14]. The non-radiative decay method, which has been used to study transient and irreversible spectral diffusion processes, yields also a broad distribution of phonon frequencies and gives rise to similar experimental processes as e. g. the heat pulse technique. On the other hand, the IR absorption 78
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties Deutsche Forschungsgemeinschaft (DFG) Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
6.2
Experimental setup for burning and detecting spectral holes
method where the sample is illuminated with narrow band IR light allows us to generate phonons of high density and of a comparatively small energy distribution in the direct neighbourhood of the optical probe. The contribution of the broad band black-body background radiation can be minimized by using an appropriate intensity of a narrow band IR source. For the optical investigation of amorphous systems, photochemical hole burning (PHB) is a high-resolution technique and a powerful tool to examine local sites in polymers at low temperatures. Dye molecules embedded in polymers at low concentration exhibit strongly inhomogeneously broadened absorption bands in a conventional spectroscopic experiment. Selecting an ensemble of these dye molecules with narrow band laser light, which gives rise to a photoreaction, leads to a sharp spectral dip at the laser frequency, the so-called spectral hole. Due to the high resolution, this method is appropriate for detecting small perturbations in the matrix such as a transient change of the phonon distribution or permanent matrix rearrangements in the vicinity of the dye molecules. Such rearrangements can be due to a configurational change of parts of the polymer main chain or due to a reorientation of small molecules (water), which may be embedded in the matrix in addition to the dye molecules. In the following, two different experimental conditions are discussed which are based on the absorption of IR light. Both experimental situations lead to a change of the local interaction of the optical probe with its local matrix environment. In the first part the phonons are treated as a time-dependent temperature bath. In the second part experiments are discussed where local groups of the matrix are selectively addressed by IR radiation of very low intensity. The observed enhanced spectral diffusion shows a characteristic dependence on the IR frequency and coincides with only a few of the numerous IR absorption bands. A quantitative description of this new process together with the identification of the resonant vibrations is given within a simple kinetic model.
6.2
Experimental setup for burning and detecting spectral holes
The experimental setup for burning and detecting spectral holes with a narrow band laser is described elsewhere [16]. The experiments were performed at 1.8 K in a bath cryostat in which the sample was immersed in superfluid helium. As samples we used different polymeric matrices such as polymethylmethacrylate (PMMA), polyamide (PA), polyethylene, polystyrene, etc. doped with free base phthalocyanine (H2Pc) at low concentration (10 –2 mol%). Their preparation is described in Ref. [16]. The water content of the samples has been controlled by heating them under vacuum conditions. Because of its strong hydrophilic nature PMMA has a high capacity of water absorption at atmospheric conditions. In order to investigate the influence of the water molecules the experiments, described in Section 6.4, were carried out either with a nearly water-free PMMA sample or with a PMMA sample containing water molecules under equilibrium conditions. The optical windows of the cryostat consisted of crystalline BaF2 which allowed the study of hole burning spectra in the lowest vibrational band of the electronic S0 ? S1 transition of H2Pc at about 0.69 mm as well as under IR illumination between 2 mm and 11 mm. In this setup, the broad band background radiation with an emission 79
6
Optically Induced Spectral Diffusion in Polymers Containing Water Molecules
maximum at 10 mm has an integrated intensity of about 300 mW/cm2 at the location of the sample. As IR radiation sources we used a Globar with a maximum emission intensity near 3 mm and different CO2 laser lines around 10 mm. An acousto-optic modulator was used for reducing the intensity of the CO2 laser down to 1 W/cm2 and for switching with times some microseconds. The radiation of the Globar was dispersed in a monochromator and with different dielectric filters to a bandwidth between 0.05 mm and 0.2 mm with typical intensities around 100 mW/cm2.
6.3
Reversible line broadening phenomena
In this Section the mechanisms of reversible line broadening phenomena under continuous or pulsed irradiation conditions are discussed. Phonons are generated by illuminating the polymeric sample with the light of a single CO2 laser line. The wavelength of the CO2 laser was selected to generate high frequency phonons only within a small penetration depth of less than 10 mm. The dye molecules, which are homogeneously distributed throughout the sample thickness of some 100 mm, are a local probe for the increase in the phonon density during the IR irradiation. Due to the thermalisation processes, phonons show a broad energy distribution. This change of the phonon distribution gives rise to an increased effective temperature under cw irradiation conditions. Because of the high cooling rate of the surrounding superfluid helium, typical irradiation intensities of about 1 W/cm2 were used to obtain a significant increase in the phonon density. The influence on the homogeneous line width of the dye molecules can be investigated in a zero burning fluence experiment, i. e. the limit of the hole width is given by extrapolating to zero burning fluence. The lower curve in Fig. 6.1, which was measured without any IR irradiation, yields an extrapolated value for the line
Figure 6.1: Zero burning fluence experiment under different phonon density conditions in polyamide doped with H2Pc (see text).
80
6.3 Reversible line broadening phenomena width of about 375 MHz. The two upper curves show increased hole widths due to an enhanced phonon generation either by IR irradiation during the burning process only(triangles) or during the detection process only(circles). The profile of a spectral hole can be written as Z1 A
!
!L /
nz
!0
!L z
!
!0 d!0 ;
1
1
where nz (o' – oL) is the site distribution function of the molecular line contributing to the hole spectrum and z (o – o') is the absorption profile of a single molecule. On the time scale of this experiment (up to some minutes) the spectral hole profiles are mainly reversible (see also Fig. 6.2 and Fig. 6.3). Therefore, the homogeneous line profile function z (o) is much more affected by the altered phonon density than the site distribution function nz (o). Thus, it is obvious that the two upper curves in Fig. 6.1 are similar and give for the hole width nearly the same zero burning fluence value of 440 MHz. Both curves are well separated from the lower curve which was obtained without any additional phonon generation. Applying an effective temperature model the difference between these two zero burning fluence values corresponds to a temperature change of about 0.3 K within the sample. Because of the strongly inhomogeneous experimental conditions during phonon generation and phonon diffusion through the sample, however, each different subensemble of dye molecules experiences a different distribution in phonon energies and, as a consequence, a bath description in terms of one single temperature is not appropriate. By using different op-
Figure 6.2: Lower part: Spectral profile of a photochemical hole with and without phonon generation during the detection process. Upper part: Directly measured difference between these two profiles as obtained by modulating the IR light at a frequency of 150 Hz.
81
6
Optically Induced Spectral Diffusion in Polymers Containing Water Molecules
Figure 6.3: Decay of the transmission maximum during a laser pulse.
tical energies for the phonon generation, it is in principle possible to study the contribution of different matrix vibrations on the dephasing of a single absorber function z (o). The time scale of the hole burning experiments discussed above is several minutes. It is about 3 orders of magnitude larger than the typical thermal relaxation time of the sample as given by
d 2 c ;
2
where d is the dimension (thickness) of the sample, c the specific heat, r the mass density, and k the thermal conductivity. For the temperature of our experiment (1.8 K) and a typical sample dimension of 1 mm the thermal relaxation time t is calculated to 10 ms. A consequence of this short relaxation time is shown in Fig. 6.2 and Fig. 6.3. The two spectral holes in the lower part of Fig. 6.2 were obtained with and without phonon generation during the hole detection process. A modulation of the IR light during the detection process results in a continuous switching between these two hole profiles. In the upper part of Fig. 6.2, a modulation frequency of 150 Hz was used during the scan of the optical frequency. The switching time between the two hole profiles is of the same order of magnitude as the thermal relaxation time. The continuous switching between the two hole profiles is detected with lock-in technique. The result shows that the phonon generation and relaxation process is sufficiently fast to follow the 150 Hz modulation of the IR light. A time-resolved experiment is shown in Fig. 6.3. The optical transmission at the peak of a hole spectrum was monitored while the sample was being illuminated with a single CO2 laser pulse of 600 ms duration. The decay of the transmission signal cannot be described with a single exponential function. It is well represented by the superposition of two exponentials with a fast contribution t1 & 10 ms and a slow contribution t2 & 150 ms. The fast contribution is close to the thermal relaxation time at the temperature of the helium bath. The slow contribution contains all the inhomogeneous experimental conditions as mentioned above, in particular a temperature gradient inside the sample that gives rise to a variation of the thermal relaxation time. Since the amplitude of the fast contribution is larger by a factor of 4, the switching time between the two curves in Fig. 6.2 is mainly determined by the fast contribution t1. 82
6.4
Induced spectral diffusion
The experimental results of this Section show that optically generated phonons can be used to study the transient broadening of the optical line shape of a single absorber. In principle, the dependence on the phonon frequency can be studied in such an experiment. In the same experiment the time-resolved dynamics of phonon diffusion and phonon decay in polymeric systems can be investigated via the dephasing mechanism of the optical probe.
6.4
Induced spectral diffusion
In this Section we will demonstrate that the resonant absorption of IR light in a polymeric system can also lead to irreversible line broadening phenomena. Even at a very low irradiation intensity (a typical value is 100 mW/cm2) the investigation of specific types of relaxation processes is possible. Such low irradiation intensities do not increase the temperature of the matrix by more than 0.01 K and hence yield no measurable dephasing effect via a change of the phonon distribution. Because of the irreversible nature of the optical line broadening, such phenomena are usually ascribed to spectral diffusion processes. The IR induced spectral diffusion is caused by addressing specific moieties within the matrix via the excitation of local vibrations. In the following weakly bound water molecules are studied. When a change in the spatial configuration takes place during the relaxation process of a water molecule the optical transition of a neighbouring dye molecule will be affected. Optical absorbers in glasses are therefore very sensitive probes for local rearrangements and are suited for a microscopic study of this topic. Due to the hydrophilic nature of the matrix the investigated guest host system, PMMA doped with H2Pc, contains a comparatively large number of water molecules (up to one volume percent). The experimental procedure is the following: A spectral hole is burned and its spectrum is repeatedly recorded while the sample is being exposed to IR radiation of a certain wavelength at a constant irradiation intensity. Figure 6.4 shows the typical increase of the hole
Figure 6.4: Time evolution of the hole broadening induced by different IR wavelengths. From top to bottom: 2.80 mm, 2.86 mm, 2.90 mm, 2.96 mm, and 3.04 mm.
83
6
Optically Induced Spectral Diffusion in Polymers Containing Water Molecules
Figure 6.5: Transmission spectra of PMMA without (curve A) and with (curve B) natural water content. The quotient C = B/A shows the H2O absorption lines. Dependence of the hole broadening on the IR wavelength after t0 = 35 min (curve D) due to induced spectral diffusion.
width and the strong dependence of this behaviour on the IR wavelength. To illustrate the pronounced spectral selectivity, one value of each hole broadening curve at t0 = 35 min is plotted in Fig. 6.5 (curve D) versus the corresponding IR wavelength. The data points do not simply reflect the absorption behaviour of the sample. Although the PMMA matrix has many absorption bands between 2 mm and 11 mm (curve B in Fig. 6.5), there are only two distinct wavelengths which give rise to induced spectral diffusion processes. The small concentration of water molecules in the matrix gives rise to non-saturated absorption bands near 2.8 mm and 6.1 mm. They correspond to the fundamental stretching and bending vibrations of H2O, respectively. The difference spectrum (curve C in Fig. 6.5) of water free and water saturated PMMA yields the exact position of the H2O absorption bands, which are in good agreement with the resonances found in our experiment. On an expanded scale in Fig. 6.5, even the asymmetric shape of the inhomogeneous H2O band is reflected by the experimental data. Replacing protonated with deuterated water yields a red shift of the IR resonances which also agrees very well with the corresponding absorption spectra of heavy water. In analogy to the tunneling model, which is based on the assumption of two potential minima, we assume two stable sites for each water molecule (Fig. 6.6). Since the experimental results indicate that a permanent change in the matrix occurs transitions between the two sites are only allowed via the first excited vibrational states. Using such a simple four-level model for the underlying kinetics, we are able to explain the temporal behaviour of the induced spectral diffusion in a quantitative fashion. 84
6.4
Induced spectral diffusion
Figure 6.6: Energy level scheme of a H2O molecule in a two-site model.
The two ground states are labelled 1 and 2, the first vibrationally excited states 3 and 4, respectively. The IR induced and the spontaneous transition rate for each site are denoted by the Einstein coefficients B and A, respectively. In our notation, B is proportional to the irradiated IR intensity. Transitions between the two sites are denoted by the conversion rate k. The transitions shown in Fig. 6.6 lead to a system of 4 coupled linear rate equations. Since we used in our experiments very low intensities of the IR light, we consider only the limit B P A where the population of the excited levels 3 and 4 is negligible at all times. The result is a time-dependent number of flips between the two ground states nf = n1?2 + n2?1 . Only those flips are taken into account which contribute to a change in the total configuration of all water molecules with respect to the time t = 0 of the hole burning process. According to Reinecke [7] the width of the spectral diffusion kernel Do is proportional to the number of flips nf . Though using the approximation that each flip yields the same contribution to Do, when taking into account the decrease of the IR intensity across the sample thickness as well as a random orientation of the H2O molecules, the result for Do can only be calculated numerically. Using a Taylor expansion, however, a very good analytical approximation of the width of the induced spectral diffusion is given by N0 !
t G 1 4pk
2Bpf t 1 b
b!
:
3
The difference between this formula and the numerical result is less than 5 %. pf is the flip probability of a single H2O molecule as defined by pf = k /(A + k), b is a dimensionless geometry parameter of about 2, N0 is the density of the H2O molecules and G is the coupling constant between dye and water molecules. Equation 3 describes very well the dependence of the broadening of a spectral hole on the number of absorbed IR photons (Fig. 6.4). It is possible to perform a test of the model. Equation 3 contains only two fitting parameters, the flip probability pf, and the coupling constant G. These two parameters can be obtained from a single experiment. An example is given in Fig. 6.7. One set of the experimental data (filled circles) is obtained at an IR intensity of 80 µW/cm2. The solid line is a fit according to Eq. 3 and yields a flip probability of 18 % and a coupling constant of G = 3.6 7 10–44 Jcm3. (Note: the slight difference with respect to the data given in reference [17] is due to the extension of the model by including the random orientation of the water molecules and the decrease of the IR intensity according to Beer’s law.) With these two values it is possible to calculate the 85
6
Optically Induced Spectral Diffusion in Polymers Containing Water Molecules
Time (s) Figure 6.7: Test of the model. Hole broadening versus IR irradiation time for three different IR intensities. The solid line is a fit, the dashed lines are calculated according to Equation 3.
time dependence of the induced spectral diffusion for any other irradiation intensity. The dotted lines in Fig. 6.7 are calculated with Eq. 3 for the obtained values and the intensities 230 µW/cm2 and 20 µW/cm2. The experimental results under these conditions, represented by the squares and triangles, are in very good agreement with the theoretical predictions. The high quantum yield of 18 % for the flip of a single water molecule becomes obvious by a comparison of the number of absorbed IR photons (typically 1014 s –1) with the total number of water molecules (about 1018) in the sample volume. Since a strong induced spectral diffusion is observed within several minutes, a local reorientation process with a high quantum yield must be involved. The coupling strength between water and dye molecules can be estimated by taking into account the static dipole moment of the H2O molecule. A reorientation of the H2O molecules affects the dye molecules by the concomitant change in the local electric field at the location of the chromophores. Assuming a mean distance of 15 Å and inserting data from Stark effect experiments [18], one obtains a value of approximately 2 7 10 –44 Jcm3 for the coupling constant which is very close to our fitted value. Together with the resonant phenomena discussed above, this yields a detailed microscopic understanding of this special kind of interaction. We can conclude that spectral diffusion induced by the absorption of IR photons in the frequency range 2–11 µm is a resonant process. Its microscopic origin is a spatial rearrangement of weakly bound water molecules after vibrational excitation. Assuming a configurational model with only two possible sites for each water molecule and applying the theoretical model of spectral diffusion by Reinecke [7], an analytical description of the resonant hole broadening behaviour can be obtained. A variation of the IR irradiation intensity shows that the model description yields excellent agreement with the experimental data.
86
References
References
1. 2. 3. 4. 5. 6.
7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
S. Jahn, K. P. Müller, D. Haarer: J. Opt. Soc. Am. B, 9, 925 (1992) M. Deye, P. Esquinazi: Zeitschr. für Phys. B, 39, 283 (1989) J. Friedrich, D. Haarer: Angew. Chem., Int. Ed. Engl,. 23, 113 (1984) C. A. Walsh, M. Berg, L. R. Narashiman, M. D. Fayer: Chem. Phys. Lett., 130, 6 (1986) J. M. Hayes G. J. Small: Chem. Phys., 27, 151 (1978) G. J. Small: Persistent nonphotochemical hole burning and the dephasing of impurity electronic transitions in organic glasses, in: V. M. Agranovich R. M. Hochstrasser (eds.): Spectroscopy and Excitation Dynamics of Condensed Molecular Systems, North-Holland, Amsterdam, p. 515 (1983) T. L. Reinecke: Sol. Stat. Com., 32, 1103 (1979) H. Maier, D. Haarer: J. Lumin., 64, 87 (1995) R. Greenfield, Y. S. Bai, M. D. Fayer: Chem. Phys. Lett., 170, 133 (1990) D. A. Parshin: Phys. Sol. State, 36, 991 (1994) U. Bogner: Phys. Rev. Lett., 37, 909 (1976) T. Attenberger, K. Beck, U. Bogner: in: S. Hunklinger, W. Ludwig, G. Weiss (eds.): Proc. of the 3rd Int. Conf. on Phonon Physics, p. 555 A. A. Gorokhovskii, G. S. Zavt, V. V. Palm: JETP Lett., 48, 369 (1988) W. Richter, Th. Sesselmann, D. Haarer: Chem. Phys. Lett., 159, 235 (1989) W. Richter, M. Lieberth, D. Haarer: JOSA B, 9, 715 (1992) G. Schulte, W. Grond, D. Haarer, R. Silbey: J. Chem. Phys., 88, 679 (1988) K. Barth W. Richter: J. of Lumin., 64, 63 (1995) R. B. Altmann, I. Renge, L. Kador, D. Haarer: J. Chem. Phys., 97, 5316 (1992)
87
7
Slave-Boson Approach to Strongly Correlated Electron Systems Holger Fehske, Martin Deeg, and Helmut Büttner
7.1
Introduction
The problem of understanding high-temperature superconductivity has been a challenge to theoreticians from a wide variety of fields. Many theoretical investigations have been carried out in order to identify the mechanism of this fascinating phenomenon as well as to establish the canonical model itself [1]. Although excellent progress is being made in deducing a consistent description of the high-temperature superconductivity systems from a first-principles theory, at present no microscopic theory can account for their unconventional normal-phase data in its entirety. The interplay of charge and spin dynamics in the normal state seems to hold the key to the understanding of the physical mechanism behind high-temperature superconductivity in the cuprates. Both macroscopic measurements of transport and magnetic properties as well as microscopic measurements probing the charge and spin excitation spectra are fundamental in establishing the anomalous normal-state properties of these materials [1]. The nature of spin excitations of the high-temperature superconductivity cuprates has been experimentally studied by means of nuclear magnetic/quadrupole resonance (NMR/ NQR) and inelastic neutron scattering (INS) techniques clarifying the persistence of strong antiferromagnetic (AFM) correlations in the normal and superconducting states [2]. Detailed investigations of the wave-vector dependence of the low-frequency spin fluctuation spectrum have revealed remarkable differences between the YBa2Cu3O6+x and La2–xSrxCuO4 families at low doping level x; the dynamic structure factor S (~ q, o) keeps its maximum at (p, p) in the YBCO system, while in LSCO, the peaks are displaced from the commensurate position to the four incommensurate wave-vectors p (1 + q0, 1), p (1,1 + q0), where q0 F 2x. On the other hand, the analysis of the NMR data shows that the relaxation rates on the planar Cu sites are similar in all materials. Two striking features are associated with 63T 1–1 : a) as a result of strong local spin fluctuations on 63Cu sites it is enhanced by one order of magnitude over the oxygen rate 17T 1–1 ; and b) in sharp contrast to the Korringa-like behaviour at the planar 17O sites, its temperature dependence does not follow the Korringa law [2]. 88
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties Deutsche Forschungsgemeinschaft (DFG) Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
7.1
Introduction
The physical origin of the contrasting q~-dependence of the spin fluctuation spectrum is still under discussion. Millis and Monien [3] have argued that the spin dynamics and, in particular, the temperature dependence of the spin susceptibility ws (T) in LSCO are caused by a spin density wave instability, whereas in the YBCO family they are due to in-plane AFM fluctuations and a novel non-Fermi liquid spin singlet pairing of electrons in adjacent planes. On the other hand, the magnetic properties are intimately related to the energy band dispersion of the non-interacting system within a Fermi liquid based framework, i. e., in this way the observed spin dynamics can be attributed to different Fermi surface (FS) geometry of LSCO-type and YBCO-type, respectively. Along this line, details of the spin fluctuation spectrum are studied using a nearly antiferromagnetic Fermi liquid approach by Monthoux and Pines [4]. From a more microscopic point of view, the important effects of FS shape on the magnetic properties were confirmed by Si et al. [5] within a large Coulomb-U auxiliary boson scheme and by Fukuyama and co-workers [6] on the basis of a resonating valence bond (RVB) slave-boson mean-field approach to the extended t-J model. Furthermore, Ito et al. [7] have recently reported that the charge transport in the CuO2 plane is determined by dominant spin scattering, i. e., the spin dynamics are manifest in the extraordinary transport properties of the high-temperature superconductivity cuprates as well. Encouraged by these findings it is the aim of this report to study magnetic and transport phenomena of high-temperature superconductors probably in terms of the most simple effective one-band model describing both correlation and band structure effects, the socalled t-t'-J model: Ht
t0 J
t
X hi;ji;
c~yi c~j
t0
X hhi;j ii;
c~yi c~j J
X hi j i
S~i S~j
n~ i n~ j 4
:
1
Ht–t'–J acts in a projected Hilbert space without double occupancy, where c~
y i = P y 1 ~ n~i is the electron annihilation (creation) operator, Si 2 0 c~i ~ 0 c~i 0 ; and n~ i c~yi c~i . J measures the AFM exchange interaction, t and t' denotes hopping processes between nearest-neighbour (NN; Ai,jS) and next nearest-neighbour (NNN; AAi,jSS) sites on a square lattice. Compared to the original t-J model the t'-term incorporates several important effects near half-filling. Starting from a rather complex three-band Hubbard or Emery model [8] for the CuO2 planes, quantum cluster calculations [9] have revealed that the relative large direct transfer between NN oxygen sites (tpp < tpd /2) leads to a sizeable NNN hopping t' in the context of an effective one-band description. More recently the t'-term has been introduced to reproduce the FS geometry observed in ARPES experiments [6, 10–12]. Fitting the quasi-particle dispersion relation c
y i
1P
"k~
2t
cos kx cos ky
4t0 cos kx cos ky ;
2
involved in Eq. 1 to experimental and band theory results yields t in the order of 0.3 eV and, e. g., for the case of YBCO, t' & –0.4 t [4, 5, 13]. Moreover, Tohyama and Maekawa [12] have emphasized that a t-t'-J model with t' > 0 can be used to describe the electron-doped systems, e. g. Nd2–xCexCuO4 (NCCO). In this case one has to shift the momentum k~ ? k~ + (p, p) [12], i. e., within a band-filling scenario one obtains a hole pocket-like FS centred at (p, p)-point which shrinks with increasing doping [14]. Finally, as pointed out by Lee [15], 89
7
Slave-Boson Approach to Strongly Correlated Electron Systems
in a locally AFM environment doped holes can propagate coherently only on the same sublattice without disturbing spins. Therefore, the t'-term coupling the same sublattice becomes crucial for the low-lying magnetic excitations. This clearly is a correlation effect related to the NNN hopping processes.
7.2
Slave-boson theory for the t-t'-J model
Apart from numerical techniques, the slave-particle methods have been employed extensively in studying the effects of electron-electron correlation in the (extended) t-J model [6, 16–21]. The two commonly used approaches, the NZA slave-boson [16, 17] and slave-fermion [18] schemes, however, yield quite different results concerning the (mean-field) ground-state phase diagram and spin/charge excitations for this model [22]. As yet, the relationship between both types of slave-field theories is not well understood. Within the NZA formulation, for example, the Hamiltonian can be solved by a mean-field approximation with the (uniform) RVB order parameter. A serious difficulty of this approach is the absence of AFM correlations [18, 22], e. g., in the half-filled case, the lowest energy state, the energy of which is considerable higher than numerical estimates indicate, does not satisfy the Marshall sign rule and fails to show the expected long-range Néel order [23]. On the other hand, the mean-field slave-fermion schemes [18] are known to give reasonable results for the spin susceptibility as well as for the spin correlation length [21, 22], but also suffer from neglecting important correlation effects, especially the fermion charge degrees of freedom are not described sufficiently well. In contrast, the fourfield slave-boson (SB) technique, introduced by Kotliar and Ruckenstein (KR) [24] in the context of the Hubbard model, has the advantage of treating spin and charge degrees of freedom on an equal footing. Starting from the scalar KR SB representation of the Hubbard model, one may generate an intersite exchange interaction via a loop expansion in the coherent state functional integral [19]. However, the effective t-J Lagrangian, derived in this way, contains only an Ising interaction term, i. e., important spin-flip exchange processes are neglected. As we shall see below, to bosonize the complete exchange interaction term one has to use the spin-rotation-invariant (SRI) extension of the KR SB theory from the beginning [25, 26].
7.2.1 SU(2)-invariant slave-particle representation For the sake of definiteness, we return to the extended t-J Hamiltonian (Eq. 1), that may be cast into the form Ht
t0 J
X i;j
90
~ yi ~j tij
J X ~ y ~ ~ y ~ i i j j : 4 hi j i
3
7.2
Slave-boson theory for the t-t'J model
In order to bring out the SU(2) symmetry of the system, we have used in Eq. 3 a spinor ~ yi
~ yi ; ~ i ~ i are built up representation, where the one-row [one-column] matrices y y ~ ~ by the projected fermion creation [annihilation] operators i c~i i c~i : denotes the contravariant [covariant] four-component vector (m = 0, x, y, z) of Pauli’s matrices. To preserve SRI, we apply to Ht-J the manifest [SU(2) 6 U(1)] invariant SB scheme [26, 27] based on the SRI SB approach developed for the Hubbard model by Li et al. [28]. Accordingly, we define scalar boson fields e({) and bosonic matrix operators p({) (representing i i empty and singly occupied sites, respectively) and pseudofermion spinor fields C({) in the i following way: j0i i eyi jvaci ; j i i
X
yi pyi jvaci :
4
The unphysical states in the extended Fock space of pseudofermionic and bosonic states are eliminated by imposing two sets of local constraints: Ci
1 eyi ei 2Trpyi pi
1 0;
5
expressing the completeness of the bosonic projectors, and y y C
2 i i i 2pi pi
o 0 ;
6
relating the pseudofermion number to the number of p-type slave-bosons (hereafter underbars denote a 262 matrix in the spin variables). Obviously, double occupancy has been projected out. In the transformation of the fermionic spinor fields analogous to [26, 28], ~ ! zi i ; i
7
the non-linear bosonic hopping operators zi o
2pyi pi
1=2 y ei 1
eyi ei 2Trpyi pi 1=2 pi
1
eyi ei o
2p~ yi p~ i
1=2
8
({) yield a correlation-induced band renormalization where p~ ({) irr' = rr'pi, –r',r . Exploiting the SU(2),O(3) homomorphism, the matrix operators pi may be decomposed into scalar (sing~i = (pix , piy, piz) components as let) p({) o and vector (triplet) p
p
y i
1X p
y ; 2 i
9
where the pim obey the usual Bose commutation rules pi; pyj0 ij 0 . Consequently, the pseudofermions i
i ; i T satisfy anticommutation relations of the form f i ; yj0 g ij 0 . 91
7
Slave-Boson Approach to Strongly Correlated Electron Systems
This way the interaction term is converted due to ~ i ! 2Tr pyi p ; ~ yi i
10
where, more explicitly, the locally defined particle number and spin operators are just n~ i ! n~ i
pi 2Trpyi pi
P
pyi pi ;
1 S~i ! S~i
pi Trpyi ~pi
pyio p~i p~yi pio 2
ip~yi p~i :
11
Actually, the components of the bosonized SB spin operator act as generators of rota
2 tions in spin space, i. e., S~i
pi satisfies the spin algebra. Since C(1) commute with i and C i the SB t-J Hamiltonian, the constraints in Eq. 5 and Eq.P 6 can be ensured by introducing the
2 1 time-independent Lagrange multipliers i
1 and
2 i i . As a consequence, the Hamiltonian of the t-J model (Eq. 3) in terms of the slave-boson and pseudofermion operators has to be replaced by HSB t t0
J
P i;j
tij yi zyi zj j
P i
i
1 Ci
1
J
P
Tr pyi pi
hi j i
2 Tr
2 i Ci :
Tr pyj pj
12
In the physical subspace, HSB t–J possesses the same matrix elements for the basis states (Eq. 4) as the original t-J Hamiltonian (Eq. 3) for the purely fermionic states. To verify this directly, special attention has to be paid to the bosonization of AFM exchange interaction term
~ yi ~i
y P ~ j 2
c~yi c~yj c~ c~ ~ j j i
c~yi c~yj c~j
c~i
;
13
which includes besides the Ising exchange contributions #i "j $ #i "j the spin-flip processes #i "j $ "i #j . Let us emphasize that the matrix elements of the spin-flip terms are not reproduced in the scalar KR SB approach [19]. By contrast, within the SRI SB approach it is a straightforward exercise to show that these contributions can be expressed in terms of the p({) irr' : X 0
pyi pyj0
pj0 pi
i j 1 i 4
j :
14
Thus our SRI SB scheme (Eqs. 4–12) provides a consistent bosonization of the extended t-J model. 1 Note that additional constraints do not exist. Especially, if the constraint p~i p~i 0 is added [25] the spin algebra is not satisfied.
92
7.2
Slave-boson theory for the t-t'J model
7.2.2 Functional integral formulation To proceed further, it is convenient to represent the grand canonical partition function for the redefined SB Hamiltonian (Eq. 12) in terms of a coherent-state functional integral [29] over Grassmann fermionic and complex bosonic fields as R Z d L
1
2 ; De ; e Dp p d d e 0 Z D ; ; L
P
1 T ei
@
1 i ei 2Tr pi
@ i o i
P hi;ji P i
Trpi ~pi Tr pj ~pj
i
@
Tr pi pi Tr pj pj
T
2 pTi i
1 i
P i z zj j : o i
2 i ti j i i;j
15
16
Apart from the above symmetry considerations the SB functional-integral formalism reveals additional global R and local gauge invariances [25, 30–32]. The action S 0 d L
is invariant under the following site and time-dependent phase transformations (yi (t)wi (t)), i. e., under the local symmetry group SU(2) 6 U(1), i ! e
ii i i
e i ;
e i ! ei e
ii
pi ! pi e
i
; i
17
;
provided that the (five) Lagrange parameters act as time-dependent gauge fields, l(1) i (t) and l
2 i
t, absorbing the time derivatives of the phase factors: _ !
1
1 i i i i ; i
2 i
2 i ! e i e
i
i
i _ i i_i o :
18
Next, in the continuum limit, the radial gauge is introduced by representing the Bose fields by modulus and phase, ei jei jei'
ei ; pi
1 X i
p pi e i ; 2
19
93
7
Slave-Boson Approach to Strongly Correlated Electron Systems
(for a time-discretized version of this gauge fixing, see [31]). Exploiting the gauge freedom of the action, we can now fix the five real-valued coefficients yi (t), wim (t) to remove five phases (ji (t), fim (t)) of the Bose fields ei, pim in the radial gauge. As a consequence, all the Bose fields {ei, pim} become real, in contrast to the Hubbard model, where one SB field remains complex [25, 32]. At this point P one should notice that the particle number and spin operators are changed into n~ i
pi p2i ; S~i
pi pio p~i (the previous notations ei, pio and p~i now denote the radial parts of the corresponding Bose fields). Then, using the familiar identity for Gaussian integrals over Grassmann fields [29], R
G 1 0
; e D
G 1
eTr ln
;
20
the fermionic degrees of freedom can be integrated out and we obtain the following exact representation of the grand canonical partition function Z
R
Sef f
D e
21
in terms of the real-valued bosonic fields fia,
2
2
2 ; i
1 ; pix ; ix ; piy ; iy ; piz ;
2 i
i
ei ; pio ; io iz :
22
In Eq. 21 the effective bosonic action Seff takes the form Sef f
R
d
( P i
0
J
P
2
1 i ei
hi j i
pio p~i p~j pjo
P
1 i
2
2 io pi
2pio p~i ~
2 i
1 2
p p~2i
p2jo p~2j 4 io
i
1
)
h i 1 0 Trij;0 ; 0 ln Gij; 0
; ;
23
where the (inverse) SB Green propagator is given by 1 0 Gij; 0
;
h
@
2 io 0
tij
zyi zj 0 ; 0
1
ij :
i ~0 ij
~
2 i
0
24
It may be remarked that since the bosons are taken to be real, their kinetic terms, being proportional to the time derivatives in Eq. 16, drop out due to the periodic boundary conditions imposed on Bose fields (fia (b) = fia (0)). Strictly speaking it follows from this property that all the Bose fields do no longer have dynamics of their own [25].
94
7.2
Slave-boson theory for the t-t'J model
7.2.3 Saddle-point approximation The evaluation of Eq. 23 proceeds via the saddle-point expansion 2, where at the first level i of the bosonized action Seff with respect of approximation we look for an extremum S
to the Bose and Lagrange multiplier fields fia : @Sef f ! 0 ) S Sef f i i : @ i
25
i is determined to give the lowest free energy The physically relevant saddle-point F (per site) ft
t0 J
=N n ;
26
where at given mean electron density n = 1 – d, the chemical potential m is fixed by the requirement n
1 @
N @
27
S= denotes the grand canonical potential). Obviously, an unrestricted minimization of (O the free energy functional is impossible for an infinite system. To keep the problem tractable, we use the ansatz ~ i m~ m ui ;
u~i
cos Q~ R~i ; sin Q~ R~i ; 0
28
~ i 2S~i 2pio p~i . Following earlier analyses of spiral states for the local magnetization m for the Hubbard model [34] the unit vector u~i is chosen as a local spin quantization axis pointing in opposite directions on different sublattices. Thereby, the order parameter wavevector Q~ is introduced as a new variational parameter to describe several magnetic ordered states: PM, FM (Q~ = 0), AFM (Q~ = (p,p)), and incommensurate (1,1)-spiral (Q~ = (Q, Q)), (1,p)-spiral (Q~ = (Q,p)) and (0,1)-spiral (Q~ = (0,Q)) states. Note that since fluctuations of the charge density are not incorporated the scalar Bose fields are homogeneous: ei = e, pio = (1) (2) (2) po, l(1) i = l , and lio = lo . The vector fields exhibit the same spatial variation as the mag
2 ~ ~ o)-representation netization: p~i p~ ui and li l
2 u~i . Then transforming Eq. 23 in the (k; the trace in Seff can be easily performed to give the free energy functional
2 Mean-field approximations to the functional integral are achieved by replacing the bosonic fields by their time averaged values. For the Hubbard model, a comparison of the paramagnetic and antiferromagnetic SB solutions with quantum Monte Carlo results shows that such a mean-field like approach yields an excellent quantitative agreement for local observables [33] and therefore can give a qualitative correct picture with relatively little effort.
95
7 ft
t0 J
Slave-Boson Approach to Strongly Correlated Electron Systems
2 2
1
e2 p2o p2 1
2 2
2 po p n o
po p 1 2
po p2 2 J p2o p2
cos Qx cos Qy 2
1 X ln1 N ~
nk ~ ;
29
1
30
k
where nk ~ exp f
Ek ~
g 1
holds. The renormalized single-particle energies Ek ~
are obtained by diagonalizing the kinetic part of Eq. 23 [26]. Requiring that ft-J be stationary with respect to the variation of the magnetic order vector, the wave-vector Q~ can be obtained from the extremal condition sin Qx;y
@Ek 1 1 X ~ nk : ~ 2 2 Jpo p N ~ @Qx;y
31
k
If one substitutes Eq. 31 together with f a (obtained from the solution of the coupled self-consistency equations (Eq. 25)) into Eq. 29, the free energy of the t-t'-J model is obtained as ft
t0 J
J 2 m
cos Qx cos Qy 4 1 X ln 1 e
Ek ~ N ~
2n2
2 m
2 o n
;
32
k
where the quasi-particle energy takes on the form
2 Ek ~
1
"k~ "k~ Q~ =y o h 2
"k~ "k~ Q~ 2 =y m
"k~ "k~
=y Q~
2 2
i1=2
33
with m = 2 po p and y = (1+d)2 – m2.
PM = (e, po, l(2) In particular, at the spatially uniform paramagnetic saddle-point, F o , (1) l ; 0, 0; 0, 0; 0, 0) the remaining bosonic fields e2 ;
96
p2o 1
;
1
2 3 2 "~
0 ; 1 2
2 o
2 "~
0
1 2
1
J ;
34
7.2
Slave-boson theory for the t-t'J model
are explicitly given in terms of J, d and a single energy parameter "~ (0) defined by "~
~ q
2 X " N ~ k~
q~
Ek~
35
k
(at T = 0). Here, the quasi-particle energy Ek~ is Ek~ 2"k~=
1
2 0 ;
36
and m can be determined from d = 1 – free energy becomes simply 2 f
PM t t0 J z "~
0
1 J
1 2
2 N
P
2
1
k~
Ek~. Finally, in this approximation, the
:
37
7.2.4 Magnetic phase diagram of the t-t'-J model In the numerical evaluation of the self-consistency loop we proceed as follows: at given model parameters J and d, we solve the remaining saddle-point equations for m, l(1), l(2) o and l(2) together with the integral equation for m using an iteration technique. Then, in an outer loop, the order parameter wave-vector Q~ is obtained from the extremal Eq. 31 by means of a secant method. Convergence is achieved if all quantities are determined with relative error less than 10 –6. Note that our numerical procedure allows for the investigation of different metastable symmetry-broken states corresponding to local minima of the variational free energy functional (Eq. 32). At first, let us consider the case t' = 0, i. e., the pure t-J model. The resulting groundstate phase diagram in the J-d plane is shown in Fig. 7.1. For the case d = 0 (Heisenberg model), we obtain an AFM ground state. At J = 0, the FM is lowest in energy up to a hole concentration of 0.327, where a first-order transition to the (1, p)-spiral takes place; above d = 0.39, we find a degenerate ground state with wave-vector (0, p). At d = 0.63 the PM becomes the lowest in energy state (second-order transition). This coincides with the U ? ? SB results of the Hubbard model (J = 4t2/U) [35]. The PM-FM instability occurs at dPM-FM = 0.33 [32]. In contrast, the slave-fermion phase diagram [20] exhibits a much larger FM region. This can be taken as an indication that correlation effects are treated less accurate within the slave-fermion mean-field scheme [20, 36]. For finite exchange interaction J the (1,1)-spiral is the ground state at small doping. With increasing d a transition to the FM takes place, which becomes unstable against the (1, p)-spiral at larger doping concentrations. For J/t > 0.08, we find a transition from (1,1)-spirals to (1, p)-spirals at about d & 0.2. In Figure 7.1, the dotted line separates the (1, p)-spiral state from the region, where the (1, p) and (0, p) states are degenerate. If we admit only homogeneous phases, the phase boundaries (1,1)-spiral ⇔ (1, p)-spiral at d & 0.2 and (1, p)/(0, p)-phase ⇔ PM at d & 0.63 remain nearly unchanged at larger values of J. However, analyzing the thermodynamic stability of 97
7
Slave-Boson Approach to Strongly Correlated Electron Systems
Figure 7.1: SB ground-state phase diagram of the t-J model.
the saddle-point solutions (i. e. the curvature of ft J
d), one observes a tendency towards phase separation into hole-rich and AFM regions above the ‘diagonal’ solid curve in the left upper part of the J-d phase diagram (see below). Figure 7.2 displays the variation of the extremal spiral wave-vector Q~ as function of doping. At d = 0, Q~ = (p, p) indicates the AFM order. At low doping the (1,1)-spiral order vector decreases approximately linear. With decreasing J the AFM exchange is weakened, consequently the deviation of the order vector from (p, p) increases. The discontinuities reflect the first-order transition from (1,1)-spirals to (1, p)-spirals. For J/t = 0.05 the transition to the FM state takes place, with Qx jumping down to zero. The (1, p)-spiral wave-vector shows a monotonous decrease of Qx until at d = 0.63 the transition to the PM (Qx = 0) occurs. Comparing the magnitude of the theoretical order vector of the spiral solutions to results from inelastic neutron scattering experiments on La2–xSrxCuO4 [37], we find good agreement for an exchange interaction strength of J/t = 0.4 (which seems to be a reasonable value with respect to the strong electron correlations observed in the high-temperature superconductors).
~ as a function of doping d (compared with exFigure 7.2: The x-component of the spiral wave-vector Q periments [_] on LSCO [37]).
98
7.2
Slave-boson theory for the t-t'J model
Next, we investigate the ground-state properties of the t-t'-J model, where we fix J/t = 0.4. Comparing the free energies of several (homogeneous) symmetry-broken states, we obtain the SB phase diagram shown in the t'/t-d plane in Fig. 7.3. Obviously, we can distinguish two regions. In the parameter region |t'/t| ^ 0.2, Figure 7.3 resembles the groundstate phase diagram of the pure t-J model (Fig. 7.1), i. e., we found large regions with incommensurate spiral order. However, compared to the pure t-J model, the t'-term stabilizes Néel order in a finite d region near half-filling. The increasing stability of AFM configurations can be intuitively understood because the t'-term moves electrons without disturbing the Néel-like background [12]. For larger ratios |t'/t| we have a completely different situation. In this parameter regime only commensurate states (AFM, FM, PM) occur, where for t' < t'c = –1.4 t we obtain the AFM state for all d. Note the rather large differences to the value of t'c obtained within a semiclassical (1/N)-expansion [38]. By varying the exchange coupling J, the phase boundaries in the t'/t-d plane are not much affected, e. g. for J/t = 1 and t'/t = –0.4, the transitions AFM ⇔ (1,1)-spiral and (1,1)-spiral ⇔ FM take place at d = 0.17 and d = 0.6, respectively. We would like to point out that the main qualitative features of our SB phase diagram do confirm recent studies of magnetic long-range order in the t-t'-J model [38, 39].
Figure 7.3: Restricted SB ground-state phase diagram of the 2D t-t'-J model at J/t = 0.4.
In Fig. 7.4 we plot the order parameter wave-vector as a function of doping at t'/t = +0.16 and t'/t = –0.4. The behaviour of Qx reflects a series of transitions AFM ⇔ (1,1)spiral ⇔ (1, p)-spiral ⇔ PM. The corresponding (sublattice) magnetization abruptly changes at the (1,1)-spiral ⇔ (1, p)-spiral first-order transition. From Fig. 7.4 the asymmetry between hole (t' < 0) and electron doping (t' > 0) becomes evident. In contrast to recent HartreeFock results for the Hubbard model [40] we found the AFM phase near half-filling for both electron-doped and hole-doped cases (provided t' ( 0). In the absence of t ' hopping, for arbitrarily small doping the AFM is found to be unstable against the (1,1)-spiral phase (cf. Fig. 7.3). Obviously, the AFM correlations are strongly enhanced by a positive t'-term, which is also in qualitative agreement with exact diagonalization studies of the t-t'-J model [12] and confirms the experimental findings for the electron-doped system NCCO [41, 42]. Note that the stability region of the AFM phase agrees surprisingly well with the combined 99
7
Slave-Boson Approach to Strongly Correlated Electron Systems
~ away from half-filling for the negative vaFigure 7.4: The x-component of the SB spiral wave-vector Q lues t'/t = –0.16 (solid) and t'/t = –0.4 (long-dashed), i. e., hole doping (d > 0), and for the positive one t'/t = 0.16 (dashed), i. e., electron doping.
phase diagram for La2–xSrxCuO4 and Nd2–xSrxCuO4 obtained from neutron scattering [43] and muon spin relaxation measurements [44], respectively. For the YBCO parameter t'/t = –0.4, the AFM disappears around d = 0.1 whereas in the phase diagram of YBCO, determined by neutron diffraction [45], this transition takes place at about x = 0.4 oxygen content. However, there exits strong evidence that at least up to x = 0.2 no holes are transferred from CuO chains to CO2 planes. As we have already noted, the phase diagrams in Fig. 7.1 and Fig. 7.3 result from the relative stability of various homogeneous states. On the other hand, there are arguments for the existence of inhomogeneous, e. g., phase-separated states in the t-J and related models [46]. Using very different methods, it was realized by several groups [47–51], that at large J/t the ground state of the t-J model separates into hole-poor (AFM) and hole-rich regions. Unfortunately, in the physically interesting regime of small exchange coupling (J/t ~ 0.2– 0.4) and low doping level this point is still controversial. To gain more insight into the phenomenon of phase separation in t-J-type models of strongly correlated electrons it seems to be important to investigate the effect of an additional NNN hopping term t' as well. Therefore we study the free energy as a function of hole density d, where a (concave) convex curvature indicates local thermodynamic (in)stability implying a (negative) positive inverse iso@2 f –1 thermal compressibility k–1 = n2 @n < 0, the domain of the two-phase regime is deter2 : If k mined performing a Maxwell construction for the anomalous increase of the chemical potential m by doping. The results of our analysis of thermodynamic stability are depicted in Figs. 7.5 a and 7.5 b for the t-J and t-t'-J model, respectively. The boundary of phase separation for the t-J model is given by the solid curve in the J-d plane shown in Fig. 7.5 a. As can be seen from Fig. 7.1, the different phase separated domains are built up by the (AFM) states at half-filling (d = 0) and the corresponding hole-rich state on the right boundary of the respective region. At J = 0, where we recover the U ? ? result of the Hubbard model [32], the free energy is a convex function Vd, i. e., our SRI SB theory does not support recent arguments [47] for phase separation in this limit. In the opposite limit of large J, complete charge se100
7.2
Slave-boson theory for the t-t'J model
Figure 7.5: Phase diagram of the t-(t')-J model including phase separated states. The phase separation boundary (solid curve) for the t-J model is shown in the J-d plane (a). We include the transition lines of Refs. [49] (dashed), [50] (dotted), and [51] (chain dashed). The triangles are the Lanczos results of Ref. [47]. The phase diagram of the t-t'-J is calculated in the t'/t-d plane at J/t = 0.4 (b). Here the twophase region consists of AFM and (1,1)-spiral states. For further explanation see text.
paration takes place for J/t > JPS /t = 4.0, which seems to be an essentially classical result. From exact diagonalization (ED) we obtain JPS /t = 4.1 + 0.1 [52] compared with 3.8 derived by means of a high-temperature expansion [51]. We note that the homogeneous magnetic phases are always unstable close to half-filling (provided J/t > 0). This is in qualitative agreement with results obtained from ED studies [47] as well as from semiclassical [49] or renormalization-group calculations [50]. Also plotted in Fig. 7.5 a are the results of the hightemperature expansion method [51], where phase separation may occur only above a critical exchange J/t = 1.2 as d ? 0, contrary to all the other approaches. The line separating the two-phase region from the stable states was determined by Marder et al. [49] within a semiclassical theory to vary as J/t = 4 d2 whereas our theory yields an approximately linear dependence at small d. We believe that, due to an improved treatment of spin correlations in our approach, the region of incomplete phase separation is reduced. The instability towards phase separation at small J can be taken as an indication that charge correlations may play an important role as well. The dotted lines of zero inverse compressibility in Fig. 7.5 b show that also for |t'/t| > 0 there is a finite range of d over which the (1,1)-spiral is locally unstable. Similar results were recently obtained by Psaltakis and Papanicolaou [38]. But it is important to stress that in our theory the AFM state is locally stable for both signs of t'. In addition, based on the Maxwell construction, we can show that near half-filling the AFM state remains also globally stable against phase separation for the t-t'-J model (cf. Fig. 7.5 b, where the two-phase region is bounded by the solid lines). At larger values of |t'/t| 6 0.5, the phase-separated region is due to the first-order nature of the transition AFM ⇔ (1,1)-spiral (dashed curve) [38]. Finally to demonstrate the quality of our SRI SB approach, for the t-J model the expectation value of the kinetic energy is compared with the results from ED for a finite 16site [48] (36-site [52]) lattice in Fig. 7.6. We find an excellent agreement between SB results and ED data. Obviously, this result does not depend on the interaction strength J, i. e. the SB 101
7
Slave-Boson Approach to Strongly Correlated Electron Systems
Figure 7.6: Expectation value of the kinetic energy as a function of doping at several interaction strengths J in comparison to ED results for the 16 (36) site lattice [48, 52].
theory well describes important correlation effects. Note that AHtSt–J /t is directly related to the effective transfer amplitude of the renormalized quasi-particle band, which is taken as input for the calculation of transport coefficients in the following section.
7.3
Comparison with experiments
7.3.1 Normal-state transport properties As one of the main normal-state puzzles of the CuO2 based high-temperature superconductors, the anomalous transport properties, in particular the temperature and doping dependence of the Hall resistivity RH (T, d) [53–56], has been under extensive experimental and theoretical study. Quasi-particle transport measurements suggest a small density of charge carriers and hence a small pocket-like Fermi surface (FS) [57]. On the other hand, direct angle-resolved photoemission (ARPES) probes of the FS [14, 58] yield a large FS which satisfies Luttinger’s theorem and might be well described by (LDA) band structure calculations [13, 59]. In principle, this contrasting behaviour is found for hole-doped (La2–xSrxCuO4, YBa2Cu3O6+x) and electron-doped (Nd2–xCexCuO4) copper oxides as well. Adopting the hypothesis of Trugman [60], most of the normal-state properties of the cuprates may be explained by the dressing of quasi-particles due to magnetic interactions and the subsequent modification of their dispersion relation. Then, once the quasi-particle band Ek ~ has been obtained, the Hall resistivity RH = sxyz /sxxsyy can be calculated in the relaxation time approximation, using standard formulas for the transport coefficients: 102
7.3 Comparison with experiments
@nk e2 X ~ u k ; ~ u k ~ 2 @E k V ~ ~ k k
(38)
@u k ~ @nk e X ~ u k : ~ " uk ~ 4 @k @Ek k cV ~ ~ 3 2
k
Here nk ~ is given by Eq. 30, V denotes the volume of the unit cell, elkg is the completely antisymmetric tensor, and u k ~ =@k . Note that RH does not depend on the relaxa~ @E k tion time t. To make the discussion more quantitative, let us now consider the doping dependence of RH (d,T ) in terms of the t-t'-J model using the saddle-point and relaxation time approximations, where FS and correlation effects are involved via the renormalized SB band Ek ~ (Eq. 33). As we have pointed out above, in our approach the SB quasi-particle band dispersion Ek ~ has to be determined in a self-consistent way at each doping level d. This should be in contrast to the NZA SB mean-field approach to the t-t'-J model of Chi and Nagi [61] where, in the J ? 0 limit, the calculation of transport properties is based on the simple replacement "k~ ! "~k~ = –2 td [(cos kx + cos ky) + 2 (t'/t) cos kx cos ky] of the non-interacting band dispersion (Eq. 2). Figure 7.7 shows the theoretical Hall resistivity as a function of carrier density in comparison to experiments on LSCO [53], YBCO [54] and NCCO [55, 56]. In the LSCO and NCCO systems, the concentration of chemically doped charge carriers in the CuO2 planes (d) definitely agrees with the composition (x) of the substitutes Sr and Ce. This simple relation, however, no longer holds for YBCO, i. e., the number of holes transferred into the planes does not increase linearly with the oxygen content. Indeed, the magnetic properties indicate d & 0 up to x = 0.2 [45]. In order to compare our theoretical model with the RH (x) data found on oxygen-doped YBCO, we use the relation d = (x – 0.2)/2 [62].
Figure 7.7: Doping dependence of the Hall resistivity for hole-doped (left panel) and electron-doped (right panel) systems. The slave-boson results for J/t = 0.4 and different ratios t'/t = 0 (solid), –0.16 (dashed), –0.4 (chain dotted) and t'/t = 0.16 (dotted) are compared with experiments on LSCO (_) [53] (at 80 K), YBCO (O) [54] (at 100 K) and NCCO (Z) [55, 56] (at 80 K), respectively. The inset shows the temperature dependence of RH for t'/t = 0 at d = 0.1 (short dashed) and d = 0.15 (dotted).
103
7
Slave-Boson Approach to Strongly Correlated Electron Systems
Figure 7.7 clearly demonstrates the importance of the NNN transfer term t' for a consistent theoretical description of the experimental Hall data. For J/t = 0.4, an excellent agreement with experiments on LSCO and NCCO, including the sign change of RH (d) at a very similar value, can be achieved using the parameter values t'/t = 0 and t'/t = 0.16, respectively. It should be noted that in the case of LSCO we obtain t'/t = 0 from RH (d) while LDA calculations yield a ratio t'/t = –0.16 [13, 59]. For YBa2Cu3O6+x, where the experiments [54] give RH > 0 up to x = 1, a negative t'-term suffices to give the correct tendency to RH (d). Using t'/t = –0.4, our theory yields a sign change of RH at d & 0.7. The strong increase (decrease) of the positive (negative) Hall coefficient as d ? 0 can be attributed to the formation of small hole (electron) pockets in the FS, which is a correlation effect. A recent analysis of resistivity saturation in LSCO [57], based on Boltzmann transport, has been taken as an indication of a small FS as well, however, the existence of a pocket-like FS is still a subtle and unresolved issue. We found that the temperature dependence of RH (d,T ) is at least in qualitative agreement with experiments on LSCO (see inset Fig. 7.7 (left panel)). Note that the quasi-particle dispersion Ek ~ exhibits extremely flat minima implying the presence of a new small energy scale D. Therefore, when the temperature becomes comparable to D the hole pockets are washed out and a sign change of RH occurs (cf. Fig. 7.7 at fixed d (inset)). 3 The FS of the interacting system are shown in Fig. 7.8 for typical ratios t'/t at J/t = 0.4 and d = +0.1, where the diagonal (1,1)-spiral phase is lowest in energy. As observed for t-J and Hubbard models as well [64], we obtain small hole (or electron) pockets with a volume ! |d|. The calculated FS are very anisotropic. As |d| increases, the pockets grow, until the FS topology changes completely at a critical doping value dc (RH (dc) = 0 (cf. Fig 7.7), reflecting the transition from hole to electron carriers for t'/t < 0 and vice versa for t'/t > 0.
Figure 7.8: Quasi-particle Fermi surface in the (1,1)-spiral phase at J/t = 0.4 for d = 0.1 (hole-doped system) and d = –0.1 (electron-doped system).
3 We recently learned of a related exact diagonalization study of Dagotto et al. [63], where, similar in conclusion, the doping and temperature dependence of RH was calculated using a strongly renormalized flat quasi-particle dispersion.
104
7.3 Comparison with experiments We want to point out that the renormalization of the quasi-particle band Ek ~ strongly depends on both interaction strength J and doping level d [62] which, in fact, calls into question the frequently used rigid band approximation. Due to the strong coupling of spin and charge dynamics the characteristic energy scale for the coherent motion of the charge carriers is J and not t (provided t > J).
7.3.2 Magnetic correlations and spin dynamics In this Section we try to understand the INS and NMR experiments on the basis of the SRI SB mean-field theory. Using a generalized RPA expression for the spin susceptibility and assuming that the AFM correlations are spatially filtered by various q~-dependent hyperfine form factors, we focus, in particular, on the spin dynamical properties in the paraphase of the t-t'-J model. Our starting point is an RPA-like form for the exchange-enhanced spin susceptibility [5, 6, 65] o
~ q; !
s
~ q; ! 1
J
cos qx cos qy o
~ q; ! 2
:
39
The irreducible part 4 2 X ~ k~ q~; i!nm G
k; i!n G
N ~
o
~ q; i!m
40
k;n
~ i!n i!n E ~ ~ 1 describing noncontains the (dressed) SB Green propagators G
k; k interacting electrons with the renormalized band structure Ek~
d (Eq. 26). Once the dynamic spin susceptibility has been obtained both, INS measurements and NMR experiments, can be explored. Probed by INS from the fluctuation-dissipation theorem the q-dependent and o-dependent spin structure factor is related to the dynamical susceptibility by S
~ q; !
1 p1
1 Im s
~ q; ! exp
k!
41
On the other hand, the nuclear spin-lattice relaxation rate aT –1 1a (a =k,k), e. g. for a field Ha applied parallel to the c-axis, given by [66] a
T1k T
1
1 kB 1 Xa 2 Im s
~ q; ! F?
~ q lim 2 2 B k !!0 N q~ k!
42
4 on = 2 np/b [om = (2 m + 1) p/b] denote the fermionic [bosonic] Matsubara frequencies.
105
7
Slave-Boson Approach to Strongly Correlated Electron Systems
and the transverse spin-spin relaxation rate, T –1 2G, for the RKKY coupling of the nuclear Cu spins, given by [67] 2
T2G2
0 12 3 X X 2 c @1 41 2 Fk2
~ q s
~ q F 2
~ q s
~ qA 5 ; N q~ k 8k
2B 4 N q~
43
provide local, atomic site (a = {63, 17} specific information. Here mB denotes the Bohr magneton and the constant c = 0.69 is the natural-abundance fraction of the 63Cu isotope. The form factors are given for 63Cu and the planar 17O nucleus as [68] 63
F
~ q A 2B
cos qx cos qy ;
44
17
F
~ q 2C cos
qx : 2
45
Together with the anisotropy of the Cu relaxation rates the measurements of the Knight shift, a K B a2 n k limq~!0 a F
~ q s
~ q; ! 0, have been used to determine the hyperfine coupling constants Aa, B and C on the basis of the Mila-Rice Hamiltonian [69]. Following Ref. [66] we take A|| & –4B, Ak& 0.84B, C & 0.87B and B & 3.3610–7 eV, where 63 gk = 7.5610 –24 erg/G and 17gk = 3.8610 –24 erg/G. As we know, the RPA susceptibility contains an unphysical instability of the paramagnetic phase at some particular wave-vector q~ below a critical doping dc as signaled by the zero in the denominator of Eq. 39 at o = 0 (Stoner condition). Therefore the use of Eq. 39 only makes sense if the system is far from the magnetic instability, i. e., d > dc , where for J = 0.4 we have dc (t'/t = –0.16) H 0.27 and dc (–0.4) H dc (0) H 0.17.
7.3.3 Inelastic neutron scattering measurements We begin with a discussion of the RPA dynamical spin structure factor S (~ q; o. The q~dependence of S (~ q; o along the main symmetry axis of the Brillouin zone is shown in Fig. 7.9 at J = 0.4 and hole density d = 0.3 for different ratios t'/t. Here the temperature is T = 35 K and the frequency ko = 0.010 eV. For LSCO-type parameters (t'/t = 0, –0.16) we found four pronounced incommensurate peaks located at the points p (1+qo, 1), p (1,1+qo). The incommensurate modulation wave-vectors move with increasing doping level d away from the corner of the Brillouin zone along the directions (1, p) or (p, 1) (square lattice notation). Note, that the incommensurate peak position obtained from a three-band RPA calculation of S (~ q; o [5] can be parametrized consistent with the experimental observation that q0 H 2 x [37], while all the effective one-band RPA approaches [6, 10] yield an incommensurability scaling rather as q0 H d. A more detailed investigation of the LSCO-type q~-scans show that the q~-variation of S (~ q; o is mainly governed by that of w o
~ q; o and, in accordance with experiments [70], the in106
7.3 Comparison with experiments commensurate peaks considerably broaden when temperature or energy transfer are increased [71]. By contrast, the same plot for YBCO-type (t'/t = –0.4) parameters shows a broad and nearly T-independent [71] maximum around the (p, p)-point [72] (Fig. 7.9) which, due to the flat topology of w o , mainly reflects the q~-dependence J (~ q) = J (cosqx + cosqy) (cf. Eq. 39). In this way, our calculations confirm recent arguments [5, 6, 73] for the importance of band structure (Fermi surface geometry) effects in explaining the difference between observed LSCO and YBCO spin dynamics. Nevertheless, whether the incommensurate signals arise from an intrinsic magnetic structure or whether they result from the formation of domains (charge superstructures) in the LSCO system remains unanswered by INS [2, 68].
Figure 7.9: Dynamic magnetic structure factor S (~ q; o is plotted along the (1,1) and (1, p)-directions of the Brillouin zone for different ratios t'/t.
In a next step, we calculate the longitudinal or spin-lattice relaxation rate, T –1 1 , using the hyperfine form factors (Eq. 45). In Figure 7.10 the temperature dependences of 63T –1 1k and 17 –1 T 1k (inset) are shown for d = 0.35 and t'/t = –0.4 in comparison to experiments [74] on fully oxygenated YBCO materials (x = 1). Although our theory does not succeed in giving the correct amplitude of aT –1 1k the qualitative features of the NMR data are described surprisingly well. Obviously, the broad magnetic peak in S (~ q; o at the AFM wave-vector Q~AFM = (p, p) strongly enhances the relaxation rate on Cu sites while, due to a geometrical cancellation (17Fa (~ q Q~AFM 0, the corresponding oxygen rate is rather insensitive to nearly commensurate AFM fluctuations and therefore is governed by the long wavelength part q~ 0 of the spin susceptibility [2]. For 63Cu the nominal Korringa ratio S : (1/T1TK2S) (KS denotes the spin part of the Knight shift) is at least one order of magnitude larger. As can be seen from Fig. 7.10, for YBa2Cu3O7-type parameters, a Korringa (1/T1 ! T ) dependence (dotted line) holds at both Cu and O sites below T* ~ 120K, demonstrating the existence of a characteristic temperature T* as well as in all other near-optimum Tc compounds [2, 75]. T* is in good agreement with the coherence energy scale suggested experimentally [75]. Above T* the 17O NMR relaxation remains linear whereas the 63Cu relaxation time does not follow the Korringa law. 107
7
Slave-Boson Approach to Strongly Correlated Electron Systems
17 –1 Figure 7.10: Spin-lattice relaxation rates 63T –1 T 1k (inset) as a function of temperature. SB results 1k and ((+); t'/t = –0.4, d = 0.35) are compared with experiments (D) on YBa2Cu3O7 [74].
In the oxygen-deficient compound YBa2Cu3O6.6, 1/(63T1kT) shows a broad maximum at about 150 K (Fig. 7.11), which reflects a strong deviation from the canonical Korringa behaviour. In the normal-state regime the theoretical results agree even quantitatively with the experimental 63Cu NMR data [76]. At this point it is important to stress that the present theory incorporates considerable band renormalization effects already via w 0
~ q; o, especially at low doping level. Thus a rather moderate strength J = 0.4 of the AFM exchange interaction yields the experimentally observed enhancement of 63T1k . In striking contrast to the optimally doped YBCO system, the Korringa relation is no longer satisfied for the planar 17O nucleus sites in the underdoped material (cf. inset Fig. 7.11). Instead, a different behaviour 17 T1kT 17 KS = const was suggested to hold down to Tc [76]. The unconventional T-scaling of 17 T1k has been taken as a signature for another important feature of the normal-state spin dynamics, the so-called spin-gap behaviour [2].
Figure 7.11: 63Cu and planar 17O relaxation data (^) for underdoped YBa2Cu3O6.6 [76] are plotted vs temperature. Theoretical results (6) are given at t'/t = –0.4, and d = 0.2.
108
7.4
Summary
Complementary measurements of the transverse spin-spin relaxation rate, T –1 2G , have provided further insights into the drastic change in the magnetic properties when passing from the overdoped to the underdoped regime [67, 77]. As experimentally observed, we found that T –1 2G increases (decreases) with increasing (decreasing) hole doping (temperature) [78]. In order to detect the opening of a spin-pseudogap as a function of T, a powerful technique is to measure the ratio T2G /T1T [79] which is nearly constant above ~ 200 K for deoxygenated YBa2Cu3O6.6 [67]. The calculated temperature dependence of this quantity is shown in Fig. 7.12 together with recent experimental results [67]. Most notably, the opening of a spin-pseudo gap at T* ~ 135 K [79], i. e. well above Tc, is clearly seen as a decrease of below T*. Note that for YBa2Cu3O7, as predicted by Fermi liquid theories, the ratio T22G /T1T is approximately constant above 150 K [2].
Figure 7.12: T2G /(63T1kT ) in YBa2Cu3O6.6 (^) as measured by Takigawa [67] compared with the SB data (6) at t'/t = –0.4, and d = 0.2.
7.4
Summary
In this work we have used a spin-rotation-invariant SB approach to investigate magnetic and transport properties of the 2D t-t'-J model. Our main results are the following (see also [81]): a) We present a detailed magnetic ground-state phase diagram of the 2D t-(t')-Jmodel, including incommensurate magnetic structures and phase separated states. At finite t', a main feature of the phase diagram, we would like to emphasize, is the existence of an AFM state away from half-filling, which is locally and also globally stable against phase separation. This result agrees with the experimentally observed AFM long-range order in the weakly doped LSCO and YBCO compounds. In contrast, for the simple t-J model we observe no AFM long-range order at any finite doping due to phase separation. 109
7
Slave-Boson Approach to Strongly Correlated Electron Systems
b) The next nearest-neighbour hopping process (t') incorporates important correlation and band structure effects near half-filling. In particular, the t'-term can be used to reproduce the FS geometry of LSCO, YBCO, and NCCO. Also the NNN hopping provides a possible origin for the experimentally observed asymmetry in the persistence of AFM order of holedoped and electron-doped systems. c) The quality of the SRI SB approach was demonstrated in comparison with exact diagonalization results available for the t-J model on finite square lattices with up to 36 sites. In this case, the SB method yields an excellent estimate for the quasi-particle band renormalization. d) Within the saddle-point and relaxation time approximation, our SB calculation of the Hall resistivity in the t-t'-J model provides a reasonable explanation of the experimentally observed doping dependence of RH on both hole-doped (La2–xSrxCuO4, YBa2Cu3O6+x) and electron-doped (Nd2–xCexCuO4) copper oxides. e) Using a generalized RPA expression for the spin susceptibility and assuming that the AFM correlations are spatially filtered by the hyperfine form factors, we have calculated the temperature dependences of spin-lattice and spin-spin relaxation rates for planar copper and oxygen sites. The results agree qualitatively well with various NMR experiments on YBa2Cu3O6+x. In addition, we can attribute the contrasting q~-dependence of the magnetic structure factor S (~ q; o seen in INS experiments for LSCO-type and YBCO-type systems to differences in their fermiology. Recently, our theory was improved to include (Gaussian) fluctuations beyond the paramagnetic saddle-point approximation [27, 80]. We derived a concise expression for the spin susceptibility ws (~ q; o of the t-t '-J model which does not have the standard RPA form. Then we were able to show that the instability line obtained from a divergence of ws (~ q; 0 is in agreement with the PM ⇔ spiral state phase boundary in the saddle-point phase diagram, which in fact proves the consistency of both approaches [27].
References
1. For a recent overview of the experimental and theoretical situation see: Proceedings of the Int. Conf. M2S HTSC IV, Grenoble 1994, in: Physica C, 235–240 (1994) Proceedings of the Int. Euroconf. on Magnetic Correlations, Metal-Insulator-Transitions, and Superconductivity in Novel Materials, Würzburg 1994, in: J. Low. Temp. Phys., 3/4 (1995) 2. A.P. Kampf: Physics Reports, 249, 219 (1994) 3. A.J. Millis, H. Monien: Phys., Rev. Lett. 70, 2810 (1993) 4. P. Monthoux, D. Pines: Phys. Rev. B, 49, 4261 (1941) 5. Q. Si, Y. Zha, K. Levin, J.P. Lu: Phys. Rev. B, 47, 9055 (1993) 6. T. Tanamoto, K. Kuboki, H. Fukuyama: J. Phys. Soc. Jpn., 60, 3072 (1991) T. Tanamoto, H. Kohno, H. Fukuyama: J. Phys. Soc. Jpn., 62, 717 (1993)
110
References
7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56.
T. Tanamoto, H. Kohno, H. Fukuyama: J. Phys. Soc. Jpn., 62, 1455 (1993) T. Tanamoto, H. Kohno, H. Fukuyama: J. Phys. Soc. Jpn., 63, 2739 (1994) T. Ito, K. Takenaka, S. Uchida: Phys. Rev. Lett., 70, 3995 (1993) V.J. Emery: Phys. Rev. Lett., 58, 2794 (1987) M.S. Hybertsen, M. Schlüter, N. E. Christensen: Phys. Rev. B, 39, 9028 (1989) P. Bénard, L. Chen, A.-M. S. Tremblay: Phys. Rev. B, 47, 15 217 (1993) M. Lavagna, G. Stemmann: Phys. Rev. B, 49, 4235 (1994) T. Tohyama, S. Maekawa: Phys. Rev. B, 49, 3596 (1994) J. Yu, S. Massida, A.J. Freeman: Phys. Lett. A, 122, 198 (1987) D.M. King et al: Phys. Rev. Lett., 70, 3159 (1993) P.A. Lee: Phys. Rev. Lett., 63, 680 (1989) D.M. News, M. Rasolt, P. Pattnik: Phys. Rev. B, 38, 6513 (1988) Z. Zou, P.W. Anderson: Phys. Rev. B, 37, 627 (1988) D. Yoshioka: J. Phys. Soc. Jpn., 58, 1516 (1989) H. Kaga: Phys. Rev. B, 46, 1979 (1992) S.K. Sarker: Phys. Rev. B, 47, 2940 (1993) J. Igarashi, P. Fulde: Phys. Rev. B, 48, 998 (1993) Y.M. Li, D.N. Sheng, Z. B. Su, L. Yu: Phys. Rev. B, 45, 5428 (1992) S.D. Liang, B. Doucot, P.W. Anderson: Phys. Rev. Lett., 61, 365 (1988) G. Kotliar, A.E. Ruckenstein: Phys. Rev. Lett., 57, 1362 (1986) R. Frésard, P. Wölfle: Int. J. Mod. Phys. B, 5&6, 685; Erratum ibid. 3087 (1992) M. Deeg, H. Fehske, H. Büttner: Europhys. Lett., 26, 109 (1994) M. Deeg, H. Fehske: Phys. Rev. B, 50, 17 874 (1994) T. Li, P. Wölfle, P.J. Hirschfeld: Phys. Rev. B, 40, 6817 (1989) J.W. Negele, H. Orland: Quantum Many-Particle Systems, Addison-Wesley, Reading, MA (1988) T. Jolicoeur, J.C.L. Guillou: Phys. Rev. B, 44, 2403 (1991) E. Arrigoni et al.: Physics Reports, 241, 291 (1994) M. Deeg, H. Fehske, H. Büttner: Z. Phys. B, 88, 283 (1992) L. Lilly, A. Muramatsu, W. Hanke: Phys. Rev. Lett., 65, 1379 (1990) R. Frésard, M. Dzierzawa, P. Wölfle: Europhys. Lett., 15, 325 (1991) E. Arrigoni, G.C. Strinati: Phys. Rev. B, 44, 7455 (1991) B. Möller, K. Doll, R. Frésard: J. Phys. Condens. Matter, 5, 4847 (1993) D. Boies, F.A. Jackson, A.-M.S. Tremblay: Int. J. Mod. Phys. B, 9, 1001 (1995) S.-W. Cheong et al.: Phys. Rev. Lett., 67, 1791 (1991) G.C. Psaltakis, N. Papanicolaou: Phys. Rev. B, 48, 456 (1993) H. Mori, M. Hamada: Phys. Rev. B, 48, 6242 (1993) A.P. Kampf, W. Brenig: J. Low Temp. Phys,. 95, 335 (1994) V. Chechersky et al.: Phys. Rev. Lett., 70, 3355 (1993) M. Matsuda et al.: Phys. Rev. B, 45, 12 548 (1992) G.M. Luke et al.: Phys. Rev. B, 42, 7981 (1990) T.R. Thurston et al.: Phys. Rev. B, 40, 4585 (1990) J. Rossat-Mignod et al.: Physica B, 180 & 181, 383 (1992) V.J. Emery, S. A. Kivelson: Physica C, 209, 597 (1993) V.J. Emery, S. A. Kivelson, H. Q. Lin: Phys. Rev. Lett., 64, 475 (1990) H. Fehske, V. Waas, H. Röder, H. Büttner: Phys. Rev., B 44, 8473 (1991) M. Marder, N. Papanicolaou, G. C. Psaltakis: Phys. Rev. B, 41, 6920 (1990) E.V.L. de Mello: J. Phys. Condens. Matter, 6, 117 (1994) W.O. Puttika, M. Luchini, T.M. Rice: Phys. Rev. Lett., 68, 538 (1992) V. Waas, H. Fehske, H. Büttner: Phys. Rev. B, 48, 9106 (1993) H. Takagi et al.: Phys. Rev. B, 40, 2254 (1989) E.C. Jones et al.: Phys. Rev. B, 47, 8986 (1993) H. Takagi, S. Uchida, Y. Tokura: Phys. Rev. Lett., 62, 1197 (1989) S.J. Hagen, J.L. Peng, Z.Y. Li, R.L. Greene: Phys. Rev. B, 43, 13 606 (1991)
111
7 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81.
112
Slave-Boson Approach to Strongly Correlated Electron Systems
H. Takagi et al.: Phys. Rev. Lett., 69, 2975 (1992) J.C. Campuzano et al.: Phys. Rev., Lett. 64, 2308 (1990) W.E. Pickett: Rev. Mod. Phys., 61, 433 (1989) S.A. Trugman: Phys. Rev. Lett., 65, 500 (1990) H. Chi, A.D.S. Nagi: Phys. Rev. B, 46, 421 (1992) H. Fehske, M. Deeg: Solid State Commun., 93, 41 (1995) E. Dagotto, A. Nazarenko, M. Bonsinsegni, Phys. Rev. Lett., 73, 728 (1994) S. Sarker, C. Jayaprakash, H.R. Krishnamurthy, W. Wenzel: Phys. Rev. B, 43, 8775 (1991) R. Grempel, M. Lavagna: Solid State Commun., 83, 595 (1992) H. Monien, P. Monthoux, D. Pines: Phys. Rev. B, 42, 167 (1990) M. Takigawa: Phys. Rev. B, 49, 4158 (1994) V. Barzykin, D. Pines, D. Thelen: Phys. Rev. B, 50, 16 052 (1994) F. Mila, T.M. Rice: Physica C, 157, 561 (1989) T.E. Mason, G. Aeppli, H.A. Mook: Phys. Rev. Lett., 68, 1414 (1992) H. Fehske, M. Deeg: J. Low. Temp. Phys., 3/4, 425 (1995) J. Rossat-Mignod et al.: Physica C, 185–189, 86 (1991) P. Littlewood, J. Zaanen, G. Aeppli, H. Monien: Phys. Rev. B, 48, 487 (1993) M. Takigawa et al.: Physica C, 162, 853 (1989) P.C. Hammel et al.: Phys. Rev. Lett., 63, 1992 (1989) M. Takigawa et al.: Phys. Rev. B, 43, 247 (1991) Y. Itoh et al.: J. Phys. Soc. Jpn., 61, 1287 (1992) M. Deeg: PhD thesis, Universität Bayreuth, (1995) C. Berthier et al.: Physica C, 235–240, 67 (1994) M. Deeg et al., Z. Phys. B, 95, 87 (1994) H. Fehske, Spin Dynamics, Charge Transport, and Electron-Phonon Coupling Effects in Strongly Correlated Electron Systems, Habilitationsschrift, Universität Bayreuth (1995)
8
Non-Linear Excitations and the Electronic Structure of Conjugated Polymers Klaus Fesser
8.1
Introduction
Conjugated polymers have attracted quite substantial research activities during the last 15 years [1]. On one hand these materials promise interesting technological applications most of which are related to the possibility of a reversible charging and decharging of these systems. These include various battery designs as well as storage of (charged) pharmaceuticals which in turn can be released in a controlled way by application of an electrical current. For the design of non-linear optical components they play an important role as organic materials due to their processibility and fine tuning of their physical properties via suitable side groups. Recently light-emitting diodes made from these systems have made conjugated polymers to possible candidates for the construction of thin displays [2]. On the other hand a theoretical description of the whole class of these materials poses interesting questions which are worth to study on their own right. As essentially quasi onedimensional systems they give rise to the hope that many of these questions might be answered analytically. The main problems are the nature of the insulator-metal transition observed during doping and the origin of the intragap states, which are mainly responsible for the relaxation processes relevant for the light-emitting properties. We have addressed both aspects within this project and this article is organized accordingly. In Section 8.2 we present the theoretical model stressing the relevant physics and discuss the related fundamental symmetries. Then in Section 8.3 we adopt the view that the doping process mainly introduces disorder into these systems and calculate various properties (density of states, optical absorption) from this assumption. In Section 8.4 we investigate the non-linear excitations responsible for the intragap states in more detail and close in Section 8.5 with an outlook on still open problems.
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties Deutsche Forschungsgemeinschaft (DFG) Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
113
8
8.2
Non-Linear Excitations and the Electronic Structure of Conjugated Polymers
Models
Common to all conjugated polymers is the existence of a carbon backbone with an alternating sequence of (short) double and (long) single bonds. The various systems, as polyacetylene, polyparaphenylene, polypyrrole, polythiophene, polyparaphenylenevinylene, etc., differ only – from a physcist’s point of view – in the side groups and other chemical structures which are energetically far away from the Fermi energy. Emphasizing the general aspects of these systems, these chemical details are safely neglected if one restricts to properties which are mainly due to the electrons around the Fermi energy. Thus the parameters in the simple model presented below should be regarded as effective parameters including some aspects of the interactions which are otherwise neglected. Beside some biological systems such as b-retinol where a finite chain of a conjugated polymer is present (and thus the properties of this polymer might be of relevance for the biological functions of this material), there may be other, related systems where similar theoretical concepts are or can be applied. These include the metal-halogen (MX) chains where instead of one, as in the conjugated polymers, now two electron bands govern the essential physics. On the same level the polyanilines can be modelled where the phonons of the conjugated polymers have to be replaced by the librons, i. e. oscillations of the quinoid/benzoid rings. Finally, a true one-dimensional modification of carbon, carbene with alternating single and triple bonds, can also be understood along these lines. So far we have only mentioned the construction of adequate models for the investigation of physical properties. However, there are competing approaches to address the same set of questions. One of these approaches uses sophisticated quantum-chemical codes [1 c,d] to calculate structure and electronic states of these materials. Although such a procedure may be able to reproduce the observed properties of a specific material quite well it is very difficult to obtain information about general trends and physical mechanisms. Therefore we did not follow this route. Another well-established method for calculating ground-state properties, namely the local density functional, has been used to some extent [3]. The drawback of this method, however, is the finite size of the system which can be calculated within a reasonable amount of computer resources. Therefore only a few questions have been addressed via this approach. Therefore, we shall argue in favor of a simple model which is capable to include the most essential physics. Assuming the sp2-hybridisation of the carbon atom leaves one p-electron per atom the others being incorporated into the bonds as s-electrons. It is the physical behaviour of this single p-electron which is responsible for all the interesting effects. Since there is only this one electron per site the polymer would be metal-like in its ground state, in contrast to nature where one finds an energy gap of the order of 1 eV. In one dimension, however, these electrons are unstable when they are coupled to the lattice. Due to the Peierls effect, scattering off 2 kF phonons, a gap opens right at the Fermi energy in accordance with the observations. In addition, electron-electron interaction contribute to the size of this gap [4] stabilizing the semiconductor ground state even further. A single-particle model along these lines has been put forward by Su et al. [5] in the early stages of conjugate polymer research. Its parameters although, derived from a simple picture, should 114
8.2
Models
nevertheless be interpreted as effective parameters including parts of the electron-electron interaction which is otherwise neglected. It has been shown [6] that for some ground-state properties such redefinition can indeed be performed. It turns out that there is a typical length scale in this model vF /D (with the Fermi velocity vF and the electronic gap 2 D) which is considerably larger than the interatomic spacing a. Thus for most physical properties of interest a continuum approximation is valid. This stresses the generic aspects of the whole class of these materials even more. In rescaled variables this model now reads H
XZ s
dx
s
xf
i3 @x 1
xg
s
x
1 2
Z
dx2
x
1
Here c (x) is a two-component spinor describing electrons moving to the left (right) along the one-dimensional polymer, s is the spin index which, except for external fields, is not relevant here. The s3 term is the kinetic energy originating from the hopping of electrons between neighbouring sites. D (x) is the lattice order parameter where a constant D (x) = D0 describes a uniform dimerization of the lattice. The s1 term is the electron-phonon coupling responsible for the Peierls distortion. Finally we have an elastic energy for the lattice. The kinetic energy is neglected within an adiabatic approximation. All quantities have been scaled in order to have a single coupling constant (l & 0.2). At this stage we postpone the non-linear excitations of this model to a subsequent Chapter, we only discuss some of the symmetries of this model which are of relevance also for these excitations. First we note that the form of the electronic part of the Hamiltonian (Eq. 1) has exactly the form of a (relativistic invariant) Dirac operator in one dimension. This correspondence has been exploited [7] successfully in obtaining solutions of this model from results known in models of elementary particles. We remark here that similar analogies can be made for specific forms of the Fermi surface also in higher dimensions. Thus the connection between solid state physics and quantum field theory models can be used for a better understanding of both. In addition there are two symmetries which can be directly related to observable quantities. The charge-conjugation symmetry, which can be expressed as H being invariant under c ? s2 Kc (K complex conjugate operator), relates particle and hole states and thus is responsible for the single particle spectrum being symmetric with respect to the Fermi energy, which has been set to zero in H (Eq. 1). A more hidden property is the supersymmetry of the electronic part Hel (Eq. 1), 0 Hel ; Hel ; 3 :
2
We have shown [8] that this more formal property is responsible for the asymmetry of the optical absorption peaks of transitions involving intragap states. Both symmetries are absent in the real systems under consideration, but for most questions this breaking of symmetries is merely a question of quantity rather than of importance for the existence of these localized states. Finally we mention that Eq. 1 has been derived as a model for polyacetylene where the ground-state order parameters Do and –Do yield the same energy. This is not the case for 115
8
Non-Linear Excitations and the Electronic Structure of Conjugated Polymers
the other polymers of this class. A simple extension of this model, introduced by Brazovski and Kirova [9], corrects this. As a result all the solutions of Eq. 1 can be carried over [10] to this more general model where only the location of the intragap states are now parameters which can be adjusted to a specific material of interest. In consequence the model (Eq. 1) can be considered as the most simple generic model of conjugated polymers.
8.3
Disorder
A major focus of theoretical investigations is the experimental observation that the electronic structure and consequently the physical properties change drastically upon doping. Here we restrict ourselves to the question how these properties are affected through a random force introduced by these impurities. For simplicity we only consider isoelectronic doping, i. e. the total number of carriers remains unchanged and the dopants introduce only scattering centers for the electrons. In consequence two different types (site or bond impurities) of scatterers can be identified. The first gives rise to a random contribution to the on-site energy of an electron whereas the latter modifies the hopping integral between neighbouring sites. In the spirit of the continuum approximation we are thus lead to consider only backward (Eq. 3) or forward (Eq. 4) scattering. Himp Ub Himp Us
PR j
PR j
dx
dx
x1
x
x
x1
x
x
xj ; xj ;
3
4
xj denotes the (random) position of an impurity. Since this problem cannot be solved exactly we have to redraw to approximate methods. We have used three main routes. In the first method we employ the first Born approximation for the impurity self-energy [11]. This enables us to formulate equations of motion for the full space-dependent Green functions and thus consider the influence of disorder on the non-linear excitations as well [12] (see next Chapter). Furthermore, the replacement of Eq. 3 and Eq. 4 by random (Gaussian) fields Z Himp
dx
x
n o 1 Vb=s
x
x ; 1
5
which is correct within the Born approximation, allows the determination of the Green function and higher correlation functions via a functional integral technique [13]. The averaging procedure is formulated through the introduction of additional Grassmann variables in a 116
8.4
Non-linear excitations
supersymmetric way. We note that this supersymmetry is not related to the one discussed earlier. Using the algebraic properties of these variables the average can be performed and the resulting functional integral can be calculated via a transfer method in one dimension. We note that this procedure still works when two coupled chains are considered but cannot be done exactly in any higher dimension. As result we obtain the density of states and also information about the extension of the (in principle) localized wave functions via the Thouless formula. It turns out [14] that for moderate disorder a typical realistic chain length of approximately 200 units is smaller than the localization length of the band states not too close to the band edgesunverständlich. Thus for such systems these states can indeed be considered as extended states even in the presence of disorder putting various models which treat the propagation of electrons along one chain as metal-like on a firmer basis. This method can easily be extended to higher dimensional systems which means that also the coupling of polymer chains can be taken into account. Thus we are able to calculate, within a saddle-point approximation, the optical absorption coefficient for a two-dimensional film. For an orientation along the chains we find [15] a typical disorder induced broadening of the absorption edge together with a shoulder on the low-energy side due to neighbouring chains coupling. Both features agree quite well with experimental findings. Concomitantly, the absorption perpendicular to the chain direction is featureless. In the second method, going beyond Born, we examined the density of states within the coherent potential approximation (CPA) which takes into account multiple scattering processes. One might think that on this level impurity states are introduced in the gap. However, we find [16] that the existence of such localized impurity states strongly depends on the relative strength of site vs. bond impurity. Only states in the gap due to disorder can be found if the site amplitude |Us| is stronger than the bond amplitude |Ub|. Since CPA is an effective medium theory this result might be questionable in one dimension. In the third method, in order to check the CPA result, we performed a numerical simulation [17] where for a given random distribution the electronic eigenvalues have been calculated numerically and the results been averaged over a large number of realizations. It turned out that the CPA results could be reproduced quite satisfactorily thus establishing the different role of both types of impurities.
8.4
Non-linear excitations
Given the single particle states according to Eq. 1 with a uniform order parameter Do one might expect that the lowest excited state corresponds to the lowest level in the conduction band being occupied thus requiring the amount 2 Do in energy, since the half-filled case considered here the valence band with E < –|Do| is full and the conduction band E > |Do| is empty. The model (2.1), however, exhibits the property that an additional carrier modifies the order parameter locally, yielding a non-homogeneous D (x), and at the same time creates through a rearrangement of the band statesadditional state(s) deep in the gap. Mathematically this behaviour is due to the fact that Eq. 1 is a non-linear model, the non-linearity re117
8
Non-Linear Excitations and the Electronic Structure of Conjugated Polymers
sulting from the requirement that the total energy functional AH {D (x)}S has to be stationary with respect to a variation of the order parameter D (x). This gives rise to a self-consistency equation where this order parameter is governed by the occupied electronic states,
x
P
occ
x 1
x :
6
Various exact solutions to this problem are known. The most prominent one is the kink (soliton) for the case of polyacetylene being, p
x o thx= 2 :
7
As already mentioned, this kink does not exist for the other polymers of this class because the ground state is non-degenerate in Do. Therefore the simplest non-linear excitation in a more general sense is the bound kink-antikink pair (polaron) which is characterized by two localized electronic states in the gap at +oo (e. g. oo /Do & 0.5 for polythiophene). In addition there exist periodic solutions, e. g. the kink lattice with a periodicity determined by the concentration of excess charges. Physically these states lower the total energy of the system through an inhomogeneous order parameter D (x), which raises the energy (cf. Eq. 1) and a much larger compensation through the intragap states, which altogether give a smaller value of the total energy than the simple single-particle picture. For the polaron this gain in energy amounts to Ep /2 Do = 0.98 < 1 for polythiophene. For trans-polyacetylene this number is 0.90. One can now envisage the processes which are involved in generating visible light. A sufficiently strong electric field can promote a single electron into the lowest unoccupied conduction band state. This state is unstable and relaxes on a fast time scale (femtosecond) into a polaron-like state which then can recombine to the ground state under the emission of radiation. A full microscopic understanding of all the processes involved is only possible if for the dynamics of the lattice the degrees of freedom are fully taken into account as well as the residual Coulomb interactions. One step in this direction has been made within this project by Bronold [18]. In his doctoral thesis he treats on an equal footing electron-electron (exciton) and electron-phonon (polaron) interactions. The coupling of this system to short laser pulses gives rise to characteristic changes of position and shapes of absorption/emission lines in optically stimulated emission and inverse Raman scattering experiments. As these effects have a very short time scale (femtoseconds), experiments are difficult to perform and a comparison with existing theoretical predictions is not convincing. Nevertheless, one expects that this line of approach will finally give a detailed understanding of the functioning of organic light-emitting diodes. Having gained some insight into the non-linear mechanisms giving rise to localized electronic (intragap) states, how dopants, mentioned in the previous Chapter, might influence these states. Two alternatives are feasible: a) the non-linear aspect dominates, i. e. the picture developed so far is still valid but only some details are modified due to the disorder. On the basis of the Born approximation we have indeed calculated [12] the electronic structure of the kink solution in the presence of impurities and found that the spatial extension of this structure is enlarged when the doping 118
8.5
Perspective
concentration is raised. In accordance with the closure of the gap at a critical concentration we find that the width of the kink tends to infinity at this value. The more interesting case of the polaron, however, could not be solved satisfactorily due to numerical instabilities, which could not be avoided. For details see Ref. [12]. b) the more subtle case of the competition between this non-linear mechanism and the multiple scattering processes off the impurities, which is treated in terms of the T-matrix or the CPA gives also rise to localized states. The experimental observation that the formation of these states is independent of microscopic details leads to the conclusion that the non-linear aspect is the dominant one. A fully self-consistent treatment of this problem with an impurity located at xo and (for simplicity) a kink at x1 gives complicated coupled integral equations [19] which have not been solved. A recent investigation for the case of a kink lattice shows both mechanisms working quite independently, however, the method employedwelche Methode ? does not give a full self-consistent solution.. Summing up this Chapter we note that the non-linear excitations play a dominant role in various physical applications of conjugated polymers. But a full understanding of the interplay of various mechanisms giving rise to these localized states has not been reached yet.
8.5
Perspective
The potential technical applications have stimulated a myriad of experimental and theoretical studies. It is obvious that similar investigations have also been performed, mostly along different lines, by various groups. The disorder aspect, responsible for the observed metalinsulator (or semiconductor) transition, in conjunction with a kink (soliton) or polaron lattice has been treated by many authors [20] in all kinds of approximate approaches. All these studies resulted in the same prediction that such an M-I transition would occur at the experimentally observed dopant concentration level. It is now clear from the foregoing Chapter that only a fully self-consistent treatment will give a satisfactory answer to this question. But since the applications for conjugated polymers envisaged at present focus on optical properties rather than electronic transport this question has been lost out of sight. Still, the nature and dynamics of localized electronic states in these materials must be fully understood. In addition, from an application point of view, these polymers appear to be similar to conventional semiconductors. The recent proposal [21] that the non-linear excitations actually modify the conventional picture, e. g. at the interface between a polymer and a metal (or conventional semiconductor) space charge regions (depletion layers) differ from an inorganic semiconductor, finds renewed interest because transistors made from conjugated polymers are feasible and of interest for the integration of optical and electronic components. From a theoretical point of view the functional integral techniques promise interesting opportunities to make contacts to other areas of research. Universal properties have been dis119
8
Non-Linear Excitations and the Electronic Structure of Conjugated Polymers
covered [22] in the absorption spectrum, discussed earlier, as well as in the distribution of energy levels in certain disordered systems [23]. We propose that there is a close connection between both aspects. A careful treatment of the underlying correlation functions, including a more general type of disorder than discussed here, has to be performed. We expect that the result will lead to new universality classes which technically spoken will show up in different supersymmetric non-linear s models. Work along this line is in progress. In summary, conjugated polymers pose interesting problems, both for applications and pure theoretical studies. Both aspects have matured during the past 15 years but still questions of a more general nature are left unanswered.
Acknowledgements
The author is indebted to all his collaborators for enlightening and stimulating discussions as well as fruitful collaborations. Thanks to A.R. Bishop, F. Bronold, H. Büttner, D.K. Campbell, K. Harigaya, U. Sum, Y. Wada, and M. Wolf.
References
1. a) A.J. Heeger, S. Kivelson, J.R. Schrieffer, W.P. Su: Rev. Mod. Phys. 60, 781 (1988) b) T.A. Skotheim (Ed.): Handbook of Conducting Polymers. Dekker, New York (1986) c) J.L. Brédas, R. Silbey (Eds.): Conjugated Polymers. Kluwer, Dordrecht (1991) d) W.R. Salaneck, I. Lundström, B. Ranby (Eds.): Conjugated Polymers and Related Materials. Oxford University Press, Oxford (1993) e) Proc. Int. Conf. Science Technology of Synthetic Metals: ICSM ’90, Synth. Met. 41–43 (1991); ICSM ’92, Synth. Met. 55–57 (1993); ICSM ’94, Synth. Met. 69–71 (1995) 2. J.H. Burroughes et al.: Nature 347, 539 (1990) 3. P. Vogl and D.K. Campbell: Phys. Rev. Lett. 62, 2012 (1989); Phys. Rev. B 41, 12 797 (1990) 4. For a review see: D. Baeriswyl, D.K. Campbell, S. Mazumdar, in: Conjugated Conducting Polymers, H. Spiess (Ed.), Springer Series in Solid State Sciences 102, 7 (1992) 5. W.P. Su, J.R. Schrieffer, A.J. Heeger: Phys. Rev. B 22, 2099 (1980) 6. D. Baeriswyl, E. Jeckelmann, in: Electronic Propteries of Polymers, H. Kuzmany, M. Mehring, S. Roth (Eds.), Springer Series in Solid State Sciences 107, 16 (1992) 7. D.K. Campbell and A.R. Bishop: Nucl. Phys. B 200, 297 (1982) 8. U. Sum, K. Fesser, H. Büttner: Ber. Bunsenges. Phys. Chem. 91, 957 (1987) 9. S. Brazovskii and N. Kirova: Pis’ma Zh. Eksp. Teor. Fiz 33, 6 (1981) [JETP Lett. 33, 4 (1981)] 10. K. Fesser, A.R. Bishop, D.K. Campbell: Phys. Rev. B 27, 4804 (1983) 11. a) K. Fesser: J. Phys. C 21, 5361 (1988) b) K. Iwano and Y. Wada: J. Phys. Soc. Jpn. 58, 602 (1989)
120
References 12. F. Bronold and K. Fesser, in: Nonlinear Coherent Structures in Physics and Biology, M. Remoissenet and M. Peyrard (Eds.), Springer Lecture Notes in Physics 393, 118 (1991) 13. K.B. Efetov: Adv. in Phys. 32, 53 (1983) 14. M. Wolf and K. Fesser: Ann. Physik 1, 288 (1992) 15. M. Wolf and K. Fesser: J. Phys. Cond. Matter 5, 7577 (1993) 16. K. Harigaya, Y. Wada, K. Fesser: Phys. Rev. Lett. 63. 2401 (1989); Phys. Rev. B 42, 1268 and 1276 (1990) 17. K. Harigaya, Y. Wada, K. Fesser: Phys. Rev. B 43, 4141 (1991) 18. F. Bronold: Doct. Thesis, Univ. Bayreuth (1995) 19. K. Fesser: Prog. Theor. Phys. Suppl. 113, 39 (1993) 20. a) E.J. Mele and M.J. Rice: Phys. Rev. B 23, 5397 (1981) b) G.W. Ryant and A.J. Glick: Phys. Rev. B 26, 5855 (1982) c) S.R. Philpott et al.: Phys. Rev. B 35, 7533 (1987) d) E.M. Conwell, S. Jeyadev: Phys. Rev. Lett. 61, 361 (1988) 21. a) S.A. Brazovskii, N. Kirova: Synth. Met. 55–57, 4385 (1993) b) G. Paasch and T.P.H. Nguyen: unpublished (1995) 22. K. Kim, R.H. Mckenzie, J.W. Wilkins: Phys. Rev. Lett. 71, 4015 (1993) 23. B.D. Simons and B.L. Altshuler: Phys. Rev. B 48, 5422 (1993)
121
9
Diacetylene Single Crystals Markus Schwoerer, Elmar Dormann, Thomas Vogtmann, and Andreas Feldner
9.1
Introduction
Polydiacetylenes can be grown as macroscopic polymer single crystals [1–3]. This property is unique. They comprise one linear polymer axis and can have spatial extensions of up to several millimetres or more in all three spatial directions (Fig. 9.1). The covalent chemical bonds along the polymer axis make them mechanically strong along the corresponding crystallographic axis. Their Young’s modulus is about a quarter of the Young’s modulus of steel [4, 5] and their tensile strength along the polymer axis has been reported to exceed that of steel [6]. Weak bonds of van der Waals-type perpendicular to the polymer axis are responsible for extremely low dimensional – generally one-dimensional – macroscopic electronic properties of these polydiacetylene single crystals. They are insulators, they can become pyro- or ferroelectric, and they show large optical non-linearities. These electronic and optical properties are primarily determined by the p-electron system along the polymer axis. Most polydiacetylenes differ only in the substituents R and R' (Scheme 9.1). But the electro-
Figure 9.1: Photo of macroscopic paratoluylsulfonyloximethylene-diacetylene (TS6) single crystals under polarized light. Monomer (top), polymer (bottom).
122
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties Deutsche Forschungsgemeinschaft (DFG) Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
9.1
Introduction
Scheme 9.1: Polydiacetylene in its isomeric structures: acetylene-type (a) and butatriene-type (b). R and R' are the substituents for different diacetylenes (see also Tab. 9.1).
nic structure of their all-trans planar carbon chain is at first approximation of acetylene-type (a) rather than of butatriene-type (b). Both isomeric structures contain a non-interrupted p-electron system along the carbon chain. The term polydiacetylene is somewhat puzzling, at least for a physicist. However, it becomes quite clear if one takes into account the structure of the diacetylene monomer (Fig. 9.2). A typical substituent R is paratoluylsulfonyloximethylene (Scheme 9.2). The diacetylene with R = R' = paratoluylsulfonyloximethylene is termed TS6 (sometimes TS). It was shown by G. Wegner and his co-workers in a series of works, published in the early 1970s [1, 7], that large molecular crystals can be grown from a solution of TS6, e. g. in acetone, and that these monomer diacetylene crystals can be converted by a topochemical (or solid state) 1,4-addition reaction to the polydiacetylene single crystals (Fig. 9.2). The crystal structures of TS before and after the reaction have been investigated in detail by Kobelt and Paulus [8], Bloor et al. [8], and Enkelmann [9] and are sketched in Fig. 9.3 and in Tab. 9.1.
Figure 9.2: The monomer diacetylene crystal is converted by a topochemical or solid state reaction to the polydiacetylene single crystal.
123
9
Diacetylene Single Crystals
Scheme 9.2: Diacetylene monomer with the substituents R = R' = paratoluylsulfonyloximethylene (TS6).
Figure 9.3: Monomer and polymer crystal structure of TS6 deduced from X-ray data [9]. Note the reactive carbene at the chain end.
Both crystal structures are monoclinic with two monomers or monomer units of different orientation per unit cell (Fig. 9.3 sketches only one of these two.). The chemical bond between two carbons of nearest neighbour diacetylenes (1,4-addition) results in the linear polymer chain and the small change in the lattice parameter along the b-axis prevents the destruction of the macroscopic single crystal during the topochemical reaction. Several surfaces of diacetylene crystals have been studied by atomic force microscopy (AFM) in order to investigate both, the single crystal surface structure and solid state reactions at the surface [129, 131]. 124
9.1
Introduction
Table 9.1: TS monomer and polymer crystals (monoclinic, space group P21/c) [9].
TS monomer TS monomer TS monomer TS polymer TS polymer
T/K
a/Å
b/Å
c/Å
b/ 8
Dx g/cm3
120 295 295 295 120
14.61(1) 14.60 14.65(1) 14.993(8) 14.77(1)
5.11(1) 5.15 5.178(2) 4.910(3) 4.91(1)
25.56(5) 15.02 14.94(1) 14.936(10) 25.34(2)
92.0(5) 118.4 118.81(3) 118.14(4) 92.0(5)
1.46 1.40 1.40 1.483 1.51
The topochemical reaction can be induced thermally and/or photochemically and/or by electron-beam irradiation. For TS the thermal conversion versus time (Fig. 9.4) is strongly temperature dependent and highly non-linear. The thermodynamics of the integral reaction has been investigated extensively by Bloor et al. [10], Eckhardt et al. [11], Chance et al. [11], and others. The reaction diagram (Fig. 9.5) for TS shows that the dark reaction is thermally activated and has an activation energy of 1 eV per monomer. It is exothermic with a polymerization enthalpy of 1.6 eV per addition of one monomer. The entire reaction is irreversible and the TS6-polydiacetylene (PTS) crystals are not solvable in ordinary solvents. During the solid state reaction almost all properties of the diacetylene crystals change drastically, e. g. the transparent monomer crystals are converted to polymer crystals with highly dichroic, strongly reflecting surfaces which contain the b-axis. In transmission the polydiacetylene crystals can only be investigated as thin films (Fig. 9.6). Their absorption spectrum, e. g. for TS, clearly shows the vibronic spectrum due to the single, double and triple bonds of the polymer chain [12]. Batchelder et al. [13] investigated extensively the optical absorption, reflection, and Raman spectra of TS single crystals. While these experiments were directed towards the study of the electronic excitations of the bulk and their coupling to the vibrations, Sebastian and Weiser [14] investigated the
Figure 9.4: Time conversion curves for the thermal polymerization of PTS at 60 8C (.), 70 8C (#), and 80 8C (d) [9].
125
9
Diacetylene Single Crystals
Figure 9.5: Reaction diagram for the thermal polymerization of TS (solid curve) and photopolymerization of 4BCMU (dashed curve) [10, 11].
Figure 9.6: Absorption spectra of a thin TS diacetylene single crystal for light polarized parallel and perpendicular to the polymer axis b (T = 300 K). Monomer (M) and polymer (P) absorption.
defects by electroabsorption. As an example Fig. 9.7 shows the absorption and the electroabsorption spectra, which they have analyzed in great detail. Since about 1980 and especially during our work for the Sonderforschungsbereich 213 we have synthesized several new diacetylenes, the substituents of which are shown in Tab. 9.2 [15]. For selected diacetylene single crystals we have investigated: 126
9.1
Introduction
Figure 9.7: Absorption a and electroabsorption Da of photoproducts in PTS-monomer as a function of ko. Full curve: experimental spectra, dashed curves: fit by Lorentzian and a charge transfer model for Da [14].
a) in Section 9.2 the elementary steps and structures during the solid state photopolymerization by transient optical spectroscopy, by electron spin resonance (ESR), and by electron nuclear double resonance (ENDOR); b) in Section 9.3 the application of the photopolymerization of thick diacetylene single crystals as a very effective holographic storage process; c) in Section 9.4 the tailoring of diacetylenes as ferro or pyroelectric crystals, which do not demand considerable efforts for the poling processes and which show good thermal stability; d) in Section 9.5 the optical non-linearity of second and of third order and their application for an optical device with femtosecond time resolution. The present paper is a review of our work with diacetylene single crystals.
127
9
Diacetylene Single Crystals
Table 9.2: A survey of the investigated diacetylenes [15]. Survey of the investigated diacetylenes R1
C
C
C
C
R2 ability to polymerize
Name
R1
DNP
O2 N CH2 O
TS
CH2 O SO2
PD-TS
CD2-TS
FBS
IPUDO
therm.
γ
= R1
+++
-
= R1
+++
+++
= R1
+++
+++
= R1
+++
+++
= R1
+++
+++
=R
-
+++
-
+
-
+
+
+
R2 NO2
CH3
D CD2 O SO2 D H CD2 O SO2 H
D CD3 D H CH3 H F
CH2 O SO2
CH3 O (CH 2 ) 4 O C NH CH CH3
1
NP/PU
CH2 O
NO2
O CH2 O C NH
NP/4-MPU
CH2 O
NO2
O CH2 O C NH
NP/MBU
CH2 O
NO2
DNP/MNP
O2 N CH2 O
NO2
O H CH2 O C NH C CH3 (-)(S) or (+)(R) O2 N CH2 O CH3
DNP/PU
O2 N CH2 O
NO2
O CH2 O C NH
DNP/4-MPU
O2 N CH2 O
NO2
O CH2 O C NH
DNP/DMPU
O2 N CH2 O
NO2
DNP/MPU
O2 N CH2 O
NO2
TS/FBS
CH2 O SO2
CH3
CH2 O SO2
F
+++
CH2 O SO2
F
CH2 O SO2
CF3
+++
FBS/TFMBS
128
CH3
+++
-
-
CH3
-
-
CH3 O CH2 O C NH CH3
-
-
-
-
O H CH2 O C NH C CH3 (-)(S) or (+)(R)
9.2
9.2
Photopolymerization
Photopolymerization
9.2.1 Carbenes The aim of this Chapter is to review our spectroscopic work towards the analysis of both, the electronic structures and the dynamics of the intermediate reaction products (Fig. 9.2), during the photopolymerization, i. e. after the excitation of the solid state reaction by light. For the entirely thermally activated solid state polymerization detailed spectroscopy of the intermediate states turned out to be difficult or not very efficient. One exception was the identification of carbenes as reactive species during the thermal solid state polymerization of TS6 (Fig. 9.8). In contrast to radicals with one non-bonded electron carbenes have two non-bonded electrons. It has been shown for the first time by Wassermann et al. [16] that the electronic ground state of pure methylene (:CH2) is a triplet state, where the total spin quantum number S is 1. Because of their non-centrosymmetry most molecular triplet states show a splitting into three components even in the absence of an external field. This splitting is due to the R
R
C
R
C
C
C
R
C
C
C
C
C
C
R
R C
C
R
R
R
R
C
R C
C
C
C
R
C
C
C R
R
2hν
C C
R
C C
C R C
R
kT
C C
R
C C
C C
R
kT
C
C
R
R
R
R
R
kT
C R
C
C
R
R
R
R
R
R
R
C
R
R
R
C
C
R
C C
C
R
C
R
C
R
C C C
C
C
C R
R
C R
C C
C
C
C
C C
R
C
C
C
C R
R
C
C
R
C
C
C R
C C
R
R
kT
C
C
R
C C
C
R
C R
C
C C C
C
R
C
C
C
C
C
C R
R
C
C
C
C C
R
kT
C
C
C
C C
R
C
C
C
R
C
C
C
C
C
C C
R
R
C
C
R
C
C
C
C
C C
C
C
C
C
C R
R
C R
R
C
C
C
C R
C
R
C
C
C
C
C C
R
C
C
C C
C
R
R
kT
C C
C C
R
C
C
R
C C
R
C
R
C C
C C R
R
R
C R
C C
C R
C C
C
C C
C
C R
C
C
R
R
C C
C
C R
C C
C
C
C
C
R
R
C
C
C R
C
C R
C C
R
C
C
C C
C
R
C
C
R
R
C
C
R
R
C
R
C
C
C
R
C
C
C C
R
C R
C
C C
R
C C
C
C
C R
C C
R
C R
C
C
R
C
R
R
C
C R
C C
R
R
C
C C
R
C C
R C
C C
R
C
R
C
C R
C R
C
C
R
C
R
C C
C
C C
R
C C
R
R
R
C C
C
C C
R
R
C
C
C C
R
R
C
C
C C
R
C
C
C R
R
C
C
R
R
Figure 9.8: UV photopolymerization of TS6: The monomer crystal is irradiated with an UV-flash. The dimer is formed from the monomer by a photoreaction, then a series of thermally activated monomer addition reactions leads via the diradicals (DR) and dicarbenes (DC) to the polymer.
129
9
Diacetylene Single Crystals
magnetic dipole-dipole coupling of the two unpaired electrons and is called zero-field splitting. In an external magnetic field the resulting fine structure of the ESR spectra is highly anisotropic, i. e. it strongly depends on the direction of the external magnetic field with respect to the orientation of the tensor describing the magnetic dipole-dipole interaction. This interaction is of course an intramolecular property and therefore single crystals are ideal candidates for the measurement and the quantitative analysis of molecular triplet states by ESR. As originally shown by the work of Hutchison and Mangum [17], and van der Waals [18] the ESR spectrum of a molecular triplet state is described by the spin Hamiltonian THs [19] T
^
^
^
Hs B Bo g S D S 2z E
S 2x
^
S 2y ;
1
where mB is Bohr’s magneton, B0 the external magnetic field, g the spectroscopic splitting factor, S^ the spin operator for a spin with total spin quantum number S = 1, S^ u with u = x, y, z are the components along the principal axes of the magnetic dipole-dipole interaction tensor. D and E represent the two independent values of this tensor, the trace of which is zero. They usually are called zero-field splitting parameters: D
o 3 2 2 r2 z2 g B h i; 4p 4 r5
2
E
o 3 2 2 y2 x2 g B h i; 4p 4 r5
3
r is the distance of the two unpaired electrons and x, y, and z are the components of r along the principal axes of the magnetic dipole-dipole interaction tensor of these two electrons. Only for systems of cylindrical symmetry (around z) the zero-field splitting parameter E vanishes (E = 0). And only for spherical symmetry (Ax2S = A y2S = Az2S = 1/3 Ar2S) the entire fine structure is zero. The values of D and E for the triplet carbenes as detected during the purely thermal solid state polymerization [20] are shown in Tab. 9.3 in comparison with typical values of different molecular triplet states. By these values and especially by the strong anisotropy of the fine structure in the ESR spectrum these carbenes (as sketched in Fig. 9.8) are clearly identified as reactive species during the pure thermal solid state polymerization of TS. They will play a major role in the electronic structures of the intermediate products during the photopolymerization as described in the following paragraphs. Table 9.3: Zero-field splitting parameters for the reactive species during the thermal solid state polymerization of TS [20], pure methylene [16], diphenylmethylene [21], and benzene [22] in their first excited triplet state.
TS :CH2 :C(C6H5)2 C6H6
130
D = cm –1 hc
E = cm –1 hc
0.2731 0.6636 0.39644 0.1581
–0.0048 0.0003 –0.01516 –0.0046
9.2
Photopolymerization
9.2.2 Intermediate photoproducts Further experiments for analyzing the electronic structure of intermediate states were not very successful, until Sixl et al. [23] and Bubeck et al. [24] published their first low-temperature spectroscopic experiments on partially photopolymerized TS crystals. Thereby, the monomer crystal is cooled to 4.2 K in the dark. Then it is irradiated with UV light (l ^ 310 nm) for a short period. After this procedure a large number of different species show up in both, the optical absorption and the ESR spectrum. They persist at helium temperature. Subsequent annealing in the dark produces further intermediate reaction products which are also identified by their optical or ESR spectra. Finally, further irradiation with visible light, which is absorbed only by the intermediate reaction products, produces still more and different reaction products. In a comprehensive series of investigations [25–47] most of these intermediate products have been identified and classified. Moreover, the mechanisms of their production and their reaction kinetics have been analyzed. According to Sixl, there exist three different series of intermediate products: 1. The diradical-dicarbene series: DR2, DR3, DR4, DR5, DR6, DC7, DC8, DC9, DC10, DC11 … polymer; 2. The asymmetric carbene series (AC); 3. The stable oligomer series (SO). The DR-DC series leads directly from the monomer to the polymer. It is initiated and processed in the following simple and clear way. The monomer crystal is irradiated with an UV flash and subsequently rests in the dark. The flash excites the monomers and produces dimers (DR2). These react by a thermally activated step by step addition of monomers, as illustrated in Fig. 9.8. In the following paragraphs we will show for a few selected examples how these results have been achieved and we will present details of both, the electronic structures and the dynamics. The AC and the SO series are produced by additional irradiation with light, i. e. these are photoproducts which do not necessarily arise during the solid state polymerization of TS6. Although they are an important part of the entire variety of structures in partially polymerized TS6 crystals, we will not treat them in this review. They have been described extensively by Sixl in his papers cited above and in Ref. [48].
9.2.3 Electronic structure of dicarbenes 9.2.3.1
Electron spin resonance of quintet states (5DCn)
As an example for the electronic structure analysis of the intermediate states with ESR, we will review below the ESR spectra of the dicarbenes DC7 … DC13 [28, 32, 42]. They are characterized by their fine structure and temperature dependence. Figure 9.9 shows the ESR 131
9
Diacetylene Single Crystals
T=10 K ν =9,46 GHz
T Q Q
ESR-Signal
T Q
T
Q
Q T
0
Q
Q
T
Q
T
200
400
600
Magnetic Field B0 /mT Figure 9.9: The ESR spectrum of perdeuterated TS after irradiation for 1000 s. The signals marked with T arise from triplet states and those with Q from quintets. The T-lines are microwave saturated. The magnetic field B0 is oriented parallel to the z-axis of the quintet fine structure tensor [28].
spectrum of a perdeuterated TS crystal which has been irradiated with UV light (313 nm) of a mercury high-pressure arc lamp (HBO 200) at 4.2 K for about 1000 s. All ESR lines in Fig. 9.9 labelled with Q are due to dicarbenes in their quintet state (S = 2). Prior to the UV irradiation no ESR signal is observed. The temperature dependences of the ESR signals (Fig. 9.10) show one common feature: the ESR intensities vanish for T ? 0, i. e. the ESR signals are thermally activated and the ground state is spinless. But the temperatures for the maximum intensities are different for each ESR signal, indicating different activation energies.
10
Signal Intensity
1
5
T/K
2
0,5
0
0
0,2
0,4
1/T / K -1
0,6
0,8
Figure 9.10: Temperature dependence of the ESR intensities of dicarbenes DC9, DC10, and DC11. The calculated lines have been fitted by Eq. 9 with the activation energies De*SQ for the quintet states [33, 47].
132
9.2
Photopolymerization
The resonance fields for fixed microwave frequencies are strongly anisotropic. Figure 9.11 shows as an example this anisotropy for the dicarbenes DC9, DC10, DC11, and DC12, respectively. The external field has been rotated to the polymer backbone plane. Each anisotropy belongs to one distinct temperature dependence. The following model (Fig. 9.12 and inset Fig. 9.13) describes quantitatively the anisotropic fine structure and the temperature dependence. The ground state (|S>) of each dicarbene in first order is a singlet state with the total spin quantum number S = 0. The excited state (|Q>) in first order is a quintet state with S = 2. The excitation energy is DeSQ . The quintet state is built up by the electronic coupling of two triplet carbenes at both ends of the oligomer via exchange interaction. R12 is the distance between the two triplet carbenes. If the singlet-quintet splitting DeSQ is large as compared to the magnetic dipole-dipole coupling (zero field splitting) within the triplet carbenes, the total spin quantum numbers S = 0 and S = 2, respectively, are good quantum numbers, i. e. the quintet state is pure. The spin Hamiltonian for this case has been analyzed by Schwoerer et al. [34].
Figure 9.11: Angular dependence of the resonance fields B0 of the 4 dicarbene structures DC9, DC10, DC11, and DC12 in perdeuterated TS-diacetylene crystals. The crystal is rotated so that the external magnetic field B0 is in the plane of the polymer backbone. The b-axis is the direction of the polymer chain, y and z are the principal axes of the fine structure tensor. The curves are fitted to the experimental points by computer calculations. A and B indicate the two magnetically equivalent directions of the molecular orientation within the monoclinic unit cell. Dots: experimental values; lines: calculated by Q 0 HS (Eqs. 4–7) [32, 42].
Figure 9.12: Dicarbene configuration. The dicarbene molecule consists of n diacetylene units with two identical triplet carbene chain ends. Dt and Et are the triplet fine structure parameters of the S = 1 carbene species. j gives the orientation of the triplet fine structure z-axis with respects to the crystal baxis.
133
9
Diacetylene Single Crystals
Figure 9.13: Experimental values for the singlet-quintet splitting DeSQ for seven different dicarbenes DC7 … DC13 [42].
If DeSQ is smaller than or somewhere in the order of the magnetic dipole-dipole coupling then the singlet and quintet states are mixed. The spin Hamiltonian QHS for this general case has been derived in the elegant work of Benk and Sixl [35]: Q
1 HS gB B0
S^ 1 S^ 2 "SQ
S^ 1 S^ 2 2 HDD S 6
4
The first term of Eq. 4 represents the electronic Zeeman term, S^ 1 and S^ 2 the spin operators for two triplet carbenes, and the second term represents the electronic exchange interaction. If S^ 1 and S^ 2 couple to a quintet state (S = 2), (S^ 1 + S^ 2 )2 = S2 = S (S + 1) = 6. If they couple to a singlet, S = 0. Therefore, this term directly results in the energy level scheme, indicated in the inset of Fig. 9.13. The pure singlet and the pure quintet states are split by DeSQ which turns out to be the characteristic property of each dicarbene. The third term of Eq. 4 represents the magnetic dipole-dipole coupling of the two triplet carbenes: ^
2 HDD S D
S1z ^
^
^ 1 2 S1 E
S21x 3 ^
^
X S 1x S 2x XA S 1y S 2y
^
^
S21y D
S22z ^
^ 1 ^2 S2 E
S22x 3
^
^
^
S22y ^
^
^
X
1 A S 1z S 2z Xa
S 1z S 2y S 1y S 2z
5
D and E are the fine structure parameters of the identical triplet carbene chain ends; x, y, and z are the principal axes of the corresponding fine structure tensors. The intercarbene magnetic dipolar interaction is represented by the parameter 134
9.2
Photopolymerization
X g2 2B
0 =4pR123 :
6
The geometrical factors A and a are only dependent on the orientation j of the fine structure tensor z-axis with respect to the b-axis of the crystal. A1
3 sin2 ';
3 sin 2' : 2
7
The diagonalization QHS of yields the allowed ESR transitions, their resonance fields, and their dependence on the orientation of the external magnetic field B0. At a first sight, the number of fit parameters seems to be high: D, E, g, the orientation of the triplet fine structure tensor, R12, and DeSQ. But besides R12 and DeSQ these parameters are well-known from earlier ESR experiments on triplet carbenes [20, 49, 50]. Therefore, R12 and DeSQ are the only free to fit parameters. Figure 9.11 shows the result of the fit. The ESR anisotropy was calculated by exact diagonalization of QHS with the fitting parameters R12 and DeSQ. In all cases the fit is almost perfect. Furthermore, it turns out that the intertriplet magnetic dipole-dipole interaction does not influence the results if R12 exceeds 12 Å. Therefore, only the parameter DeSQ remains to fit for each dicarbene. The obvious differences between the anisotropies of the different dicarbenes (Fig. 9.11) are due to DeSQ only! The result is shown in Fig. 9.13 where DeSQ is decreasing exponentially with increasing number n of monomer units: "SQ "0SQ e
R=R0
;
8
with R = n 1m (1m is the length of the monomer unit within the oligomer). The slope in Fig. 9.13 yields R0 = 5.4 Å & 10 a0 (a0 = Bohr radius). The origin of the abscissa in Fig. 9.13 was deduced by Neumann [47]. But even without the knowledge of this origin the value of R0 as compared to a0 shows that the triplet carbene is highly delocalized and not at all restricted to the end of the oligomer, as one might think because of Fig. 9.12. The model also gives the temperature dependencies of the dicarbene signals. Because in first order the ground state is spinless, the ESR intensities I of the quintet states (beyond Curie’s law) must be thermally activated: I /
1 5 exp
"SQ =kT : T
9
The lines in Fig. 9.10 are calculated with Eq. 9 by fitting De*SQ. For the dicarbenes DC8, DC9, DC10, and DC11, both values, De*SQ and DeSQ, are determined respectively. Within the experimental uncertainty they are identical [32, 42]. This latter result is an excellent proof for the dicarbene model because De*SQ and DeSQ were determined from two completely different properties: the anisotropy of the fine structure and the intensity of the ESR spectra. We therefore have no doubt that we really did observe dicarbenes (as sketched in Figs. 9.8 and 9.12). They are an ideal modelling substance for short linear oligomers of diacetylenes which are perfectly oriented in the single-crystal lattice. The longest dicarbene (DC13) observed has according to our model a length of (R = 1364,9 Å) 64 Å! 135
9 9.2.3.2
Diacetylene Single Crystals
ENDOR of quintet states
The most important result of the preceding Chapter is the delocalization of the spins S = 1 of the two triplet carbenes which is necessary for their coupling over a distance of up to 64 Å (!) to the well defined quintet states. It was therefore attractive to measure and analyze the ENDOR spectrum of at least one quintet dicarbene. ENDOR should detect at least the protons of the CH2 groups of the substituents if the nuclear spins of these protons are hyperfine coupled with the electron spin of the triplet carbene. As compared to ENDOR with an electron spin of S = 1/2, the complication is the high anisotropy of the five electronic Zeeman levels Qu (u = 1 … 5) of the quintet state (Fig. 9.14). Not only their energy separation, i. e. the ESR transition fields (Fig. 9.11), are strongly dependent on the direction of the external field. Also their effective spin Seff, i. e. the expectation value of the spin, is dependent on the direction and on the strength of the external field B0. Therefore in this case the orientation quantum number ms is an unsuitable quantum number not only because of the large zero field splitting, as expressed by D, but also because of the singlet-triplet mixing, as expressed by DeSQ . As described in the preceding paragraph this problem has been solved with high accuracy, Hartl et al. [36] measured the ENDOR spectra and their anisotropies for one quintet dicarbene (5DC10), which is accessible most comfortably at T = 4.2 K (Fig. 9.10) and for which the total spin quantum number (S = 2) is a good quantum number. The aim of these experiments was to determine the hyperfine coupling constants with the above-mentioned protons and subsequently to extract the electron spin density from these values, i. e. the delocalization of the triplet carbene quantitatively.
Figure 9.14: Quintet state with an external field Bo interacting with one proton (I = 1/2). All allowed ESR transitions (a–d) and NMR transitions (1–5) are shown. By ENDOR, in first order, only the two NMR transitions directly connected to the observed ESR line are detectable [36].
136
9.2
Photopolymerization
The spin Hamiltonian QHS;i of a quintet dicarbene coupled to one individual proton, numbered i, is Q
^ i Ii : Hs;i Q H0s gI K B0 I^ i SA
10
The first term in Eq. 10 is given by Eq. 4 and the second is the nuclear Zeeman energy. I^ i is the nuclear spin operator for nuclear spin 1/2. The third term is the hyperfine interaction of the individual proton i, as defined by the hyperfine tensor Ai . S^ is the total electron spin operator, S^ S^ 1 S^ 2 . As nuclear dipole-dipole interaction can be neglected we will omit the index i in the following. The nuclear terms in Eq. 10 are small as compared to the electronic terms. Therefore, we treat them in first order perturbation theory, taking the solutions of QH0S as basis. QH0S has five quintet eigenstates |Qu S , u = 1, 2, 3, 4, 5. For very high fields they become the high field states |Qm S, for which S^ z |Qms S = kms |Qms S, ms = +2, +1, 0, –1, –2. But for the fields used in ordinary ESR spectrometers the electronic Zeeman energy is in comparison to the fine structure not large and therefore the electron spin is not quantized along the external field B0 [39]. This results in a strong |Qu S-dependence on the direction of B0 with respect to the crystal axes. From QH0S the effective spin D E ^ jQu Sueff Qu j S
11
can be calculated exactly [34]. It is this electron spin which interacts via A with the proton spin. The first order perturbation theory of the nuclear terms calculates the shift Dnu of the individual proton Larmor frequency with respect to the free proton Larmor frequency nF (ENDOR shift): hDnu = hnu – gI mK |B0 | ,
(12)
(gImK |B0 |/h = nF). The result of the calculation of the Larmor frequencies nu of the hyperfine coupled protons is [36, 43]: hu jSuef f A
gI K B0 j :
13
For a quintet state one should observe 5 different ENDOR lines per proton. This is illustrated in Fig. 9.14 where the observed NMR transitions in the ENDOR experiment are indicated by the ciphers (n) = (1), (2), (3), (4), and (5); the first order ESR transitions are indicated by the lower case letters (a), (b), (c), and (d). Fig. 9.15 shows four ENDOR spectra as detected via the ESR transitions (a), (b), (c), and (d), respectively. Four protons i = 1, 2, 3, and 4, respectively, are clearly separated from the free proton frequency nF. In the vicinity of nF a large number of weakly coupled protons are visible. They also have been resolved by expansion of the NMR frequency scale. For a few strongly coupled protons we are able to detect all five NMR transitions (1) to (5). This is a further and definite proof that we do observe quintet states. 137
9
νF
3(2)
Diacetylene Single Crystals νF
2(2) 1(2)
(c) Bo II Y νF = 14.193 MHz
(a) Bo II X νF = 9.570 MHz 2(1)
1(4)
3(4)
2(4)
4(4)
0
νF
(b) Bo II Y νF = 10.911 MHz
4(2)
0
10
40
30
20
10
3(2)
2(2)
0
2(4)
(d) Bo II X νF = 15.553 MHz
4(4) 3(4)
1(2)
20
NMR Frequency ν/MHz
30
20
10
νF
30
0
8
12
16
20
Frequency ν/MHz NMR ν / MHz FREQUENCY NMR
Figure 9.15: ENDOR spectra as detected by the ESR transitions (a–d); nF is the free proton frequency. (1) to (5) label the NMR transitions illustrated in Fig. 9.14, and 1, 2, 3 … number the individual protons. The external field Bo is oriented along the y or x-axis of the fine structure tensor [36].
The ENDOR shift anisotropy is shown in Fig. 9.16 for the strongly coupled protons i = 1, 2, 3, and 4. This anisotropy is mainly due to the anisotropy of the effective spin Sueff . The lines were calculated (via Eqs. 12 and 13) by fitting Ai . In total we have analyzed 22 protons, the hyperfine tensors Ai of which are presented in Tab. 9.4. Two features can be
Figure 9.16: ENDOR shift anisotropies for the four strongest coupled protons i = 1, 2, 3 and 4, detected via the ESR transition (b). x, y and z are the principal axes of the fine structure tensor. The experimental values were taken for the rotation of B0 in the yz-plane and in the zx-plane, respectively. The ENDOR shifts Dn are calculated for (b)i(2) transitions (drawn out) and the (b)i(3) transitions (dashed), respectively [36].
138
9.2
Photopolymerization
Table 9.4: Complete hyperfine tensors for 22 protons, calculated by fitting the experimental anisotropy in a least-squares method. The Aij are diagonalized principal values of the fitted tensor, a is the isotropic coupling constant, Bjj the dipolar anisotropic tensor, F, Y, and c are Euler angles of the hyperfine tensor axes relative to the fine structure axes [36]. i
Axx MHz
Ayy MHz
Azz MHz
a MHz
Bxx MHz
Byy MHz
Bzz MHz
F degr.
Y degr.
C degr.
Assignment
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
16.298 11.870 2.348 1.905 1.788 0.407 1.456 0.700 –0.791 –0.992 0.219 0.098 –0.819 0.075 1.969 –0.066 –0.456 2.350 –0.311 –0.777 –2.070 –1.072
18.953 14.290 3.737 1.797 0.710 0.804 0.089 0.620 0.981 1.424 –0.079 0.161 1.321 –0.120 –1.045 –0.165 –0.051 –2.533 –0.237 –0.289 1.303 –0.735
16.747 10.683 5.667 1.253 –0.391 0.703 –0.119 0.076 0.520 0.148 0.400 0.271 –0.275 0.239 –0.889 0.235 –0.203 –0.473 –0.442 –0.143 –1.288 –0.652
17.333 12.281 3.917 1.652 0.702 0.638 0.475 0.465 0.237 0.194 0.180 0.177 0.076 0.065 0.012 0.002 –0.203 –0.219 –0.330 –0.403 –0.685 –0.820
–1.035 –0.411 –1.569 0.253 1.086 –0.231 –0.981 0.234 –1.028 –1.185 0.039 –0.079 –0.895 0.011 1.958 –0.067 –0.254 2.569 0.019 –0.374 –1.385 –0.252
1.620 2.009 –0.180 0.146 0.007 0.166 –0.387 0.155 0.744 1.230 –0.259 –0.016 1.245 –0.185 –1.057 –0.166 0.254 –2.314 0.093 0.114 1.988 0.085
–0.586 –1.598 1.749 –0.399 –1.093 0.065 –0.594 –0.389 0.284 –0.045 0.220 0.094 –0.351 0.174 –0.901 0.234 0.000 –0.254 –0.112 0.260 –0.603 0.168
13.6 –39.2 28.0 74.9 56.0 10.3 26.3 57.4 –23.4 –23.2 –67.5 21.5 –6.7 12.6 –68.2 –24.1 91.1 –20.4 –23.5 –36.0 6.0 4.7
–51.8 76.4 80.5 101.2 –16.4 10.1 –25.4 –15.3 –80.7 1.6 111.6 90.6 –5.7 7.3 –7.4 –5.2 114.5 9.5 91.3 –30.2 –49.3 18.6
2.8 –12.8 –6.9 4.6 –21.8 –36.9 –4.4 8.0 –15.4 –67.2 41.8 –25.7 –6.0 –38.9 49.5 38.5 27.5 –53.6 89.0 18.1 6.5 –27.0
CH2 CH2 CH2 CH2 CH2 CH2 CH2 CH2 CH2
CH2 ARYL
ARYL ARYL CH2 CH2
extracted from Tab. 9.4 immediately: first, the anisotropic part of the hyperfine tensor is small as compared to the isotropic part for almost all protons, and second, several protons (i = 17, 18, 19, 20, 21, 22) show a negative value of the isotropic coupling constant a. We are able to assign 6 protons unambiguously: i = 1, 2, 3, 13, 21, and 22 (Fig. 9.17). The analysis of the hyperfine data shows that the spin density at C1 (Fig. 9.18) is only 11%, –2,2 % at C2, 17 % at C3, –6 % at C4, 7,9 % at C'1 and –2 % at C'2, i. e. the spin is highly delocalized within the oligomer (Fig. 9.18). This corresponds with the above-described exchange coupling of the two carbenes and is to our opinion a very impressive demonstration of the power of ENDOR.
9.2.3.3
ESR and ENDOR of triplet dicarbenes 3DCn
For a long time it was unclear whether the triplet state (S = 1) of the dicarbenes (3DC) does exist, and if it does whether its energy is higher or lower than the energy DeSQ of the dicarbene quintet state (5DC). Müller-Nawrath et al. [51] have shown theoretically and experimentally that the ESR transitions of the triplet states of carbenes (3C) and of dicarbenes (3DC), respectively, are mutually degenerated in diacetylene oligomers of diacetylene crystals if the 139
9
Diacetylene Single Crystals
Figure 9.17: Assignment of 6 hyperfine tensors to methylene protons at the polymerization head. The arrows indicate the directions of the strongest main value of the respective hyperfine tensors. These 6 tensors have been fitted simultaneously yielding the spin density distribution shown in Tab. 9.4 [36].
20
17
15 11 10
7,9
5 0 -5 -10
-2
C2'
-2,2 C1'
-6 C4
C3
C2
C1
Figure 9.18: Carbene spin density at the reactive polymer chain end (Fig. 9.17).
oligomers are long, i. e. if the number of added monomers n is more than 7. Therefore 3C and 3DC cannot be discriminated by ESR. Their ENDOR spectrum, however, is completely different. The ENDOR shifts Dn are related by DnDC = –1/2 DnC. The existence of several ENDOR lines, which fulfills the characteristic factor –1/2 in the above-mentioned relation, unambiguously shows the existence of the triplet state of dicarbenes. Its excitation energy is lower than the excitation energy DeSQ of the quintet states of the same dicarbene [51].
140
9.2
Photopolymerization
9.2.4 Flash photolysis and reaction dynamics of diradicals A single UV laser flash with wavelength l = 308 nm, pulsewidth 15 ns, and flash energy 1 mJ initiates photopolymerization by the production of the diradical DR2 [52]. At low temperature, i. e. 4.2 K, this photoproduct is stable. It is detected by its absorption spectrum. In the spectral range the 0-0 transition peaks at 422 nm between monomer and polymer absorption (Fig. 9.20 a). Annealing the crystal containing photoproduct DR2, prepared as described above, produces the diradicals DR3, DR4, DR5, and DR6 respectively by addition of one monomer per step (Fig. 9.8). All these diradicals can be detected by optical absorption spectroscopy, for example by cooling the crystals to low temperature in order to slow down the dark reaction (Fig. 9.20 b). Sixl et al. [48] were the first who detected these optical absorption spectra [131]. The dark reaction at low annealing temperatures has been investigated extensively by Gross [38, 46]. If the reaction is photoinitiated by a single UV laser flash at high temperatures (T > 180 K) the entire time-dependent reaction series, DR2 ? DR3 ? DR4 ? DR5 ? DR6, can be observed by monitoring the transient optical absorption of each of the products DR2 to DR6. By this experiment we were able to analyze the reaction kinetics of these thermally activated steps separately [37, 44]. As an example Fig. 9.19 shows the transient absorption
0.23
T=270 K DR2 422nm
0 0.23
DR3 514nm
0 0.29
∆OD
DR4 578nm
0 0.5
DR5 664nm
0
0
10
t / µs
20
Figure 9.19: Time sequence of the intermediate products DR2, DR3, DR4, and DR5 at T = 270 K after the UV flash which is indicated by an arrow. DOD is the change of the optical density after the UV flash [37, 44].
141
9
Diacetylene Single Crystals
Figure 9.20 a: Difference of the optical absorption spectra (DOD) of TS6 after and before one single UV pulse (l = 308 nm, t = 15 ns, E = 0.1 mJ, T = 80 K). The peak at 422 nm is due to the absorption of the Dimer DR2.
Figure 9.20 b: Appearance of DRn reaction intermediates after a single pulse irradiation at 308 nm and additional annealing. (a): 5 K, 0 min; (b): 100 K, 6 min; (c): 100 K, additional 30 min plus 120 K, 36 min; (d): 130 K, 36 min plus 140 K, 16 min [131].
at T = 270 K as detected by the change of the optical density (DOD) after the flash vs. time. Each intermediate product is detected at the maximum of its optical absorption: DR2 at 422 nm, DR3 at 514 nm, DR4 at 578 nm, and DR5 at 664 nm. The delay in the production of a subsequent intermediate is clearly demonstrated. The whole reaction passes in a 10 ms time scale at room temperature. Assuming that DR2 is produced by the flash promptly, DR3 by a dark reaction from DR2, DR4 by a subsequent dark reaction from DR3, and so on, we used a simple kinetic 142
9.2
Photopolymerization
model for the quantitative analysis. This model is described by the following equations for the concentrations ni and the rate constants Ki for the DRi, i = 1, 2, 3, 4, 5: dni Ki dt Ki K0 e
1 ni 1
Ei =kT
Ki ni ;
14 a
:
14 b
As an example, the fit of this model to the transient DR4 at 200 K is shown in Fig. 9.21. The fit does not show any significant deviation from the experimental curve.
Figure 9.21: Experimental transient and model curve according to Eq. 14 a for DR4 in perdeuterated TS at T = 200 K. The difference between experiment and model is plotted around the baseline [37, 44].
One result of these experiments is shown in Fig. 9.22. The addition reactions are thermally activated, the activation energies being about 0.25 eV per monomer, almost identical for each step (Tab. 9.5). Figure 9.22 also includes values for a product labelled V which is presumably due to long polymer chains with reactive carbene chain ends. A second result of these experiments is an estimation of the polymer yield for the photoreaction as defined by the number of polymerized monomer molecules per absorbed UV photon. Q is the quantum yield for the initiation process. Thus, the polymer yield P is the product of Q and the kinetic chain length L, i. e. the number of monomer molecules which are added to the chain after one initiation process: P = Q 7 L = 0.07 ± 0.02
(15)
The polymer yield was determined from the increase of the polymer absorption DOD due to UV irradiation [53]. If we assume a kinetic chain length of 100, which is a reasonable value, we get a quantum yield for chain initiation of 7610–4. P increases with decreasing temperature from 300 K to 180 K by nearly a factor of five. If TS is perdeuterated P also increases by a factor of 2.5 [44, 54]. Both, the temperature effect and the isotope effect on the polymer, can be explained qualitatively by an increase of the kinetic chain length due to a longer carbene chain end lifetime [37, 44, 54]. 143
9
Diacetylene Single Crystals
Figure 9.22: Temperature dependencies of the rate constants for the decays of DR2, DR5, and V. They can be described by the Arrhenius law in a range of three orders of magnitude [37]. Table 9.5: Activation energies DEi and frequency factors Ko for the reaction rates of DRi, evaluated from their Arrhenius plots [37].
DEi/ eV Ko /s –1
9.3
K2
K3
K4
K5
0.25 ± 0.03 1010 ± 1
0.26 ± 0.03 1011 ± 1
0.30 ± 0.03 1011 ± 1
0.30 ± 0.03 1011 ± 1
Holography
Diacetylenes have been subject to intense work due to their unique ability to undergo topochemical solid state polymerization, resulting in macroscopic polymer single crystals [1–3, 25, 28, 37]. Whether this reaction takes place depends on the monomer stacking distance and the tilt angle (Fig. 9.3). Both can be influenced by varying the rest groups R. The polymerization can be initiated by heat, UV radiation, X rays, or g rays and is irreversible. The optical properties, especially the absorption coefficient a and the refractive index n, are known to change dramatically during polymerization. By means of UV photopolymerization high-efficiency holographic grating on diacetylene crystals can be recorded, as was first shown by Richter et al. [55, 56]. Utilizing a fre144
9.3 Holography quency-doubled argon laser (lw = 257 nm) Richter et al. obtained surface phase gratings, due to the low penetration depth of this UV wave (Fig. 9.6). At high exposures higher diffraction orders up to five has been observed. Because of the 5 % stacking distance mismatch between monomers and polymers, Richter et al. often observed a destructive surface peel off. This problem has been shown to become less important by using longer UV wavelengths and, therefore, higher penetration depths [57]. The first aim within the Collaborative Research Centre 213 was to find a suitable technique for recording such gratings in an effective and reproducible way. Using this technique we have investigated the most important diffraction characteristics of these gratings: efficiency, thickness, angular selectivity and their dependence on exposure, sample thickness, and prepolymerization. The second aim, however, was to explain these characteristics within appropriate theoretical approach. Using this knowledge, the chain length of the polymers during UV polymerization and subsequent thermal treatment can be estimated. In the final investigation we have shown that images and even a holographic trick film can be recorded in the diacetylene crystals at room temperature. The peculiarity of the method is the difference of recording (UV) and reading (VIS) wavelengths. This difference allows prompt readout without a developing process and without perturbation of the hologram by the readout laser.
9.3.1 Theory Figure 9.23 shows the writing and reading beams for recording and replay of the simple holographic grating, respectively. Provided that they are of equal intensity two coherent UV
Figure 9.23: Schematic geometry of writing and reading of a holographic grating. Two coherent UV waves (dashed) form the grating of a spatial periodicity L. A VIS wave (solid) will generally be diffracted into different orders j.
145
9
Diacetylene Single Crystals
waves, impinging symmetrically the photoactive medium, form an intensity pattern on the surface of the photoactive material, I
x I0 cos2
p x=L :
16
L is the grating distance given by L w =2 sin w ;
17
where lw is the vacuum wavelength and yw the angle of incidence outside the medium. This intensity pattern results in a photoproduct distribution of the same periodicity, represented by a grating vector K, K 2p x^ =L ;
K jKj 2p=L ;
18
and results in a UV photopolymerization pattern, the refractive index n, and the absorption coefficient a of which can be described as n
x n0 a
x a0
1 P h0 1 P h0
nh cos
Khx ;
19 ah cos
Khx :
The importance of the Fourier coefficients besides h = 1 generally depends on reaction kinetics, exposure, and saturation effects. In general, n and a can also vary with the z coordinate. In principle a readout light beam (vacuum wavelength lr) will be diffracted by the grating resulting in different orders j as indicated in Fig. 9.23. Each of them have an amplitude Sj and a propagation vector rj. The total electric field inside the medium is a superposition of all theses waves: E
z
j1 P j 1
Sj
z s^ j exp
irj r :
20
Here s^ j are the polarization vectors. The wave vectors rj are coupled to the grating vector K via rj r0 jK :
21
This treatment was used first by Magnusson and Gaylord [58] and leads to a system of coupled differential equations for the complex amplitudes Sj : uSj cos
a0 i#j Sj uz 1 X 1 i cos Sj h
2pnh =r 2 h1 146
iah s^ j
h
s^ j Sjh
2pnh =r
iah s^ jh s^ j 0
22
9.3 Holography where is given by #j jK cos
'
jK2 r =4pn0
23
with a slant angle j between K and the x direction; j = 908 for the symmetric case shown in Fig. 9.23. This so-called coupled wave approach is a generalization of the work of Kogelnik [59], who assumed pure sine gratings read under the Bragg condition where only one transmitted and one diffracted wave is present. Kogelnik [59] gives analytical solutions for the amplitudes of the reference and the signal wave (zeroth and first order respectively) – only for the first ascent period of a growth curve, extending earlier work for the transparency region. For the efficiency Zj of the transmission grating with the thickness d j
d Sj
d Sj
d
24
he gets for j = 1 the well-known formula
d sin2
pn1 d= cos 0 sinh2
a1 d=2 cos 0 exp
2ad= cos 0 ;
25
with the replay wavelength l and the Bragg angle y0. It should be pointed out that this formula is only valid for the special case of thick (volume) gratings, which show pure Bragg behaviour. That is they possess neither higher Fourier coefficients nor a modulation amplitude n1 nor large enough a1, to produce higher diffraction orders, assuming n1 and a1 do not vary with z. The thickness of a holographic grating is often described in terms of the Q factor, defined by Q 2pd=L2 n0 :
26
Gratings with Q ^ 1 are regarded as thin those with Q 6 10 as thick.
9.3.2 Experimental setup We used diacetylene single crystal platelets, approx. 20 mm to 200 mm thick and some mm2 in area, cleaved from a parent TS6 crystal parallel to the (100) surface. The experimental setup is shown in Fig. 9.24. For writing holographic gratings we utilized a cw helium-cadmium laser (lw = 325 nm) or a xenon chloride excimer laser (lw = 308 nm). We gave preference to the recording geometry suggested by Bor et al. [62] and not to the wellknown beamsplitter geometry. The first order diffracted beams of a reflection grating R are reflected by a pair of parallel mirrors M1 and M2 (or pass a biprism instead) and are superimposed on the sample S. This geometry yields four main advantages: a) Given the fringe distance of the reflection grating, the resulting fringe distance on the sample is L = D/2 and wavelength-independent; 147
9
Diacetylene Single Crystals
b) Both writing beams are superimposed correctly because they have passed the same number of reflections. This is especially important if non-Gaussian beams are used; c) The intensities of both beams are equal and wavelength-independent; d) The setup is realizable compactly and can easily be adjusted for the use of short coherence lengths. Control of exposure is possible using an UV enhanced photodiode (De3). To read the gratings we used a 3 mW helium-neon laser (lr = 633 nm). At this wavelength the increase in refractive index can be expected to be high, whereas the absorption should not become too strong during the induction period of the polymerisation reaction (Fig. 9.4). The sample holder was mounted on the axis of a stepping motor for varying the angle of incidence. For analyzing transmitted or diffracted light a preamplifed large-area photodiode (De1) was mounted on a radius level of a second stepping motor being coaxial to the first one. Both motors include a reduction gear, giving an angular resolution of 0.06 mrad. Signal improvement was achieved by chopping the readout beam and using a second photodiode (De2) as an intensity reference. Both photodiode outputs were led to lock-in amplifiers. Two polarizers, Pol1 and Pol2, were used to attain both UV and VIS laser polarizations parallel to the sample’s b-axis.
Figure 9.24: Experimental setup. De1, De2, De3: large area photodiodes, R: reflection grating, S: sample on sample holder, Sh: beam shutter, BS: beam splitter, M: mirrors, Pol1, Pol2: polarizers (l/2 plates), Ch: chopper.
Two sample orientations relative to the grating fringes were investigated: a) UV polarization, VIS polarization, and polymer axis b lying in the plane of incidence (E mode), henceforth denoted as bk orientation; b) UV polarization, VIS polarization, and polymer axis b were perpendicular to the plane of incidence (H mode), called b|| orientation. 148
9.3 Holography The high dichroism of the crystals and their well-formed habit can be used to align their orientation under a polarizing microscope. Of course, the optical anisotropy of the crystals as well as anisotropic reaction kinetics give rise to birefringence effects.
9.3.3 General characterization Figure 9.25 shows for typical samples the maximum efficiency Z (y0) = Z as a function of the exposure. These growth curves show that an optimum can be reached between 0.5 and 1.0 J/cm2 for lw = 308 nm and between 5 and 10 J/cm2 for lw = 325 nm. For TS6 the penetration depth at 308 nm is about 110 mm, whereas at 325 nm it is about 420 mm. This difference and in addition a presumably lower quantum yield at 325 nm results in much slower growth curves for this wavelength. The decrease in efficiency is not only an effect of coupling back intensity from first to zeroth order, but also produced by an increase of absorption. Sample quality and thickness can influence these curves. A destruction of the surface is only observed if the optimal exposure is exceeded by a factor 2 to 3. This is due to the lattice mismatch of monomer and polymer stacking distance. The energy needed to reach the maximum is approximately by a factor 5 larger than at 257 nm [55, 56]. From the growth curves, i. e. from the exposure E which was needed to produce a certain efficiency Z, the holographic sensitivity S of the material given by S
p =E
27
can be estimated. For TS6 and depending on sample quality this value can range from 0.4 to 1.4 cm2/J. This is in the order of typical sensitivities for photorefractive crystals but far less
Efficiency η
0.5
0
12.5
25 0.13
0.25
0
0.065
0
0 3.0
1.5
Exposure E / Jcm
-2
Figure 9.25: Holographic growth curves of TS6 (solid curve: lw = 308 nm, d = 130 mm; dotted curve: lw = 325 nm, d = 270 mm) and IPUDO (dashed curve: lw = 308 nm, d = 260 mm). Orientation bk, L = 3.3 mm. Z (y0) = Z: efficiency for readout Bragg condition.
149
9
Diacetylene Single Crystals
than for common silver halide materials [63]. Nevertheless, the resolution of TS6 is comparable to that of common photographic materials. From the experiments of Richter et al. [55] it is known that a resolution of 2500 lines/mm is achievable. In our electron beam lithography experiments with a resolution of 5000 lines/mm has been achieved [64]. Supposing the thickness of the sample is equal to the grating thickness, we can calculate the factor Q for our gratings. At a grating distance of L = 3.3 mm this factor ranges from 2.5 to 60 for samples of thickness 25–500 mm, whereas at L = 0.8 mm Q factors of 800 can be achieved. These values are typical for thick phase gratings. Nevertheless, it should be pointed out that for example a grating with Q = 40 already shows higher diffraction orders. This confirms the arguments brought by Moharam and Young [60] that the r factor (r = L2/L2n1 n0) should be preferred when discussing grating diffraction.
9.3.4 Angular selectivity The angular selectivity is a measure for diffracted intensity when readout is performed under off-Bragg conditions or, quantitatively speaking, it is the half-width Dy of the function Z (y), where y is the angle of incidence. This property is of central interest, when discussing holographic storage media, because it limits the number of holograms/holographic gratings which can be stored simultaneously. Only as a rule of thumb, Kogelnik [59] gives Dy & Ln0 /d, whereas Magnusson and Gaylord [58] do not give any analytical expression for this quantity. Both approaches can easily be implemented on a computer to simulate diffraction properties. Assuming nh = 0 for h 6 2 and only for the first ascent period of a growth curve we find a Dy-dependence of the following simple form n0 =d
28
valid only for the first ascent period of a growth curve, because the angular selectivity curves get split beyond the first maximum [61]. The dependence of Dy on sample thickness d proved to be very well reproducible with respect to adjustment and sample quality. Figure 9.26 shows two typical measurements of Z (y) representing samples of different thickness. In contrast to the early experiments of Richter et al. [55] and of Niederwald et al. [56] these values represent an improvement of Dy by up to two orders of magnitude. Figure 9.27 shows Dy values versus sample thickness for TS6. No significant difference could be observed with respect to exposure or orientation for this grating distance. There is also no significant difference between TS6 and IPUDO with respect to angular selectivity. Measurements at L = 0.8 mm are summarized in Fig. 9.28. Within the series the minimum angular selectivity of 0.188 arose, using a 380 mm thick TS6 platelet. The hyperbolas plotted in Figs. 9.27 and 9.28 are fits of the function Dy = Ln0 /d to the data points. Averaged over all exposures n0 ranges from 1.3 to 1.7. 150
9.3 Holography
Rel. Efficiency
1
0.5
0 -30
0
30
Readout Angle q / Deg. Figure 9.26: Dependence of relative efficiency Z as a function of the incidence angle y for two TS6 samples of different thickness d.
Angular Selectivity ∆Θ / Deg.
Figure 9.27: Angular selectivity Dy as a function of sample thickness d for TS6 (open circles) and IPUDO (full circles). L = 3.3 mm, lw = 308 nm [65].
2
1
0
0
250
Sample Thickness d / µ m
500
Figure 9.28: Angular selectivity Dy as a function of sample thickness d for TS6. L = 0.8 mm, lw = 325 nm for both writing geometries, b|| (full circles) and bk (open circles) [65].
151
9
Diacetylene Single Crystals
9.3.5 Prepolymerized samples Until now we were assuming that sample thickness and hologram thickness are equal. In fact, for large thickness values d one can observe slight deviations from the hyperbola function due to the finite penetration depth of the UV light. These deviations should become more obvious for smaller penetration depths. To observe this some Dy measurements were carried out using thermally prepolymerized (up to 5 h at 70 8C before recording gratings) TS6 samples. Penetration depths for these samples varies between 18 to 80 mm at 308 nm and from 16 to 420 mm at 325 nm in the b|| orientation. The effect of prepolymerization on angular selectivity is shown in Fig. 9.29. The values for samples prepolymerized for 2 h still show for thin samples a weak thickness dependence. For thick samples Dy does not tend to decrease further. Samples prepolymerized for 5 h do not show any dependence on d, indicating a penetration depth distinctly smaller than the smallest sample thickness used. Thus, for an increasing polymer concentration the holographic gratings get thinner and thinner. The possibility of producing high efficiency volume phase gratings is restricted to fresh monomer crystals.
Figure 9.29: Angular selectivity of prepolymerized TS6 samples: 5 h (upper open circles), 2 h (full circles), and fresh crystals (lower open circles) [66].
9.3.6 Chain length, polymer profile, and grating profiles A model [66, 68] describing the spatially inhomogeneous reaction kinetics of diacetylenes must take into account a kinetic chain length L, which depends on the polymer conversion P. The monomolecular reaction in a simple homogeneous situation then is given by dP=dt L
P
1
P :
27
For the experiments described so far, L can be assumed constant during the exposure time, because only very little conversion takes place. Furthermore, for all experiments de152
9.3 Holography scribed, the factor (1 – P) can be dropped, because the experiments were carried out in the low conversion regime of the induction period and because saturation effects affect development experiments in both orientations. For the two orientations, different situations must be investigated corresponding to the different interaction of the two characteristic lengths L and L (Fig. 9.30), bjj :
dP=dt L
P cos2
px=L :
28
Here a polymer molecule grows parallel to the fringes contributing its whole chain length L to the polymer growth at point x, where it has been initiated. In the other case, a chain initiated in x contributes to the polymer growth in the whole interval [x – L/2, x + L/2]: b? : dP=dt
xL=2 R x L=2
cos2
px0 =L dx0 :
29
Integrating both Equations we see that in the second case the chain growth smears out the photoproduct distribution for a certain amount. This effect should be stronger if L/L approaches 1. For the quotient of both polymer modulations we get the ratio d,
P? sin
pL=L : pL=L Pjj
30
The resulting d values range for fresh samples from 0.85 to 0.95 and tend to decrease with increasing prepolymer content. For samples prepolymerized (5–6 hours, 3–4 % polymer) we find that d is between 0.65 and 0.80. These values correspond to chain lengths (L) of 0.15 to 0.4 mm or 300 to 800 repeat units [66].
Figure 9.30: Microscopic model to understand the interaction of the two characteristic lengths L and L: The two geometries investigated change the angle between chain growth direction and grating fringes.
153
9
Diacetylene Single Crystals
9.3.7 Multrecording In additional experiments, we succeeded in recording more than only one grating into one thick TS6 crystal. After each exposure interval, we rotated the sample for a certain amount to slant the gratings relatively to each other. Figure 9.31 shows the first order diffraction versus the angle of incidence for a 310 mm thick sample containing 42 gratings tilted stepwise by 1.08 8. We observed that gratings already present are almost not influenced by the subsequent recording processes except by the increase of total absorption, which causes a loss of efficiency less than 10 %. Thus, we did choose an exposure of only 0.07 J/cm2 for each grating. As can be seen from Fig. 9.31, the signal-to-noise ratio is bad for the last written gratings. It will get worse by decreasing the angular distance of the individual gratings but will hardly improve by increasing it [67]. A serious diffraction of the UV light by the gratings already recorded in the sample does not take place. UV diffraction efficiencies are extremely small in diacetylene crystals.
Efficiency η / 10 -3
8
21 4
4
0 -10
20
50
Readout Angle Θ / Deg.
Figure 9.31: Efficiency of the first order of a multihologramm consisting of 42 single gratings successively recorded in a 310 mm thick TS6 crystal. The gratings were recorded in the order of the numbers indicated [69].
9.3.8 Holography For the reconstruction of a real hologram with different writing and reading wavelengths, it was necessary to use a divergent reference wave. With a test platelet as object a resolution of 300 lines/mm was achieved in excellent TS6 samples (Fig. 9.32). Holograms up to 32 pictures were written (lw = 325 nm) in one crystal using the angular selectivity of the thick phase. When rotating the sample the pictures can be reconstructed (lr = 633 nm) successively (holographic trick film). The storage density of about 6.76109 cm –3, estimated from the real resolution and angular selectivity, is about one to two orders of magnitude lower than the storage density calculated from the grating distance [70]. 154
9.4
Di-, pyro-, and ferroelectricity
Figure 9.32: Reconstructed image from one of the 32 pictures of a holographic trick film [70].
9.4
Di-, pyro-, and ferroelectricity
The dielectric properties of diacetylene monomer single crystals are not strikingly different from those of other organic materials. Typically, they have permittivity values (earlier called dielectric constant er) of about 4–6. Solid state polymerization, however, results in a pronounced anisotropy of the electric permittivity with maximum values parallel to the polymer chain direction, due to the large polarizability of the extended p-orbitals. During our TOPOMAK activities the change of er , accompanying solid state polymerization, was analyzed quantitatively (Section 9.4.1.1). The results of the er analysis can be applied for the in situ monitoring of the polymer content (Section 9.4.1.2). The tailoring of pyro or ferroelectric properties of diacetylenes (Section 9.4.2) is less straightforward than might be surmised from the well-ordered arrangement of the R, R' substituents in Scheme 1, and from the fact that polar side groups can easily be introduced as substituents R and R' (Tab. 9.2). The difficulties originate from the packing of the individual polar diacetylene monomer molecules in the elementary cell of the solution-grown single crystal. This, generally, gives rise to a center of inversion symmetry for a pair of molecules thus compensating the individual molecular electric dipole moments in an antiferroelectric arrangement. During solid state polymerization spurious electric polarization has been observed [71] resulting from intramolecular distortion of originally centrosymmetric monomer units. Here we want to emphasize that our systematic TOPOMAK investigations have realized pyroelectric (Section 9.4.2) as well as ferroelectric properties (Section 9.4.3) for appropriately substituted diacetylenes.
155
9
Diacetylene Single Crystals
9.4.1 Dielectric properties of diacetylenes Typically, the electric permittivity of monomer single crystals of substituted diacetylenes ranges from 4 to 6 at room temperature. These er values are therefore larger than values of simple non-polar organic polymers, like polytetrafluoroethylene, in agreement with the existence of polar side groups. As is exemplified in Fig. 9.33, monoclinic crystals of 2,4-hexadiynylene di-p-toluenesulfonate (TS, see Tab. 9.2) shows generally a weak but non-negligible anisotropy for er [72, 75]. 8 TS T = 60°C
ε r (h)
7
(1)
6
(2)
5 4
3
(3) 0
10
20
30
40
polymerization time t(h)
Figure 9.33: Electric permittivity er (t) as a function of the polymerization time for TS crystals at 60 8C. The electric field was applied parallel to the chain direction of the monoclinic crystals (1) and in the two orthogonal directions (2) and (3). The permittivity was measured at 1 kHz for three different thin parallel-plate single crystal capacitors [75].
9.4.1.1
Correlation of polymer content and electric permittivity
Generally, a sigmoid time-conversion curve is observed for thermal solid state polymerization of diacetylenes. This behaviour was shown for TS in Fig. 9.4 and is reported for the unsymmetrically substituted 6-( p-toluenesulfonyloxy)-2,4-hexadiynyl-p-fluorobenzenesulfonate (TS/FBS, Tab. 9.2). In Fig. 9.34 a the slow conversion of the initial induction period of the solid state polymerization lasts until a polymer content of about 10 % is achieved. For higher up to complete conversion, this is followed by an autocatalytic reaction enhancement. The standard technique for the derivation of time-conversion curves is the gravimetrical analysis of a large number of crystals in a point-by-point procedure. After thermal polymerization for a well-defined period both, the soluble monomer and the insoluble polymer portions, are determined gravimetrically (Fig. 9.34 a). This technique consumes a considerable number of crystals and a substantial amount of time. Furthermore, only averaged time-conversion curves are obtained. Figure 9.34 b shows that during solid state polymerization of TS/FBS the electric permittivity parallel to the chain direction, er||, increases by a factor of about 2. The permittivity er|| is almost a linear function of the polymer content, as is exemplified in Fig. 9.34 c. We have found comparable behaviour for the substituted diacetylenes TS, FBS, FBS/TFMBS, and DNP [72–75]. Because the reorientation of the side groups during the polymerization is weak, it does not influence substantially the increase of er parallel to the chain direction. 156
9.4
Di-, pyro-, and ferroelectricity
Figure 9.34: Thermal solid state polymerization of TS/FBS (PTS is another acronym for TS). (a): Time-conversion curve derived by gravimetrical analysis. (b): Time-permittivity curve derived at 1 kHz for a thin parallel-plate single-crystal capacitor oriented with the polymer chain direction (b-axis) parallel to the electric field. (c): Correlation of conversion and electric permittivity (with time as implicit parameter) obtained by combination of (a) and (b) [73].
This proves that only the extended p-electron system of the diacetylene backbone is responsible for the enhanced polarizability. This conclusion is supported by the experimental analysis for three orthogonal directions in TS single crystals shown in Fig. 9.33 [75]. The minor variations for the two orthogonal directions (2) and (3) can be explained by the changes of the lattice parameters. In contrast, the change of the respective lattice parameters with polymer content does not suffice to explain the change of Der||. The linear relation between permittivity and polymer content is in principle surprising because there is a distribution of chain lengths of solid state polymerization, differing between induction period and autocatalytic range. The linearity, shown in Fig. 9.34 c, reveals that the polarizability of the p-electron system saturates already at chain lengths below the shortest ones occurring during thermal solid state polymerization. The experimental range for er is 1.4 to 2.2 derived for different substituted diacetylenes by Gruner-Bauer [72–75], which agrees with the observations of other groups [76– 78]. These values compare favourably with the estimate Der & 1.6 obtained for TS by simplified model calculations [75]. For these theoretical estimates the method of Genkin and 157
9
Diacetylene Single Crystals
Mednis [79] has been modified by Gruner-Bauer, extending earlier work for the transparency region [80].
9.4.1.2
Application to topospecifically modified diacetylenes
The linear relation of electric permittivity parallel to the chain direction vs. polymer content thus established can be used for the control of the polymer content of prepolymerized samples as well as for the derivation of time-conversion curves of individual single crystals in situ. We have used this technique to study the solid state polymerization of topospecifically and fully deuterated TS. Striking differences of their reactivities were reported before by Ch. Kröhnke [54]. The limited amount of samples available was sufficient for the permittivity analysis. Figure 9.35 shows the behaviour of different topospecifically deuterated derivatives of TS at T = 60 8C [73]. It should be stressed that the induction periods of single crystals with nominally the same history did not differ by more than 10 % at the same polymerization temperature. The toposelective modification of these substituted diacetylenes by the deuteration of the methylene groups close to the triple bonds of the diacetylene monomer (that are engaged in the crankshaft-type motion of the monomer molecule around its center of mass during solid state polymerization) evidently has a drastic influence on the solid state polymerization.
8
PTS T = 60°C
7 6
ε r (t)
5 8
CD - PTS T = 60°C
7 6 5 8
CD2 - PTS T = 60°C
7 6 5 0
10
20
30
polymerization time t / h
40
Figure 9.35: Electric permittivity as function of time for solid state polymerization at T = 60 8C for TS, PD-TS (fully deuterated TS), and CD2-TS (where only the methylene groups close to the triple bond of the diacetylene monomer unit are deuterated, see Tab. 9.2 a) [73].
158
9.4 9.4.1.3
Di-, pyro-, and ferroelectricity
Additional applications
Structural phase transitions accompanying solid state polymerization influence the behaviour of er (t). Thus they can not be observed but additional hints concerning the type of structural changes can be obtained. For example, a structural phase transition for DNP (Tab. 9.2) occurs at a polymer content above 95 % resulting in the loss of order perpendicular to the polymer chains [81]. The accompanying increase of the side-group mobility results in a distinct increase of the electric permittivity [75]. Even more dramatic changes of er were observed at the transition to a fibrillar structure in polar crystals of DNP/MNP (Tab. 9.2) [74, 75]. Thinking of applications outside fundamental research, the correlation of permittivity and polymer content can also be used for different kinds of ageing control. Since solid state polymerization is an activated process, the temperature-weighted time at elevated temperature – like a low-temperature radiation dose – influences the capacity of a diacetylene crystal-plate capacitor according to a well-defined characteristic history-capacity. Evidently, this can be adapted to an appropriate electronic ageing control.
9.4.2 Pyroelectric diacetylenes One fascinating goal of diacetylene materials research is the realization of polar polymer single crystals without complicated poling procedures. Substituted diacetylenes R1–C:–C:C–R2 incorporating polar side groups R1 and R2 with large but different electric dipole moments have to be synthesized. Since solid state polymerization depends on the ability of individual molecule side groups, which can perform intramolecular torsion and giving rise to an overall crankshaft-like motion, it is difficult – and was impossible for us – to predict the packing arrangement of individual monomer molecules in the solid. Frequently, the unit cell of substituted diacetylenes was observed to accommodate pairs of formula units in a centrosymmetric arrangement, thus compensating the net electric polarization. Polymorphism turned out to be an obstacle to systematic tailoring of dielectric properties of diacetylene single crystals, because different crystal structures of the same diacetylene derivative could be obtained from different, and occasionally even from the same solvent [82]. Nevertheless, Strohriegl synthesized during our TOPOMAK activities several non-centrosymmetric diacetylenes, which crystallized also in a polar phase. Typically, their permittivities were relatively large and anisotropic [83]. We restrict this report to three examples.
9.4.2.1
IPUDO
IPUDO (for the molecular structure see Tab. 9.2) is an example of a symmetrically substituted diacetylene. For the monomer as well as polymer crystals, non-centrosymmetrical orthorhombic crystal structures can be found already at room temperature [84]. The c-axis is the polar axis of the monomer crystal. IPUDO can only be polymerized by g radiation (60Co). For 85 % polymerized crystals pyroelectric properties were observed only in b direction. The permittivity of IPUDO is highly anisotropic, with maximum values of 8.5 for the 159
9
Diacetylene Single Crystals
monomer (parallel to a), or 11.6 for the 85 % polymer (parallel to b) and with minimum values (parallel to c) smaller by a factor of 2 (3.5) for the monomer (polymer) [83]. The distortion of the long side groups by the development of hydrogen bonds between neighbouring –CO–NH- groups is supposed to be responsible for the non-centrosymmetry [83, 85, 86]. Thus it is not surprising that the variation of the electric polarization of about 3610–8 Ccm –2 between room temperature and 4 K amounts to an unbalancing of only about 1% of the compensation of oppositely oriented C=O…HN dipole moments.
9.4.2.2
NP/4-MPU
NP/4-MPU (for the molecular structure see Tab. 9.2) is an example of a non-centrosymmetric diacetylene that forms polar monomer single crystals only, if it is grown from appropriate solvents, here from 2-propanol [82]. The resulting modification I is orthorhombic with the polar space group Fdd2 (Z = 16) and c as the polar axis [82]. This modification is not reactive thermally or under X-ray irradiation, because the monomer packing is outside the favourable range for solid state polymerization. The diacetylene rods make an angle of 678 with the stacking axis c , and the stacking distance d = 4.61 Å. The permittivity of this diacetylene is highly anisotropic and shows the largest value of about er & 23 for an electric field applied in the direction of the polar axis [82, 87]. Figure 9.36 shows the variation of the spontaneous electric polarization DP of NP/4MPU with temperatures between 10 K and the melting point. DP amounts to about 15 % of the electric polarization, which can be estimated from the volume density of molecular electric dipole moments of about 3 Debye (10 –29 Cm). The pyroelectric coefficient p (T) = dPS /dT = 8.8610 –10 Ccm –2 K–1 is of the same order of magnitude (smaller by 1/3) as that of the well-known and commercially used pyro and ferroelectric polyvinylidenefluoride. Therefore NP/4-MPU single crystals can be used for the detection of radiation [89], which we showed by using chopped low-power laser light as radiation source. The pyroelectric current and the total variation of the surface charge
Figure 9.36: Temperature-dependent change of the spontaneous polarization for a NP/4-MPU single crystal (modification I) along the polar c-axis for temperatures below the melting point Tm. The pyroelectric coefficient P at 300 K is also given [87].
160
9.4
Di-, pyro-, and ferroelectricity
have been used for the detection. Furthermore, a transversal piezoelectric coefficient of 1018 fCN –1 for NP/4-MPU was derived at room temperature. It is comparable with the value of a-quartz [88].
9.4.2.3
DNP/MNP
DNP/MNP (for molecular structure see Tab. 9.2) was the most successful one of Strohriegl’s syntheses [74]. It has a polar crystal structure for monomer and polymer crystals (space group P21). DNP/MNP polymerizes – thermally or exposed to UV radiation – extremely fast, because during solid state polymerization the molecules are packed optimally. The solid obtained by thermal polymerization of monomer crystals exhibits a fibrous texture, probably due to the large changes in lateral packing of the side groups. The temperature influence on the spontaneous electric polarization perpendicular to the chain axis c is shown in Fig. 9.37 for the monomer as well as the polymer crystal. The polarization varies up to room temperature by 7610 –8 Ccm –2 for the monomer crystal, with a pyroelectric coefficient of about 3.2610–10 Ccm –2K–1 at room temperature.
∆P(T) / 10 -7 C cm -2
0.0 -0.2
polymer
monomer
-0.4
DNP / MNP
-0.6 -0.8 -1.0
parallel to polar axis 0
100
T/K
200
300
Figure 9.37: Variation of the spontaneous electric polarization parallel to the polar b axis of DNP/MNP crystals with temperature. The data of DP (T) = P (T)–P (5 K) were derived by charge integration during a temperature cycle [74].
9.4.2.4
Spurious piezo and pyroelectricity of diacetylenes
Sample defects can be another origin of piezo and pyroelectric phenomena in substituted diacetylenes, which generally can be identified via sample dependence and smaller size of these effects [88]. Bloor et al. discussed such spurious pyroelectric effects, which seemed to be correlated with the occurrence of macroscopic deformations, such as screw dislocations in TS single crystals [90]. Similarly the analysis of weak polarization, caused by molecular distortion during solid state polymerization of TS, was reported by Bertault et al. [91]. We have observed comparable weak pyroelectric phenomena for TS/FBS [87]. 161
9
Diacetylene Single Crystals
9.4.3 The ferroelectric diacetylene DNP Whereas several pyroelectric diacetylenes were identified, uniform ferroelectric phases seem to be rather the exception, according to our experience with many new substitutions of diacetylenes [92]. The ferroelectric low-temperature phase of the symmetrically disubstituted diacetylene DNP, i. e. 1,6-bis(2,4-dinitrophenoxy)-2,4-hexadiyne (Tab. 9.2), turned out to be one of the rare and interesting exceptions [93–98]. On both ends DNP carries polar dinitrophenoxy groups, with an electric dipole moment of 10 –29 Cm, whose mutual twisting gives rise to the spontaneous electric polarization of the non-centrosymmetric low-temperature phase (space group P21) (Fig. 9.38). The DNP monomer crystal has besides a pyroelectric [93] low-temperature phase a ferroelectric one for T < Tc = 46 K. According to our investigations, the direction of the spontaneous polarization (Fig. 9.39) can be influenced by an external electric field [97]. Structural defects, occurring in all DNP crystals, give rise to a distribution of transition temperatures and to the existence of domains, whose polarization can not be inverted with the accessible external fields – thus behaving like pyroelectrics. Only the domains with the highest transition temperatures could be poled [97].
b
b
a
a)
b
a
b)
a
c)
Figure 9.38: Packing arrangement of DNP molecules in the monomer crystal viewed along the c-axis at three temperatures (a): T = 296 K, (b): T = 145 K, (c): T = 5 K [96].
The temperature dependence of the spontaneous polarization (Fig. 9.39) of the monomer crystal can be described in the framework of Landau’s phenomenological theory assuming a tricritical phase transition [97]. The maximum experimental value of the electric polarization of 2.4610–7 Ccm –2 compares favourably with the polarization that was calculated from the intramolecular twisting angle of the polar dinitrophenoxy groups of 5.18 determined by X-ray structural analysis at 5 K [96]. The electric permittivity (Fig. 9.40) reaches values of about er & 150 at the transition temperature. Its temperature dependence is strongly influenced by defects [97]. Thus, it is less appropriate for a comparison with theoretical predictions. In the early days of TOPOMAK, H. Schultes has already observed that the phase transition of DNP shifts to lower temperatures (Fig. 9.41–43) with increasing polymer content 162
9.4
Di-, pyro-, and ferroelectricity
3
P(T) / 10 -7 Ccm -2
DNP 2
1
0
0
10
20
30
T/K
40
50
60
Figure 9.39: Temperature dependence of the spontaneous polarization parallel to the polar b-axis for different DNP monomer single crystals [97].
75
0.3
0.2
ε r (T)
50
0.1
25
0
0
40
20
T/K
(χ ferro (T)) -1
DNP
0 80
60
polarization / 10 -7 C cm -2
Figure 9.40: Temperature dependence of the low-frequency electric permittivity (left axis) and the inverse of the ferroelectric part of the susceptibility (right axis) for a DNP monomer crystal [97].
0h
1,0
2h
0,5 5h 10 h 0
25
50
T/K
Figure 9.41: Variation of the zero-field electric polarization in crystallographic b-direction of DNP single crystal with duration of thermal polymerization at 130 8C [94] (Fig. 9.34 for conversion curve).
163
9
Diacetylene Single Crystals
ε r (T) / ε r (293 K)
8 5h
4
10 h 13 h 14 h 16 h
0
0
25
50
75
T/K
100
Figure 9.42: Influence of duration of thermal solid state polymerization at 130 8C on temperature dependence of electric permittivity of DNP [94].
1,6 40 1,2
30
0,8
X/%
20
80
40
10
0
0
∆S / J mol -1 K -1
Tc / K
50
0
0,4 0
8
4
t/h
16
8
16
12
0,0
t p /h
Figure 9.43: Variation of the transition temperature Tc (er maximum) and the conversion entropy (transition entropy change) DS with the duration tp of the thermal solid state polymerization of DNP at 129 8C [83, 87]. The inset shows the typical variation of the polymer content X with tp.
(thermal polymerization) and can not be observed in the polymer crystal [94, 98]. This was a puzzle for our early attempts to understand the ferroelectric phase transition of DNP. In the microscopic picture of the phase transition, nuclear magnetic resonance spectroscopy and relaxation of the DNP protons gave important additional information [95, 96]. The proton NMR spectrum reflects the orientation of the proton-proton axis of the methylene groups close to the central diacetylene unit via the nuclear-spin magnetic dipole interaction in a rather clear-cut way (Fig. 9.44). For fixed crystal orientation and varied temperature the spectrum of the methylene group protons of the DNP monomer proved that both DNP moieties are twisted around the central C–C single bond of the diacetylene below Tc. Since this degree of freedom is lost during the solid state polymerization the ferroelectric phase transition is suppressed. 164
9.4
Di-, pyro-, and ferroelectricity
r
H0 II b
r
H0 b
-40
0
40
-40
Rel. Frequency (kHz)
0
40
Figure 9.44: Simulated (left) and experimental (right) H NMR spectra as a function of the crystal orientation with respect to the external magnetic field, recorded at room temperature (np = 200 MHz). The crystal was rotated in steps of 58 around its long axis (a-axis), which was oriented perpendicular to the external field [96].
Additional information on the phase transition was obtained from proton-spin-lattice relaxation measured as function of the Larmor frequency, temperature, and orientation of the single crystals (Fig. 9.45). The analysis indicated the slowing down of a molecular motion on approaching the ferroelectric phase transition with the activation energy of about 0.020 eV, which is in the range of known librational and torsional modes of diacetylenes [96]. The phase transition of DNP could further be characterized by specific heat measurements for monomer and thermally polymerized single crystals (Fig. 9.46) [98]. These data support the description of the phase transition of the monomer crystals as a tricritical transition. This means it is a borderline case between a first-order and second-order phase transition, with a distribution of transition temperatures. The transition enthalpy was much lower than the corresponding order-disorder transition, in agreement with results obtained by Bertault et al. via Raman spectroscopy, which proved the importance of displacive contributions to the DNP phase transition [99].
165
9
Diacetylene Single Crystals
Spin-lattice relaxation rate (10 -3s-1 )
31 MHz
90 MHz
200 MHz
Temperature (Kelvin)
Figure 9.45: Spin-lattice relaxation rate of the protons in a DNP monomer single crystal for three Larmor frequencies. Two rate maxima can be discerned with temperature dependence explained by the model of Bloembergen, Purcell, and Pound (solid line) [96].
K -1
30
∆/C/J mol
-1
2h
2p =0h
20
10
6h
0 30
40
T/K
50
Figure 9.46: The ferroelectric contribution to the molar heat capacity of solid state polymerized DNP single crystals for different annealing times tp at 129 8C for [98].
9.4.4 Summary The increase of the electric permittivity for electric fields parallel to the polymer chain direction during solid state polymerization of diacetylenes can be used for in situ monitoring the monomer to polymer conversion of individual single crystals. The large librational am166
9.5
Non-linear optical properties
plitudes of the diacetylene moiety, required for solid state polymerization, are also the basis for the occurrence of interesting dielectric phase transitions. The tailoring of diacetylenes as ferro or pyroelectric crystals, which show good thermal stability and do not demand considerable efforts for the poling process, is a trial-and-error process, but has been realized during our TOPOMAK activities in a number of cases. Thus, material properties useful for applications of pyro- or piezoelectricity thus have been obtained.
9.5
Non-linear optical properties
9.5.1 Aims of investigation Polydiacetylenes are polymers showing a one-dimensional semiconducting behaviour. This one-dimensional structure causes exceptionally high third order non-linearities (w(3)) [100], also in off-resonant wavelength regions [101], with extremely short sub-picosecond switching times [102]. After this discovery it was believed that an optical amplifying switch (optical transistor) or even an optical computer was close at hand. At the start of our SFB 213 project the initial optimism about the application of the huge non-linearity of polydiacetylenes in optical switching was already somewhat damped. It was becoming clear, that the polydiacetylenes’ non-linear optical coefficients, despite belonging to the largest non-resonant non-linearities, were not sufficient for cascadable, intensity amplifying switches [103]. There was not much known about the mechanisms leading to the large non-linearities and different, sometimes contradicting theoretical models were proposed to describe them. On the other side, there was a demand for tailor-made materials with predictable non-linearities, absorption bands, and good optical quality. In this scenario, polydiacetylenes were nevertheless interesting, as the mechanisms of their large non-linearity form a good basis to build on. Only a few modifications were well characterised and there was a general lack of measurements with single crystals due to problems with sample preparation. Therefore, the aim was to characterise systematically the influence of side groups on the optical properties, preferably in the macroscopically ordered, stable, and reproducible framework of good and – later on – thin crystals.
9.5.2 Experimental setup Mainly two kinds of experiments will be described here, two-beam pump-probe and degenerate four wave mixing (DFWM) measurements. In a pump-probe experiment, an intense pulsed pump beam and a weaker equally pulsed probe beam of the same wavelength are focused into the sample, and the transmission 167
9
Diacetylene Single Crystals
of the probe beam is measured. The timing of the two pulses can be shifted, to make it possible to probe the decay of excitations generated by the pump beam. Two contrary effects may occur, bleaching and induced absorption. Near-resonant pump-probe lifetime measurements were performed by W. Schmid [105, 123], using the same picosecond dye laser system as for DFWM, described below. Th. Fehn further on investigated the subject in the off-resonant wavelength region between 720 nm and 820 nm, using a commercial Titan-Sapphire laser system (Coherent) with pulse lengths of ca. 120 fs, pumped by an argon ion laser (Coherent Mira). DFWM measurements can be done in a variety of geometries. Here, the forward mixing geometry is used (Fig. 9.47), where three beams, forming a right angle, are focused into the sample. This setup allows time-resolved measurements and, in contrast to third harmonic generation (THG) measurements, yields the w(3)(o;–o,o,–o) tensor which is related to an intensity-dependent refractive index, the interesting quantity for optical switching applications.
Figure 9.47: Beam geometry for DFWM measurements.
The interference pattern of the pump beams 1 and 2 forms horizontal stripes in the medium. As w(3) is directly related to an intensity-dependent refractive index, this interference pattern generates a refractive index pattern. The third probe beam is partially reflected on these horizontal planes, generating the signal beam 4. The efficiency of diffraction is related to |w(3) |. The pulse timing is adjustable, so the decay of the refractive index pattern can be probed. When the delay between the beams 2 and 3 is in the range of the laser’s coherence length, these pulses generate a diffraction grating for beam 1, resulting in an artificial raise of the signal, called coherence peak. All DFWM measurements were done with a commercially available synchronously pumped dye laser with a cavity dumper (Spectra Physics model 3500). When using Pyridine-1 as radiant dye, the wavelength can be tuned between 670 nm and 750 nm. The pulse width is about 1 ps; the repetition rate can be adjusted between single shot and 8 MHz. This is important to reduce the average heat load absorbed by the crystals. To suppress stray light signals a well-known modulation technique was used, the modulation of two pump beams with different frequencies using a chopper blade with a set of different divisions. As the DFWM signal depends on the product of the input intensities, the signal can be detected at the sum or difference of the two modulation frequencies [104], 168
9.5
Non-linear optical properties
3 Isig / I1 I2 I3 I1
0
cos !1 t 1I2
0
cos !1 t 1 I3
1 I1
0I2
0 I3 cos
!1 !2 t cos
!1 2
!2 t 2 cos !1 t 2 cos !2 t :
31
W. Schmid proposed the separated detection of w(5) effects by an extension of this modulation method. A w(5) signal without w(3) contributions can be detected at twice the sum or difference frequency [105], as the w(5) signal contains terms depending on the square of the product of the modulated beam intensities, Isig / I1 I2 I3 s k3 k35
I1 I2 I3 k5
I1 I2 I3 2 ;
32
where k3 is a function of w(3), k5 of w(5), and k35 depends on both non-linearities and their relative phase. Although this modulation method permits qualitative measurements of w(5) effects, especially determination of relaxation times, quantitative evaluation of w(5) is not possible, due to the infinite number of modulation harmonics with ill-defined relative amplitudes caused by the trapezoidal modulation form provided by a chopper blade [106]. This problem has been solved by A. Feldner. He developed a new type of modulator, working with two independently rotating polarizing foils (Fig 9.48). This type of modulator generates two harmonics in the intensity modulation spectrum with a fixed ratio of 4 : 1 (cos4ot) [106].
Figure 9.48: Scheme of intensity modulation by rotating polarizing foils (a). The vertical polarizer (b) ensures that an anisotropy of the material has no effect on the signal or the modulation spectrum.
For time-resolved pump-probe measurements of shorter relaxation times, a Titan-sapphire laser pumped by an argon ion laser is employed. In the current configuration, the wavelength can be tuned between 720 nm and 820 nm; the pulse width is about 120 fs.
169
9
Diacetylene Single Crystals
9.5.3 Theoretical approaches The simplest approach is the free electron model [107, 108]. The electrons are treated to move freely in a one-dimensional box, subject to a potential V0 cos (px/d) by the ion cores (d denotes the bond length). The polarizability a and hyperpolarizability g is derived from the 2nd and 4th order perturbation energies caused by an electrical field along the chain. This rough model does not account for local field effects nor for the alternating bond length found along the axis in PDAs. The model yields a static w(3) = 3.0610 –11 esu along the axis. Agrawal, Cojan et al. [109–111] treated explicitly the linear and non-linear optical properties of PDAs. They computed the electronic energy levels using a Hückel formalism. They computed a static w(3) = 0.7610 –10 esu for PTS and w(3) = 0.25610–10 esu for TCDU. This model neglects electron-electron interaction and therefore excitonic effects. The model yields the correct order of magnitude for w(3), but the wrong sign. The phase space filling (PSF) model was developed to describe non-resonant NLO properties in semiconductors [112, 113], especially in quantum well structures [114]. Greene et al. adapted this model to one-dimensional polymer chains [115, 116]. The model is only applicable to systems were the low energy absorption band is excitonic, as is the case with PDAs [117]. Formation of excitons is limited by the number of available electron states that are necessary to form the exciton. With an increasing number of excitons, the dipole momentum for forming a new exciton is reduced. The exciton band bleaches. As the following measurements will show, the PSF model seems to describe best the non-linear optical behaviour of polydiacetylenes. A very important prediction of this model is the proportionality of w(3) and a in the near-resonant frequency regime [115]. This behaviour was found in polydiacetylenes, strongly supporting the PSF theory [118].
9.5.4 Sample preparation For measurements well off the resonance, i. e. with wavelengths larger than 720 nm, p-TS6 crystals were prepared by thermal polymerisation of monomer crystals, grown out of a saturated solution and manually cut using a shaver blade. Thickness of these crystals varies between 40–100 µm. Later, for measurements closer to resonance, a method has been developed to grow thin mono-crystalline layers of TS6 and 4BCMU between glass substrates. To avoid crystal strains the monomer crystals have to be removed from the substrate before polymerising. The resulting crystal thickness can be made as low as 300 nm.
9.5.5 Value and phase of the third order susceptibility w(3) From DFWM measurements |w(3) | was determined for several polydiacetylenes in the nearresonant to off-resonant wavelength region, 680 nm to 750 nm (Tab. 9.6). Concurrently, the 170
9.5
Non-linear optical properties
Table 9.6: w(3) values for some different Polydiacetylenes [118, 119]. Material
Modification
|w(3) |/esu
at wavelength
PTS FBS 4BCMU 4BCMU 4BCMU
crystal crystal amorphous film thin monocrystalline film thin monocrystalline film
2610 –10 2610 –10 4610 –11 3610 –10 2610 –9
720 nm 720 nm 720 nm 720 nm 670 nm
imaginary part of w(3) can be computed from the non-linear absorption coefficient b, obtained from measurements of the intensity dependence of the sample transmission. It was found that the real part of w(3) was dominating the imaginary part by a factor of 3 [118]. This is predicted by the PSF model, from the bleaching of the exciton band. The value of w(3) is nearly identical for p-TS6 and p-4BCMU, and 4 times lower for p-IPUDO. These values were reproducible within 30 %. For p-FBS, the reproducibility was only within an order of magnitude. No difference was found between thermally and x-ray polymerised crystals [118].
9.5.6 Relaxation of the singlet exciton In p-TS6 energy relaxation times could be resolved at wavelengths below 700 nm. As the relaxation times are of the same order of magnitude as the laser pulse width (1 ps), a model function has to be fitted to the measured curve to obtain the relaxation (Fig. 9.49).
Figure 9.49: w(3) relaxation in p-TS6 at a wavelength of 680 nm and temperature of 200 K. For comparison, the dashed line gives the 3rd order autocorrelation of the laser pulse. The relaxation time of the singlet exciton can be seen as a significant broadening effect [120].
171
9
Diacetylene Single Crystals
The DFWM signal decays with half the time constant of the exciton density. For measurements of the relaxation time, the pump-probe experiments described below are to be preferred, because the intensity-dependent transmission directly follows the exciton lifetime [120]. In the off-resonant regime at 760 nm, the lifetime is about 0.5 ps, rising to more than 2 ps at 680 nm (Fig. 9.50). As with DFWM measurements, the lifetime can not be seen as an exponential signal decay, but as a broadening of the laser pulse autocorrelation signal. The lifetime
Figure 9.50: Wavelength dependence of the lifetime of the singlet exciton in p-TS6, derived from the broadening of the autocorrelation function of picosecond pulses. Different symbols denote different crystals [123].
can be estimated by fitting a model function accounting for the laser pulse shape. Th. Fehn investigated the subject, using a Titan-sapphire laser system with pulse lengths of 120 fs between 720 nm and 820 nm. Here, the exponential signal decay can be directly resolved, because T1 of the singlet exciton is much longer than the laser pulse (Fig. 9.51). Accordingly, the relaxation time can be determined more accurately. In comparison, W. Schmid’s fits tend to yield a slightly longer relaxation time, but within the error ranges the results are consistent.
9.5.7 The w(3) tensor components The w(3) tensor component parallel to the polymer chain of about 2–3610–10 esu (off resonant) is the well dominating source of the non-linear response. The DFWM signal was found to be polarized parallel to the polymer chain and the signal decreased below the detection threshold if any or all of the pump beams were polarized perpendicular to the chain. From –12 these observations w(3) esu, i. e. at least a factor perp can be estimated to be lower than 1610 172
9.5
Non-linear optical properties
Figure 9.51: The energy relaxation time of the singlet exciton is clearly resolvable with Ti-sapphire laser driven pump-probe experiments.
of 200 lower than w(3) par [121, 122]. The behaviour of the different PDAs under investigation (p-TS6, p-4BCMU, p-FBS) was very similar [118]. In the near-resonant frequency regime, a constant ratio of w(3) and the linear absorption coefficient a has been found, as predicted by the PSF model [115]. This model provides the best description of the experimental results [120].
9.5.8 Signal saturation With high pump intensities saturation of the w(3) signal was observed (Fig. 9.52). Although this general behaviour was found in all PDA samples, the finer details, e. g. the dependence on the mean intensity, varied between different measurements. It turned out that mainly three effects are responsible for the damping of the signal, whose relative influence depends on the wavelength, the crystal thickness, and the general quality of the crystals. A thermo-optical effect widens the focus diameter thus decreasing the intensity of the pump/probe beams. Induced absorption and two-photon absorption decrease this intensity too. Finally, polydiacetylenes show a significant influence of w(5) on the non-linear susceptibility, which is contrary to w(3). The induced absorption was measured concurrently to the DFWM signal by the transmission of one of the pump beams. Thus, one can easily account for this effect in the measurements. Our measurements on manually cleaved thick (50–150 µm) crystals [118] showed a stronger damping with higher repetition rates of the laser system, i. e. higher heat load but lower peak intensities. No damping was observed at off-resonant frequencies, e. g. at 750 nm. Therefore, the damping is clearly related to the heat load absorbed by the crystal, an indica173
9
Diacetylene Single Crystals
Figure 9.52: The flattening of the intensity dependence of the DFWM signal can successfully be described by a significant value of the fifth order susceptibility w(5) (inset solid curve: fit with assumed contributions of both w(3) and w(5)) [123].
tion of a thermo-optical effect. An amplification of the effect by inclusions of the solvent was proposed. In samples prepared later on with better optical quality W. Schmid observed a different behaviour [120]. The damping became stronger with lower repetition rates, i. e. higher peak intensity. No thermo-optic effect was observed. This can be well explained by the advances in crystal preparation: solvent inclusions were reduced and the crystal quality improved, resulting in a lower absorption coefficient at the same, near-resonant wavelength (e. g. 720 nm). So, the absorbed heat load and the thermo-optical coefficients were reduced concurrently. The damping effect observed by W. Schmid has to be addressed to a w(5) effect contrary to w(3) [123]. Later measurements on thin crystalline films, using an improved measurement technique, sustained this interpretation [106].
9.5.9 Spectral dispersion, phase, and relaxation of w(5) The value of w(5) has first been evaluated by Schmid [123], by fitting a polynomial to the measured curve of signal intensity versus pump laser intensity. At 748 nm, this fit yields a value of w(5) = 2.8610 –34 (m/V)4, with a phase angle of 1658 between w(3) and w(5). With the improved modulation method, a w(5) value of (1.69 ± 0.79)610–33 (m/V)4 (1.1610–16 esu) was evaluated at 729 nm [106]. This is in reasonable agreement with Schmid’s measurements, considering the different wavelengths, different measurement techniques, and different samples. After successful growth of thin (a few micrometers) p-TS6 crystals, it was possible to study w(5) signals at near-resonant wavelengths without an interfering thermo-optical effect 174
9.5
Non-linear optical properties
Figure 9.53: Wavelength dependence of w(3) and w(5) in p-TS6 single crystals. The isolated dots are experimental values obtained from a thin crystal (about 5 µm), all other dots are from a thick crystal (about 50 µm) [106].
[106]. The spectral dispersion of w(3) and w(5) has been measured with the improved modulation technique in the near resonant regime (Fig. 9.53). The w(5) amplitude increases roughly proportional to w(3), as expected from the PSF model. In this total regime, a phase between w(5) and w(3) of about 1608 was found. In the near resonant regime, the w(5) relaxation shows the same behaviour as w(3). No energy relaxation can be resolved in the off-resonant regime around ca. 720 nm, but the relaxation time increases with shorter wavelengths (Fig. 9.54). However, with even larger wavelengths, the w(5) relaxation times rise again (e. g. 2.3 ps at 748 nm) [120]. This can be successfully described proposing a three level energy scheme
Figure 9.54: Relaxation time of the w(5) response in p-TS6. At 720 nm, only an 5th order autocorrelation can be seen. At 685 nm a finite lifetime is clearly resolvable [106].
175
9
Diacetylene Single Crystals
Figure 9.55: At wavelengths between 720 nm and 750 nm, the w(5) signal relaxes in a two-step process [120].
with an even state at 3.2 eV [123]. In addition to the fast dominant process, a very slow weak relaxation has been found, with relaxation times between 20 ps and 100 ps (Fig. 9.55) [120].
9.5.10 Conclusion The polydiacetylenes’ non-linear optical behaviour was found to be in accordance with the predictions of the PSF model that assigns the non-linearity to the bleaching of the exciton band. Theoretical approaches neglecting excitionic effects clearly fail. The exciton dominates the non-linear optics of polydiacetylenes [118]. With increasing laser intensity, the w(3) signal shows significant deviations from the ideal I3-dependence. In thick crystals at near resonant wavelengths, a thermo-optic effect can be observed, which is amplified by crystal impurities [118]. However, also thin crystals show a similar damping of the w(3) signal. In contrast to the thermo-optical effect, this does not depend on the average heat load absorbed by the crystal, but on the peak intensity of the pump pulses. This effect can be described by a w(5) effect with a phase shift of 1608 against w(3), by analysis of the intensity dependence of the net signal of both effects [120] as well as by direct measurements using a new modulation technique [106]. Polydiacetylenes with different symmetrically substituted side groups differ mainly in terms of crystal quality, stability, and solubility; the influence on the third order susceptibility |w(3)| is rather small [118]. Interesting effects where found, however, in the unsymmetrically substituted diacetylene NP/4-MPU. This diacetylene forms a non-centrosymetric crystal with a significant w(2) [124, 125]. The second harmonic generation with this crystal can
176
References be phase matched, making the material very interesting for commercial applications, e. g. as a multiplying medium for a laser pulse autocorrelator [126]. In terms of optical quality, stability, and the value of w(3) polydiacetylenes still are superior to other polymers with a linear p-electron system, e. g. the polyparaphenylenevinylene investigated at our institute [127, 128, 130]. This is mainly due to their unique ability to form macroscopically ordered single crystals with aligned polymer chains.
References
1. G. Wegner: Z. Naturforschung, 24 b, 824 (1969) G. Wegner: Macromol. Chem., 145, 85 (1971) 2. H.-J. Cantow (Editor): Polydiacetylenes, Springer, (1984) 3. D. Bloor, R.R. Chance: Polydiacetylenes, NATO ASI Series, Martinus Nijhoff Publishers (1985) 4. R.J. Young: in: D. Bloor and R.R. Chance (eds.): Polydiacetylenes, NATO ASI Series, Martinus Nijhoff Publishers, (1985) 5. R.H. Baughman, H. Gleiter, N. Sendfeld: J. Polym. Sci., Polym. Phys. Ed., 13, 1871 (1975) 6. D.N. Batchelder, D. Bloor: J. Polym. Sci., Polym. Phys. Ed., 17, 569 (1979) 7. G. Wegner: Macromol. Chem., 145, 85 (1971) G. Wegner: Recent progress in chemistry and physics of poly(diacetylenes) : in: W.E. Hatfield (ed.): Molecular metals, Plenum press N.Y., 209 (1979) 8. D. Kobelt, E. Paulus: Acta Cryst., B30, 232 (1974) D. Bloor, L. Koski, G.C. Stevens, F.H. Preston, D.J. Ando: J. Mater. Sci., 10, 1678 (1975) 9. V. Enkelmann: Acta Cryst. B33, 2842 (1977) V. Enkelmann: Structural aspects of the topochemical polymerization of diacetylenes, in: H.-J. Cantow (ed.): Polydiacetylenes, Springer, 91 (1984) 10. D. Bloor: Diacetylene polymerization kinetics, in: D. Bloor and R.R. Chance (eds.): Polydiacetylenes, NATO ASI Series, Martinus Nijhoff Publishers, 1 (1985) 11. H. Eckhardt, T. Prusik, R.R. Chance: Solid State photopolymerization of diacetylenes, in: D. Bloor and R.R. Chance (eds.): Polydiacetylenes, NATO ASI Series, Martinus Nijhoff Publishers, 25 (1985) 12. H. Eichele: Diplomarbeit, Universität Stuttgart (1975) 13. D.N. Batchelder : The study of electronic excitations of polydiacetylenes and resonance raman spectroscopy, in: D. Bloor and R.R. Chance (eds.): Polydiacetylene, NATO ASI Series, Martinus Nijhoff Publishers, 187 (1985) 14. G. Weiser, L. Sebastian: Electric field sensitive defect states in fully and partially polymerized PTS, in: D. Bloor and R.R. Chance (eds.): Polydiacetylenes, NATO ASI Series, Martinus Nijhoff Publishers, 213 (1985) 15. E. Dormann: private communication 16. E. Wassermann et al.: Chem Phys. Lett., 7, 409 (1970) E. Wassermann et al.: J. Chem Phys., 54, 4120 (1971) 17. C.A. Hutchison Jr., B.W. Mangum: J. Chem. Phys., 34, 908 (1961) C.A. Hutchison Jr., B.W. Mangum: J. Chem. Phys., 29, 952 (1958) 18. J.H. van der Waals, M.S. de Groot: Mol. Phys., 2, 333 (1959) J.H. van der Waals, M.S. de Groot: Magnetic interactions Related to phosphorescence: in: A.B. Zahlan (ed.): The Triplet State, Cambridge University Press (1967)
177
9
Diacetylene Single Crystals
19. S.P. Mc Glynn, T. Azumi, M. Kinoshita: Molecular spectroscopy of the triplet state, Prentice Hall, New Jersey, (1969) 20. H. Eichele, M. Schwoerer, R. Huber, D. Bloor: Chem. Phys. Lett., 42, 342 (1976) 21. C.A. Hutchinson Jr., B.E. Kohler: J. Chem. Phys., 51, 3327 (1969) 22. G. Herzberg: Molecular spectra and molecular structure, Vol. III, Van Nostrand Reinhold Company, New York (1966) 23. H. Sixl, W. Hersel, H.C. Wolf: Chem. Phys. Lett., 53, 39 (1978) 24. C. Bubeck, H. Sixl, H.C. Wolf: Chem. Phys., 32, 231 (1978) 25. M. Schwoerer, H. Niederwald: Macromol. Chem. Suppl., 12, 61 (1985) 26. W. Hersel, H. Sixl, G. Wegner: Chem. Phys. Lett., 73, 288 (1980) 27. C. Bubeck, W. Neumann, H. Sixl: Chem. Phys., 48, 269 (1980) 28. R.A. Huber, M. Schwoerer: Chem. Phys. Lett., 72, 10 (1980) 29. W. Neumann, H. Sixl: Chem. Phys., 50, 273 (1980) 30. C. Bubeck, W. Hersel, H. Sixl, J. Waldmann: Chem. Phys., 51, 1 (1980) 31. H. Niederwald, H. Eichele, M. Schwoerer: Chem. Phys. Lett., 72, 242 (1980) 32. R.A. Huber, M.Schwoerer, H. Benk, H. Sixl: Chem. Phys. Lett., 78, 416 (1981) 33. W. Neumann, H. Sixl: Chem. Phys., 58, 303 (1981) 34. M. Schwoerer, R.A. Huber, W. Hartl: Chem. Phys., 55, 97 (1981) 35. H. Benk, H. Sixl: Mol. Phys., 42, 779 (1981) 36. W. Hartl, M. Schwoerer: Chem. Phys., 69, 443 (1982) 37. H. Niederwald, M. Schwoerer: Z. Naturforsch., 38 a, 749 (1983) 38. H. Gross, W. Neumann, H. Sixl: Chem. Phys. Lett., 95, 584 (1983) 39. H. Sixl, W. Neumann, R. Huber, V. Denner, E. Sigmund: Phys. Rev. B, 31, 142 (1985) 40. C. Bubeck: PhD thesis, Universität Stuttgart (1981) 41. W. Hersel: PhD thesis, Universität Stuttgart (1981) 42. R.A. Huber: PhD thesis, Universität Bayreuth (1981) 43. W. Hartl: PhD thesis, Universität Bayreuth (1981) 44. H. Niederwald: PhD thesis, Universität Bayreuth (1982) 45. R. Müller: Diplomarbeit, Universität Bayreuth (1982) 46. H. Gross: PhD thesis, Universität Stuttgart (1983) 47. W. Neumann: PhD thesis, Universität Stuttgart (1984) 48. H. Sixl: Spectroscopy of the intermediate state of the solid state polymerization reaction in diacetylene crystals, in: H.-J. Cantow (ed.): Polydiacetylene, Springer, 49 (1984) 49. R. Huber, M. Schwoerer, C. Bubeck, H. Sixl: Chem. Phys. Lett., 53, 35 (1978) 50. C. Bubeck, H. Sixl, H.C. Wolf: Chem. Phys., 32, 231 (1980) 51. R. Müller-Nawrath, R. Angstl, M. Schwoerer: Chem. Phys., 108, 121 (1986) 52. H. Sixl, W. Hersel, H.C. Wolf: Chem. Phys. Lett., 53, 39 (1978) 53. H. Niederwald, K.H. Richter, W. Güttler, M. Schwoerer: in Photochemistry and photobiology, Vol. 2, Harwood Academic Publishers, London, 1155 (1984) 54. Kröhnke: PhD thesis, Universität Freiburg (1979) 55. K-H. Richter, W. Güttler, M. Schwoerer: Appl. Phys., A32, 1 (1983) 56. H. Niederwald, K.-H. Richter, W. Güttler, M. Schwoerer: Mol. Cryst. Liq. Cryst., 93, 247 (1983) 57. B.E. Kohler, H.-D. Bauer, Be.E. Kohler, W. Güttler, M. Schwoerer: Chem Phys. Lett., 125, 251 (1986) 58. R. Magnusson, T.K. Gaylord: J. Opt. Soc. Am., 67, 1165 (1977) 59. H. Kogelnik, Bell Syst. Tech. J., 48, 2909 (1969) 60. M.G. Moharam, L. Young: Appl. Opt., 17, 1757 (1978) 61. D.L. Staebler, W. Burke, W. Phillips, J.J. Amodei: Appl. Phys. Lett., 26, 182 (1975) 62. Z. Bor, N. Racz, G. Szabo, A. Müller, H.-P. Dorn, Helv. Phys. Acta, 56, 383 (1983) 63. H.M. Smith (ed).: Topics in Applied Physics, Vol. 20, Holographic Storage Materials, Springer, (1977) 64. H. Niederwald, H. Seidel, W. Güttler, M. Schwoerer: J. Phys. Lett., 88, 1933 (1984) 65. H.-D. Bauer, Th. Vogtmann, I. Müller, M. Schwoerer: Chem. Phys., 133, 303 (1989)
178
References 66. H.-D: Bauer, Th. Vogtmann, M. Schwoerer: Mol. Cryst. Liq. Cryst., 183, 421 (1990) 67. Th. Vogtmann, H.-D. Bauer, I. Müller, M. Schwoerer: in: J.L. Bredas and R.R. Chance (eds.): Conjugated Polymeric Materials, NATO ASI Series E, 182 (1990) 68. H.-D. Bauer: PhD thesis, Universität Bayreuth (1990) 69. Th. Vogtmann, Diplomarbeit, Universität Bayreuth (1988) 70. W. Lorenz, Diplomarbeit, Universität Bayreuth (1990) 71. R. Nowak, J. Sworakowski, B. Kuchta, M. Bertault, M. Schott, R. Jakubas, H.A. Kolodziej: Chem. Phys., 104, 467 (1986) 72. P. Gruner-Bauer, E. Dormann: The dielectric properties of diacetylenes during thermal solid state polymerization, Second International Conference Electrical, Optical and Acoustic Properties of Polymers, Canterbury, UK, (The Plastics and Rubber Institute, London, UK), 15/1 (1990) 73. P. Gruner-Bauer, I. Müller, E. Dormann, Ch. Kröhnke: Makromol. Chem., 192, 1541 (1991) 74. P. Strohriegl, J. Gmeiner, I. Müller, P. Gruner-Bauer, E. Dormann, V. Enkelmann: Ber. Bunsenges. Phys. Chem., 95, 491 (1991) 75. P. Gruner-Bauer, E. Dormann: Phys. Rev. B, 45, 13 938 (1992) 76. M. Hurst, R.W. Munn: Chem. Phys., 127, 1 (1988) 77. M. Orczyk, J. Sworakowski, M. Bertault, R.M. Faria: Synth. Metals, 35, 77 (1990) 78. M. Orczyk: Chem. Phys., 142, 485 (1990) 79. V.N. Genkin, P.M. Mednis: Sov. Phys. JETP, 27, 609 (1968) 80. C. Cojan, G.P. Agrawal, C. Flytzanis: Phys. Rev. B, 15, 909 (1977) 81. P.-A. Albouy: Thèse 3 ième cycle, Université de Paris Sud (1982) 82. P. Strohriegl, H. Schultes, D. Heindl, P. Gruner-Bauer, V. Enkelmann, E. Dormann: Ber. Bunsenges. Phys. Chem., 91, 918 (1987) 83. E. Dormann, P. Gruner-Bauer, P. Strohriegl: Makromol. Chem., Macromol. Symp., 26, 141 (1989) 84. H. Bässler, H. Sixl, V. Enkelmann: Polydiacetylenes, Adv. Polymer Sci., Vol 63, Springer, Berlin, (1984) 85. P. Gruner-Bauer, P. Strohriegl, E. Dormann: Dielectric and pyroelectric properties of substituted diacetylenes, in: Electrical, Optical and Acoustic Properties of Polymers, (The Plastics and Rubber Institute, Conference Publication, Canterbury, UK), 11/1 (1988) 86. Hans Schultes: PhD thesis, Universität Bayreuth (1987) 87. P. Gruner-Bauer, P. Strohriegl, E. Dormann: Ferroelectrics, 92, 15 (1989) 88. M. Feile, P. Gruner-Bauer, P. Strohriegl, E. Dormann: Ferroelectrics, 106, 345 (1990) 89. P. Gruner-Bauer: PhD thesis, Universität Bayreuth (1991) 90. D.Q. Xiao, D.J. Ando, D. Bloor: Chem. Phys. Lett., 90, 247 (1982) 91. M. Bertault, M. Schott, J. Sworakowski: Chem. Phys. Lett., 104, 605 (1984) 92. H. Schultes, P. Strohriegl, E. Dormann: Z. Naturforsch., 42 a, 413 (1987) 93. G.F. Lipscomb, A.F. Garito, T.S. Wei: Ferroelectrics, 23, 161 (1980) 94. H. Schultes, P. Strohriegl, E. Dormann: Ferroel. Bulletin, 1, 36 (1986); H. Schultes, P. Strohriegl, E. Dormann: Ferroelectrics, 70, 161 (1986) 95. H. Winter, E. Dormann: The pyroelectric phase transition of „DNP“, a disubstituted diacetylene single crystal, in: A. Blumen et al. (eds.): Dynamical processes in condensed molecular systems, World Scientific Publishing Co., 214 (1990) 96. H. Winter, E. Dormann, M. Bertault, L. Toupet: Phys. Rev. B, 46, 8057 (1992) 97. P. Gruner-Bauer, E. Dormann: J. Phys.: Condens. Matter, 4, 5599 (1992) 98. G. Nemec, E. Dormann: J. Phys.: Condens. Matter, 6, 1417 (1994) 99. J. Even, M. Bertault, A. Girard, Y. Delugeard, J.-L. Fave: Chem. Phys., 188, 235 (1994) 100. C. Sauteret, J.-P. Hermann, R. Frey, F. Pradère, J. Ducuing: Phys. Rev. Let., 36, 956 (1976) 101. G.M. Carter, M.K. Thakur, Y.J. Chen, J.V. Hryniewicz: Appl. Phys. Lett., 47, 457 (1985) 102. G.M. Carter, J.V. Hryniewicz, M.K. Thakur, Y.J. Chen, S.E. Meyler: Appl. Phys. Lett., 49, 998 (1986) 103. D.R. Ulrich: Mol. Cryst. Liq. Cryst., 189, 3 (1990) 104. Ithaco Scientific Instruments IAN 37 (1985)
179
9 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120. 121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131.
180
Diacetylene Single Crystals
W. Schmid, Th. Vogtmann, M. Schwoerer: Opt. Comm., 121, 55 (1995) A. Feldner: Diplomarbeit, Universität Bayreuth (1996) J. Ducuing: in: P.G. Harper, B.S. Wherett (eds.): Nonlinear Optics, Academic Press (1977) K.C. Rustagi, J. Ducuing: Opt. Comm., 10, 258 (1974) G.P. Agrawal, C. Flytzanis: Chem Phys. Lett., 44, 366 (1976) C. Cojan, G.P. Agrawal, C. Flytzanis: Phys. Rev., B 15, 909 (1977) G.P. Agrawal, C. Cojan, C. Flytzanis: Phys. Rev., B 17, 776 (1978) H. Haug, S. Schmitt-Rink: J. Opt. Soc. Am., B 2, 1135 (1985) D.S. Chemla, D.A.B. Miller: J. Opt. Soc. Am., B 2, 1155 (1985) S. Schmitt-Rink, D.S. Chemla, D.A.B. Miller: Phys. Rev. B, 32, 6601 (1985) B.I. Greene, J. Orenstein, R.R. Millard, L.R. Williams: Phys. Rev. Lett., 58, 2750 (1997) B.I. Greene, J. Orenstein, S. Schmitt-Rink: Science, 24, 679 (1990) L. Sebastian, G. Weiser: Chem. Phys., 62, 447 (1981) Th. Vogtmann: PhD thesis, Universität Bayreuth (1993) Th. Vogtmann, W. Schmid, Th. Fehn, F. Bauer, I. Bauer, M. Schwoerer: Synth. Met., 55–57, 4018 (1993) W. Schmid: PhD thesis, Universität Bayreuth (1994) Th. Fehn: Diplomarbeit, Universität Bayreuth (1992) Th. Fehn, Th. Vogtmann, J. Hübner, M. Schwoerer: Appl. Phys. B, 59, 203 (1994) W. Schmid, Th. Vogtmann, M. Schwoerer: Chem. Phys., 204, 147 (1996) M. Braun: Diplomarbeit, Universität Bayreuth (1996) M. Braun, F. Bauer, Th. Vogtmann, M. Schwoerer: J. Opt. Soc. Am., B 14, 1699 (1997) H. Oester: Dplomarbeit, Bayreuth (1997) F. Bauer: PhD thesis, Universität Bayreuth (1996) F. Bauer, M. Schwoerer: Mol. Cryst. Liq. Cryst., 283, 131 (1996) S.N. Maganov, G. Bar, H.-J. Cantow, H.-D. Bauer, I. Müller, M. Schwoerer: Polymer Bulletin, 26, 223 (1991) Th. Schimmel, B. Winzer, R. Kemnitzer, Th. Koch, J. Küppers, M. Schwoerer: Adv. Mat., 6, 308 (1994) H.-D. Bauer, A. Materny, I. Müller, M. Schwoerer: Mol. Cryst. Liq. Cryst., 200, 205 (1991)
10
Matrix-Molecule Interaction in Dye-Doped Rare Gas Solids Thomas Giering, Peter Geissinger, Wolfgang Richter, and Dietrich Haarer
10.1
Introduction
The investigation of the electronic states of polyatomic organic dye molecules is enhanced considerably when they are doped into suitable solid host matrices [1] at low concentrations. If the interaction between host and guest molecules is weak, the guest molecule will exhibit its characteristic electronic and vibronic signature except for a possible shift of the total spectrum, starting with the zero-phonon origin. The hosts are chosen according to their absorption because they must not overlap with the guest states under investigation. This can be achieved for a variety of polymers, n-alkanes, and rare gases, so that the optical absorption spectra of these guest-host systems are dominated by the guest molecules. A natural extension of these investigations was the incorporation of guest molecules into disordered systems to serve as probes of the host properties. In contrast to n-alkane hosts, in disordered host materials the optical absorption spectra of these guest-host systems are often characterized by broad and featureless bands, the so-called inhomogeneous broadening. When a dye molecule is incorporated into a host matrix its transition energy will experience a shift due to the dye-matrix interaction. In a (hypothetic) perfect crystal all guest molecules will experience exactly the same shift, whereas in disordered hosts a distribution of local environments and therefore a distribution of transition energies is observed, which can be as large as several 100 cm –1. The inhomogeneous broadening prevents access to the homogeneous absorption lines, whose widths are related to energy and phase relaxation processes in the system. The fact that the inhomogeneous broadening is a consequence of statistically distributed local environments of the guest molecules means that it can be described by a stochastic model which dates back to Markoff [2]. For a review see Stoneham [3]. In this model, for details see next Section, the host material is dissected into matrix units, which in the case of rare gas hosts are identical with the atoms. For polymeric hosts the matrix units correspond to the respective monomer units. The inhomogeneous distribution of absorption frequencies is then given by averaging over all possible arrangements of matrix units around a cavity containing the dye molecule, which are weighted by the line shift caused by each respective configuration. The overall inhomogeneous shift is calculated by adding up the contributions of each matrix unit to the total line shift. Their contributions depend on their respective positions to the dye molecule. A modification of the interatomic distances, for example Macromolecular Systems: Microscopic Interactions and Macroscopic Properties Deutsche Forschungsgemeinschaft (DFG) Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
181
10 Matrix-Molecule Interaction in Dye-Doped Rare Gas Solids through the application of external pressure, leads to a change of the interaction and therefore to a shift of the line. This suggests that pressure effects can also be taken into account within the framework of the stochastic model [4]. However, in order to generate measurable changes of the entire inhomogeneous band, pressure changes in the range of several gigapascals are required. These high pressures alter the structure of the investigated sample significantly which also affects the homogeneous width [5]. To observe the effects of small pressure changes the spectral resolution has to be increased significantly. This can be accomplished by the experimental technique of hole burning spectroscopy, which was introduced in 1974 [6, 7]. The hole burning method, for a description see Ref. [8] and Chapter 5, allows access to homogeneous line widths which are masked by the inhomogeneous broadening. Furthermore, within the inhomogeneous band, spectral holes can serve as persistent narrow frequency markers. The changes to these frequency markers are due to: structural relaxations of disordered host materials (Chapter 5), IR induced spectral diffusion (Chapter 6), external perturbations – like electric fields – [9, 10], and pressure (Section 10.6) and can be monitored accurately over extended time periods. In the ideal case the hole width is twice the homogeneous width. Therefore the increase in spectral resolution is roughly given by the ratio of inhomogeneous to homogeneous width. This ratio, also referred to as the multiplexing factor, which is due to the possible application of the hole burning technique in data storage devices, is typically in the range of 103 to 105 [8]. This means that for producing detectable changes in a spectral hole due to external pressure the magnitude of the external perturbation can be reduced by approximately the multiplexing factor. This in turn allows sample investigations near equilibrium conditions, meaning that the displacements of the matrix units from their zero-pressure positions are small. The results of the first pressure tuning experiments reveal a red shift of the hole center, when the pressure is raised, and a blue shift, when the pressure is lowered after the burning of the hole [11, 12]. Furthermore, in both cases a hole broadening is observed. Laird and Skinner’s extension of the stochastic approach [4] for describing pressure effects was first applied to dye molecules in various polymer hosts like polyethylene (PE), polystyrene (PS), and polymethylmethacrylate (PMMA). Their prediction of a frequencydependent pressure shift was duly verified. Within 20 % the predicted value agreed with the experimental results. The apparent success of the stochastic model was surprising, because polymers meet the basic requirements of the model quite poorly. Usually it is assumed that one type of interaction, e. g. dispersive forces, between dye and matrix molecules predominates. Also, the matrix units, which in this case are the monomer units, are considered to be spherical and independent of each other yielding additive contributions to the solvent shift of the doped dye molecules, which are also assumed to be spherical. Additionally, the matrix units are assumed to be able to arrange themselves independently around the dye molecule. In the case of polymer hosts, the matrix units clearly are unable to arrange themselves independently, because they are connected by strong directional bonds. Moreover, the monomer units are often slightly polar, like PMMA. Furthermore, the validity range of the results of the stochastic model remained unclear, since in the course of the calculation two conflicting approximations with regard to the number density r of the matrix units within the interaction range of the dye molecule were made: the Gaussian approximation (valid for r ? ?) and the continuum approximation (valid for r ? 0). The latter is neglecting correlations between matrix units, which arise from their mutual steric exclusion. 182
10.2 Stochastic theory To clarify these inconsistencies we set out to systematically investigate the stochastic model, using model systems that come closest to its basic assumptions. For this reason we chose solid rare gases as host matrices. For these systems additional information is readily available [13, 14]. Furthermore, the interaction potential and its parameters are well-known for pure rare gases. Matrix parameters can be varied systematically by using different rare gases, while still retaining a structural similarity. Some rare gas mixtures show a different structure than pure rare gases (Sections 10.3 and 10.6). Through a systematic variation of the mixing ratio the impact of the involved structural transition onto the optical data can be investigated.
10.2
Stochastic theory
The above-mentioned stochastic model description considers an amorphous system of N matrix units, containing a small concentration of dye molecules. Each matrix unit will shift the electronic absorption line of the dye molecule by some amount n (R~i , where R~i is the position of the ith matrix unit with respect to the dye molecule. The contributions of all matrix units are assumed to be additive. The total inhomogeneous distribution of absorption lines can then be written as [2–4]: 1 I
N V
Z
dR~ 1 . . . dR~N P
R~ 1 ; . . . ; R~N
N X
! ~
Ri :
1
i1
V is the volume of the sample. In order to further evaluate this expression, the function P (R~1 ; . . . ; R~N ), which is the probability of finding N matrix units at positions R~1 ; . . . ; R~N , has to be specified. If correlations between the matrix units are neglected, the (N + 1) particle solute-solvent distribution function can be factorized into a product of two-particle distri~ bution functions g (R): P
R~ 1 ; . . . ; R~N
N Q n1
g
R~n :
2
This is the so-called continuum approximation, which is valid in the case of small number densities r of matrix units. Furthermore, considering only a non-polar solute and solvent, the perturbation func~ of the transition frequency is of the familiar Lennard-Jones type tion n (R) ( " ~ 4"
R
12
s R
R0
R 1
6 #
s R0
if R R0 ; (3) if R < R0 183
10 Matrix-Molecule Interaction in Dye-Doped Rare Gas Solids with the origin shifted by R0 to account for the large size differencepbetween solute and sol vent [4]. In this representation R0 + s/2 is the solute radius, while 6 2 s can be taken as the solvent diameter. p This means that a matrix unit located at the position of the potential minimum R0 + 6 2 s will shift the electronic absorption line of the dye by the potential depth e, as given in Eq. 3. In the case of r ? ?, the inhomogeneous distribution is Gaussian, which is a consequence of the Central Limit Theorem [15]. For the spatial distribution of the matrix units a simple step function was chosen [4], meaning that a matrix unit can be found anywhere outside a spherical cavity of radius R0 + Rc, but not inside. The parameter Rc determines how close a matrix unit can come to the dye molecule. p With the above conventions about solvent and solute dimensions, we have Rc = s/2 (1 + 6 2) & 1.061 s. Inserting Eqs. 2 and 3, the inhomogeneous distribution (Eq. 1) can now be evaluated, resulting in expressions relating the experimentally accessible parameters, namely the solvent shift ns (spectral position of the band maximum with respect to its gas phase position) and the full width at half maximum Gs to the local number density r and to the microscopic parameters e, R0, s, and Rc [15]. We can now turn to the question of matrix correlations, which arises because of the application of two conflicting approximations in the course of the calculation: the Gaussian approximation for large and the continuum approximation for small number densities of the matrix units. A reasonable way out of this dilemma is to retain the Gaussian approximation and to introduce matrix correlations. Here the only correlation effect considered is the principle that two matrix units cannot be located at the same position. This means that the factorization used in Eq. 2 cannot be applied. Matrix correlations can be accounted for, within the framework of the stochastic model, by introducing a three-parti~ R~0 [16–18]. Applying the Kirkwood superposition approxicle distribution function g3 (R; mation [19], ~ R~0 g
R ~ g
R~0 gS R~ g3
R;
R~0 ;
4
we introduce a solvent-solvent distribution gS (|R~ R~0 |). In analogy to the solute-solvent distribution, we insert a simple step function for gS (|R~ R~ 0 |) with the cut-off radius given by p 6 2s. As in the case of neglected correlations it is possible to derive expressions that allow the calculation of the potential depth e and the local number density of the matrix units r from spectroscopic data (Section 10.5). As mentioned in the introduction, the stochastic approach can also provide a description of the effects of external pressure on spectral holes. In our experiments, however, the pressure change is always accompanied by a simultaneous change in temperature, see Ref. [20]. Therefore, the observed hole shifts and broadenings will not only be due to pressure changes, but also to the thermal expansion of the matrix. There may also be dynamical effects such as phonon scattering and (fast) tunneling systems (TLS) relaxations, which we will not treat in this contribution. Pure pressure effects have been accounted for by the Laird and Skinner theory [4]. This theory can be conveniently expanded to also include the thermal expansion of the matrix and its influence on the dye molecule, for details see Refs. [20, 21]. In analogy to the inhomogeneous distribution (Eq. 1), the temperature-pressure kernel, i. e. the probability that 184
10.2 Stochastic theory a guest molecule with the original solvent shift n will have a new transition frequency n' after a pressure change Dp and a temperature change DT, can be written as f
0 ; p; T
Z * 1 dR~ 1 . . . dR~N P
R~ 1 ; . . .; RN N I
V N N P P
R~i 0 0
R~ i ; p; T : i1
5
i1
The observed temperature-pressure shift was found to be linear, while the concomitant hole broadening can be described by a power law. The latter is clearly dominated by dynamical processes which are affected by the change in temperature. It is now assumed that, in analogy to pure pressure effects, the hole shift and the hole broadening due to the thermal volume expansion are linear functions of the temperature change. The function n' (R~i ; Dp; DT) in Eq. 5 can then be linearized to ~ p; T
R ~
R; 0
k
! ~ R @
R p g @R 3
! ~ R @
R T : @R 3
6
k is the compressibility and g the volume thermal expansion coefficient of the matrix. It is important to note that the temperature and pressure effects have opposite signs, see Eq. 6 and Refs. [11, 22]. The further evaluation follows the lines of the calculation given in Ref. [21]. Within the above-mentioned approximations, the pressure-temperature kernel is found to be Gaussian. At this point we restrict our considerations to the evaluation of the hole shift, which is calculated as "Z
; p; T
(
~ R @
R d R g
R 3 @R 3
)( 1
~
R S 2 sS
)#
gT
kp : (7)
sS is related to the full width at half maximum of the inhomogeneous line shape GS by S sS = 2 pG , while n is the burning frequency of the spectral hole with respect to the gas-phase 2 ln 2 position of the transition frequency of the dye molecule. For the modified Lennard-Jones potential (Eq. 3) Eq. 7 has to be evaluated numerically. It can, however, be simplified by considering only a purely attractive van der Waals potential. This approximation seems reasonable, since only matrix units located in the attractive part of the intermolecular potential can cause the observed red shift in pure pressure tuning experiments [11]. We obtain for the temperature-pressure shift:
; p; T 2 kp
gT :
8
With this equation, we found a simple way to extract the pure pressure shift from our pressure-temperature data. Due to the opposite sign of pressure and temperature shifts (Eq. 8), the pure pressure shift will be even larger with the inclusion of the temperature ef185
10 Matrix-Molecule Interaction in Dye-Doped Rare Gas Solids fects, leading to a larger compressibility after including the correction. The evaluation of our experimental data leads to an additional complication. The volume thermal expansion coefficient depends, contrary to the compressibility, strongly on the temperature, even in the small temperature interval from 1.8 K to 5 K. Fortunately, for crystalline R TDT samples experimental data are available [23] which allow us to substitute gDT ? T dT' g (T') in Eq. 8. With the help of this equation it can be shown that the pressure effects should clearly dominate the experiment [20], which is confirmed by the observed red shift in pure pressure-tuning experiments [11]. The stochastic description of pressure effects can also be modified to include correlations between matrix units [20, 21]. It is interesting to note that for the purely attractive van der Waals potential, Eq. 8 will not acquire any additional terms suggesting that for the determination of the compressibility k matrix correlations will also play a minor role when the Lennard-Jones potential Eq. 3 is valid (Section 10.7).
10.3
Rare gases
Due to their simplicity rare gases have served as model systems for a long time. This means that a large data basis [13, 14, 24] is available. Of special importance for our investigations is the knowledge of the microscopic interaction potential between rare gas atoms. For its attractive region it was already calculated by F. London [25], to depend as 1/R6 upon the interatomic distance. In the well-known Lennard-Jones (6,12) potential the exponential R-dependence for the repulsive contribution is approximated by an algebraic 1/R12 term: s 12 s 6
R 4"~ : R R
9
The potential depths e~ , which are directly connected with the polarizability a and the parameters s, which determine the distance of the potential minimum, are listed in Tab. 10.1. Table 10.1: Lennard-Jones parameters ~e and s [26, 27] and polarizability a [28]. Gas
~e [cm –1]
s [Å]
a [10–25 cm3]
Argon Krypton Xenon
83.7 113 161
3.405 3.65 3.98
16.3 24.8 40.1
Another aspect is the structure of solid rare gases, which can be produced either by slow freezing the liquid or by condensation onto a sufficiently cold substrate. While the for186
10.4 Experimental mer method yields single crystals with macroscopic dimensions, condensed rare gases are polycrystalline. Electron and X-ray diffraction [29, 30] experiments show the crystals and crystallites to have a face-centred cubic structure, although theoretical simulations predicted the hexagonal close packing to be energetically favoured by about 0.01%. The situation is different for certain quench condensed rare gas mixtures, where computer simulations [31] predicted the possibility of an amorphous structure for rare gases whose atomic radii differ by at least 10 %. X-ray diffraction measurements [30] indeed revealed an amorphous structure for Ar1–xXex matrices with mixing ratios 0.2 < x < 0.7. This opens up the possibility to study the transition between the polycrystalline and amorphous state by varying the mixing ratio in a single system.
10.4
Experimental
The dye-doped rare gas matrices had to be prepared in situ. For this purpose a gas handling system was installed which provided alternatively the pure rare gases (Linde, 99.998 % purity for argon, 99.990 % purity for krypton and xenon) and rare gas mixtures. The concentrations of the mixture components were determined by monitoring the pressure in the mixing chamber during the composition process. Commercially available phthalocyanine (Aldrich, used without further purification) was sublimated and subsequently mixed with the desired matrix gas in a Knudsen effusion furnace. Following Bajema et al. it consisted of a sublimation chamber (T & 600 K) and a super-heating chamber (T & 700 K). The gaseous matrix passed through a nozzle onto a sapphire substrate being in thermal contact with the cold finger of a continuous flow cryostat. Two different cryo-systems were used, which opened a wide temperature range for investigations. In system I temperatures down to 4 K could be produced using a commercially available continuous flow cryostat (Oxford Instruments). A home-built special outer vacuum container enabled either sample deposition or optical access through the sample. The system II is our own design. It consisted of a continuous flow cryostat which was immersed into liquid helium after the sample preparation. Temperatures down to 1.5 K could be reached by pumping the helium. Furthermore, hydrostatic pressure could be exerted by sealing off the helium bath. A typical dye concentration was approximately 2610 –3 mol/l with a sample thickness of about 10 mm. To characterize the samples, absorption spectra were recorded with a monochromator (Jobin Yvon THR 1500, used resolution about 0.2 cm –1). Hole burning studies were performed using a single mode dye laser (Coherent 599, bandwidth about 3 MHz). The optical setup is described in detail in Refs. [11, 32].
187
10 Matrix-Molecule Interaction in Dye-Doped Rare Gas Solids
10.5
Inhomogeneous absorption lines
The visible absorption spectra of phthalocyanines show two regions with strong and characteristic absorption bands. The lowest electronic energy transition, called Q band, is degenerated in metallophthalocyanines due to a molecular D4 h symmetry. In free base phthalocyanines (H2Pc), however, the reduction of molecular symmetry from D4 h to D2 h, caused by the two H atoms in the inner ring, lifts the degeneracy of the Q band. It splits into two bands, which are conventionally referred to as Qx (for the lower energy component) and Qy (for the higher energy component). These two absorption bands are shown in Fig. 10.1 for a H2Pc-doped krypton matrix. At higher frequencies a number of vibronic lines appear.
Figure 10.1: Absorption spectrum of H2Pc in a krypton matrix.
Our spectra agree quite well with spectra recorded previously by Bajema et al. [33]. Due to the inherent local disorder in vapour condensed matrices [30, 34] the absorption spectra are inhomogeneously broadened. For the investigated systems Gaussian line shapes are expected from theoretical considerations [2, 3]. However, this prediction is difficult to test due to a slight asymmetry of the absorption spectra towards lower frequencies. This observation indicates a second preferential position of H2Pc molecules in the rare gas matrix and confirms earlier measurements [35]. In samples with a sufficiently high dye concentration this second site leads to a weaker set of spectral transitions shifted about 70 cm –1 towards lower energies [35, 36]. Table 10.2 compiles the spectral positions of the Qx and Qy bands as well as the widths of the Qx bands and their solvent shifts nS, i. e. the frequency shifts of the respective absorption maxima with respect to the absorption of free H2Pc molecules, as measured in a supersonic free jet [37]. From argon to xenon increasing solvent shifts and inhomogeneous widths can be observed. 188
10.5 Inhomogeneous absorption lines Table 10.2: Spectroscopic data (positions n, solvent shift nS, width GS ) of the absorption bands of H2Pcdoped rare gas matrices. Units are in cm –1. Matrix
n (Qx)
GS (Qx)
nS (Qx)
n (Qy )
Argon Krypton Xenon
14764 14664 14540
44 58 89
364 464 593
15731 15605 15466
The solvent shift measures the strength of the dye-matrix interaction which is purely dispersive for the investigated systems. When going from argon to xenon the increasing solvent shift can therefore be explained by the increasing matrix polarizability. The latter is even effective enough to compensate the reduction of the number of interacting matrix atoms. Due to their increasing size the solvent shift is produced by fewer interacting units which, according to basic statistics, enlarges the relative fluctuation of the line shifts and manifests itself in the observed increasing inhomogeneous width. However a quantitative analysis of the absorption bands requires the formalism of the stochastic theory, outlined in Section 10.2, which is able to connect the measured solvent shifts and inhomogeneous bandwidths to two microscopic parameters of the system, namely the respective number densities r of matrix units around a dye molecule and the depths e of the dye-matrix interaction potentials. While for polymer matrices the stochastic theory was to be used to determine geometric parameters [4], they are already known for our rare gas model systems from independent investigations. This enabled us to reduce the number of fit factors and calculate the r and e values as listed in Tab. 10.3.
Table 10.3: Potential parameters e and number densities r calculated with (^ r; ^e) and without (r, e) matrix correlations. rcryst : number density for pure rare gas crystals.
Argon Krypton Xenon
^e [cm –1]
e [cm –1]
^ [Å–3] r
r [Å–3]
rcryst [Å–3]
23.1 33.4 47.5
1.57 2.45 4.58
0.00975 0.00787 0.00594
0.145 0.104 0.062
0.0267 0.0222 0.0173
The widely used continuum approximation turned out to lead to the unreasonable result that r exceeds the number densities measured in bulk rare gas crystals! The physical reason for this discrepancy is that the continuum approximation does not prevent matrix units from occupying identical positions, since their mutual steric exclusion has been neglected! Therefore the calculation is based upon too many matrix units – reflected in the unrealistically large number density – giving insufficient weight to the individual matrix unit [20]. Taking matrix correlations into account, as sketched in Section 10.2, the depth of the dye-matrix interaction potential e^ is increased and the number density r^ reduced to appropriate values (Table 10.3). 189
10 Matrix-Molecule Interaction in Dye-Doped Rare Gas Solids
10.6
Pressure effects
In Fig. 10.2 the effect of hydrostatic pressure on a spectral hole, as measured with our system II, in an argon matrix is shown. A pressure increase causes a red shift and broadening of the initial hole. It has to be emphasized, however, that due to the construction of our cryostat, pressure changes were always connected with temperature changes following the helium vapour pressure curve. Therefore the measured broadening and shift of spectral holes do have thermal and dynamical contributions and are not only due to the pure pressure effect. These two (thermal and dynamical) contributions to the hole shift are separated in Fig. 10.3 for the case of argon. The measured hole shift is plotted versus the pressure increase Dp. The pure pressure contribution to the hole shift was extracted by taking into account the thermal expansion of the matrix (Section 10.2). Figure 10.3 also shows that the
Figure 10.2: Broadening and pressure shift of a spectral hole (H2Pc in an argon matrix). Trace (1): Original hole profile; Trace (2): Spectral hole after a pressure increase of 88 kPa.
Figure 10.3: Shift of hole minimum vs. pressure change: Raw data (squares) and data after temperature correction (bullets).
190
10.7
Rare gas mixtures
pure pressure shift depends linearly on the pressure increase and is larger than the total observed pressure shift due to the opposite signs of temperature and pressure effect (Eq. 8). Using Eq. 7 it is now possible to calculate the matrix compressibility k. The observed linear pressure shift towards lower energies suggests that the interacting matrix atoms are located in the attractive part of the Lennard-Jones potential. The van der Waals potential is therefore expected to be a good approximation, which was employed in the derivation of the analytical expression for the pressure dependence (Eq. 8). Indeed, Tab. 4 shows that the results obtained, using the Lennard-Jones and the van der Waals potential respectively, are identical within the accuracy of our experiment. Table 10.4: Matrix compressibilities in units of GPa –1 determined with (^ k) and without (k) correlations. Calculation using Lennard-Jones (LJ) and van der Waals (vdW) potentials.
Argon Krypton Xenon
^LJ k
kLJ
kvdW
0.2500 0.2899 0.3768
0.2429 0.2653 0.3919
0.2758 0.2895 0.4191
Of further interest is the question whether matrix correlations are of the same importance for the calculation of the matrix compressibility as they are for the potential parameter and the number density. A detailed analysis, however, reveals that for our model systems as well as for H2Pc-doped polymers PE and PS the corrections to the simple equations due to matrix correlations are smaller than the experimental error. This explains the good agreement between optically determined compressibilities, calculated conventionally without consideration of matrix correlations, and mechanically determined bulk values for polymeric systems.
10.7
Rare gas mixtures
So far experiments with H2Pc-doped pure rare gas matrices have been reported. As discussed above, the transition from a polycrystalline to an amorphous solid can be studied in a single system using condensed rare gas mixtures, on which we will focus now. The Qx absorption bands of H2Pc-doped matrices with various Ar-Xe compositions are plotted in Fig. 10.4 together with the spectra for the pure argon and xenon matrices. In contrast to the slight asymmetry of the spectra in the pure rare gas matrices, which was already mentioned above, the absorption bands in rare gas mixtures can be well-described by a Gaussian line shape. For all composition ratios the Qx bands are located within the spectral range given by the absorption lines of the two pure rare gas hosts. However, it is not possible to describe the spectra of the mixed matrices as a linear combination of the pure constituents spectra. This demonstrates that the gases are homogeneously mixed before condensation as well as in the solid matrix. 191
10 Matrix-Molecule Interaction in Dye-Doped Rare Gas Solids
Figure 10.4: Normalized absorption spectra of the Qx band of H2Pc in solid Ar1–xXex matrices. From left to right: x = 1 (pure Xe), x = 0.6, 0.4, 0.3, 0.15, x = 0 (pure Ar).
More insight into the dependence of the spectra upon the composition ratio can be gained if the frequency of the absorption maximum is plotted against the Xe concentration. This is done in Fig. 10.5, where the solvent shift is also given. The latter rises continuously with increasing Xe concentration. The solid line in Fig. 10.5 was calculated with Eq. 1 extended to mixed systems. The density of the matrix units is interpolated from the values of the pure components assuming a constant atomic volume, as described in [39]. Using this simple approach, the calculated
Figure 10.5: Position of the maximum of the Qx band (left-hand scale) and solvent shift nS (right-hand scale) of H2Pc in Ar1–xXex matrices versus Xe concentration. The solid line corresponds to the interpolation of the mass density on the basis of constant atomic volume.
192
10.7
Rare gas mixtures
curve agreed quite satisfactory with the experimental data. The slight deviation from the theoretical curve may have its origin in uncertainties of the respective matrix densities [40] and compositions due to different sublimation temperatures of argon and xenon. The quasi-homogeneous line width is obtained by extrapolating the spectral hole width to zero hole area, assuming that the latter depends linearly on the burning fluence [8]. Thus, saturation effects may be excluded [41].The quasi-homogeneous line widths are determined for all mixed Ar-Xe matrices with a sufficient optical density at a temperature of 1.8 K. The results are plotted in Fig. 10.6 versus the Xe concentration. Since the line width depends on the spectral position in the absorption band [40] the values are interpolated to the frequency of the absorption maximum of each sample. The quasi-homogeneous line width in pure argon matrices is significantly lower than in pure xenon matrices. From argon to xenon concentrations up to 50 % only minor changes for H2Pc-doped rare gas mixtures are observed with no recognizable discontinuity at the expected transition from a polycrystalline to an amorphous matrix.
Figure 10.6: Quasi-homogeneous width, as measured at the maximum of the Qx band, of H2Pc in Ar1–xXex matrices versus Xe concentration.
It is well-known that at liquid helium temperatures the fluorescence lifetime plays a minor role only for the quasi-homogeneous width whereas the dominating contributions come from fast relaxations of TLS and coupling to local modes. The experimentally observed smaller widths in argon matrices as compared to xenon matrices can therefore be attributed to the smaller TLS density, which was determined by specific heat investigations [30], and to a higher local mode frequency, measured via hole burning [20]. These two relaxation mechanisms are thought to be characteristic for disordered systems. Thus, TLS and local modes are not only present in amorphous solids but also in polycrystalline samples, in which these excitations can exist at grain boundaries as well as in the neighbourhood of im193
10 Matrix-Molecule Interaction in Dye-Doped Rare Gas Solids purities inside the crystallites. The line width behaviour at the transition to the amorphous state can therefore be attributed to the high disorder and to the porous structure that is already present in the pure matrices.
10.8
Summary
The present investigation centred on a stochastic model, which was developed to describe the inhomogeneous broadening of absorption lines of dye molecules doped into a disordered host. Furthermore, the model seems to account well for the effects of external pressure on spectral holes burned into these bands, making it possible to determine the host compressibility by performing a purely optical experiment. This model has also been widely used for dye-doped polymer hosts although they hardly satisfy the basic assumptions, making its application questionable. The aim of the present work was therefore to put the model to the test by investigating dye-doped rare gas solid systems that come closest to satisfying these requirements. In its original form the theory treats the matrix units in a continuum approximation by neglecting correlations between them. This leads to unreasonable results for microscopic parameters, which could be demonstrated for the first time in our model systems. Therefore, the theory was extended to take into account steric exclusion of matrix units. For the depth of the dye-matrix interaction potential and the local matrix density the modified theory produces physically realistic results. The behaviour is different with respect to pressure effects on spectral holes. Our investigations verified that the local matrix compressibility, measured in our experiments, is mainly sensitive to the dispersive part of the dye-matrix potential. Therefore, the details of the repulsive region of the dye-matrix potential as well as the consideration of matrix-matrix correlations cause only minor changes to the matrix compressibility, which are beyond the experimental accuracy. This results justifies previous determinations of the matrix compressibility using the original model. In order to investigate the influence of the matrix structure our experiments were extended to dye-doped rare gas mixtures of argon and xenon. In these matrices the transition, from polycrystalline pure rare gas matrices to amorphous mixed rare gas solids, can be monitored by varying the composition ratio. At the transition to the amorphous state the quasihomogeneous line width revealed no increase, which can be attributed to the high degree of structural disorder already present in the condensed pure rare gas matrices. Structural transitions can also be studied in annealing experiments. Therefore the next step in our investigations will focus on the changes of the inhomogeneous and quasi-homogeneous widths, occurring during thermal cycling experiments in rare gases and – turning to more complex systems – in dye-doped amorphous water [42]. The investigation of water matrices promises, in addition to the structural aspects, to shed some light on the importance of electrostatic interactions due to the polarity of the water molecules. 194
References
Acknowledgements
The authors gratefully acknowledge many elucidating discussions with Dr. L. Kador.
References
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36.
E. Shpol’skıˇ: Sov. Phys. Usp., 3, 372 (1960) A. Markoff: Wahrscheinlichkeitsrechnung, Teubner, Leipzig, (1912) A. Stoneham: Rev. Mod. Phys., 41, 82 (1969) B. Laird, J. Skinner: J. Chem. Phys., 90, 3274 (1989) A. Ellervee, R. Jaaniso, J. Kikas, A. Suisalu, V. Shcherbalov: Chem. Phys. Lett., 176, 472 (1991) B. Kharlamov, R. Personov, L. Bykovskaja: Opt. Commun., 12, 191 (1974) A. Gorokhovskií, R. Kaarli, L. Rebane: JETP Lett., 20, 216 (1974) J. Friedrich, D. Haarer: Angew. Chem. Int. Ed. Engl., 23, 113 (1984) L. Kador: PhD thesis, Universität Bayreuth, (1988) R. Altmann: PhD thesis, Universität Bayreuth, (1992) T. Sesselmann, W. Richter, D. Haarer, H. Morawitz: Phys. Rev. B, 36, 7601 (1987) S. Reul, W. Richter, D. Haarer: J. Non-Cryst. Sol., 145, 149 (1992) G. Pollak: Rev. Mod. Phys., 36, 748 (1964) M. Klein, J. Venables (eds): Rare Gas Solids, Academic Press, London, (1977) L. Kador: J. Chem. Phys., 95, 5574 (1991) H. Sevian, J. Skinner: Theor. Chim. Acta, 82, 29 (1992) L. Kador: J. Chem. Phys., 99, 7 (1993) L. Kador, P. Geissinger: Mol. Cryst. Liq. Cryst., 252, 213 (1994) S. Simon, V. Dobrosavljevic, R. Stratt: J. Chem. Phys., 93, 2640 (1990) P. Geissinger: PhD thesis, Universität Bayreuth, (1994) P. Geissinger, L. Kador, D. Haarer: Phys. Rev. B, 53(8), 4356 (1996) M. Sapozhnikov: J. Chem. Phys., 68, 2352 (1978) C. Tilford, C. Swenson: Phys. Rev. B, 5, 719 (1972) G. Horton: Am. J. Phys., 36, 93 (1968) F. London: Z. Phys., 63, 245 (1930) N. Ashcroft, N. Mermin: Solid State Physics, Saunders College, (1976) N. Bernardes: Phys. Rev., 112, 1534 (1958) Landolt-Börnstein: Zahlenwerte und Funktionen, in: A. Eucken (ed.): I. Band: Atom- und Molekülphysik, p. 399, Springer, (1950) S. I. Kovalenko, E. I. Indan, A. A. Khudoteplaya: Phys. Stat. Sol. (a), 13, 235 (1972) H. Menges, H. Löhneysen: J. Low Temp. Phys., 84, 237 (1991) M. Schneider, A. Rahman, I. Schuller: Phys. Rev. B, 34, 1802 (1986) P. Geissinger, D. Haarer: Chem. Phys. Lett., 197, 175 (1992) L. Bajema, M. Gouterman, B. Meyer: J. Mol. Spectry., 27, 225 (1968) K. Takeda, Y. Osamu, H. Suga: J. Phys. Chem., 99, 1602 (1995) V. Bondybey, J. English: J. Am. Chem. Soc., 101, 3446 (1979) P. Geissinger, W. Richter, D. Haarer: J. Lumin., 56, 109 (1993)
195
10 Matrix-Molecule Interaction in Dye-Doped Rare Gas Solids 37. 38. 39. 40. 41. 42.
196
P. Fitch, A. Haynam, D. Levy: J. Chem. Phys., 73(3), 1064 (1980) M. Anderson, C. Swenson: J. Phys. Chem. Solids, 36, 145 (1975) M. Loistl, F. Baumann: Z. Phys. B – Condensed Matter, 82, 199 (1991) T. Giering, P. Geissinger, L. Kador, D. Haarer: J. Lumin., 64, 245 (1995) L. Kador, G. Schulte, D. Haarer: J. Phys. Chem., 90, 1264 (1986) T. Giering, D. Haarer: J. Lumin., 66 & 67, 299 (1996)
II
Mainly Micelles, Polymers, and Liquid Crystals
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties Deutsche Forschungsgemeinschaft (DFG) Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
11
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions Heinz Hoffmann
11.1
General behaviour of surfactants
Surfactants consists of molecules with a hydrophobic and a hydrophilic part. Due to this amphiphilic nature these molecules adsorb from aqueous solutions onto interfaces, as is expressed in their name [1]. The molecules are densely packed in these adsorbed monolayers. In the aqueous bulk phase the surfactant molecules assemble above a characteristic concentration, called cmc, into micellar structures, which can be understood as interfaces in the bulk solution. The driving force for the adsorption and the aggregation is the same for both processes and is given by the hydrophobic interaction [2]. The molecular packing of the surfactant molecules in films and micelles is mainly determined by the area a which a surfactant molecule requires at the interface. In both, films and micelles, the molecules will occupy about the same area [3]. If the area a is larger than the cross section a0 of the hydrocarbon chain in its equilibrium conformation the interface of the micelle will be curved towards the hydrocarbon core. If it is the same the interface will be flat on a local scale and if it is smaller the interface will be curved the other way around. The importance of the area a for the structures of micelles was recognized by Tanford [2] and later by Ninham and co-workers [4]. From simple geometrical considerations it follows that the shape of a micelle can be expressed by the packing parameter P = ar/v, where r is the length of the hydrocarbon chain and v its volume. The packing parameter varies from 3 for globules to 1 for bilayers. As a result of their packing parameter single-chain surfactants often form globular micelles and double-chain surfactants tend to form bilayer structures. In a film at a macroscopic interface the area a of the molecule controls the thickness of the film and the order parameter of the hydrocarbon chains inside the film. If a > a0 the film is thinner than the length of the surfactant chain and water borders directly on the hydrocarbon chains of the surfactant resulting in a large interfacial tension. If a & a0 the film looks like one half of a bilayer of a real membrane. In this case the water is not in contact with the hydrocarbon chains and the interface has a low interfacial tension. In this qualitative model the parameter a controls the curvature of a micelle and the interfacial tension at an interface. Note that the differently shaped micelles are present only in surfactant solutions. When the micellar solutions are in contact with hydrocarbon the micelles will solubilize hyMacromolecular Systems: Microscopic Interactions and Macroscopic Properties Deutsche Forschungsgemeinschaft (DFG) Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
199
11
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
drocarbon and will be transformed into microemulsion droplets. Then the curvature of the droplets correlates with the interfacial tension [5]. The minimum of this interfacial tension corresponds to the maximum of the solubilization capacity of the surfactants [6, 7]. This relation has been used extensively to optimize surfactant systems for tertiary oil recovery. On the basis of theoretical work the values of interfacial tension at the minimum seems to depend on the bending constants of the films [8]. A high (low) minimum of interfacial tension would mean a high (low) bending constant.
11.2
From globular micelles towards bilayers
Amphiphilic substances with an extremely small hydrophilic group, for example aliphatic alcohols with intermediate chain lengths, are called cosurfactants. Normally cosurfactants do not form micelles in aqueous solutions but they are surface active and make up mixed micelles with normal surfactants. In cosurfactant/surfactant mixtures the mean area a per head group must vary smoothly with the mole fraction of the cosurfactant, between the value for the pure surfactant as and the value for the cosurfactant ac. For most uncharged surfactants the value as corresponds to a situation where normal globular micelles are formed while ac corresponds to a situation where inverse micellar structures of the L2 phase exist. In principle, by varying the mole fraction Xc of the cosurfactant in aqueous solutions, we should encounter all the types of micellar structures and mesophases that can possibly exist in surfactant solutions. While the area per head group can vary smoothly with Xc, it is evident that the curvature of the micellar interface cannot vary continuously from strongly convex to strongly concave for micelles of the same type. The system has to switch to differently shaped micelles and mesophases if Xc is varied. There is evidence that the systems use defects in mesophases to adjust the mean curvature that is set by Xc. Strain in a mixed system can also be avoided by concentration fluctuations. The system can form structures with two different Xc values that are in equilibrium, one with a lower Xc and one with a higher Xc than the average value. For all these reasons we find a rich variety of micellar structures and phases if Xc is varied and the total concentration is kept constant. The diversity which is encountered in such situations is shown in Fig. 11.1 which represents the phase diagram of the ternary system tetradecyldimethylaminoxide/hexanol/water (C14DMAO/C6OH/H2O) at the water-rich corner. With increasing Xc we observe the six visibly different single-phase regions L1 (micellar solution), L*1 , Lal, L*3 (vesicle phases), Lah (lamellar phase with flat lamellae), and L3 (bicontinuous sponge phase) [9]. All these phases have distinct properties. Furthermore there is a change in shape of the micelles in the L1 phase from globules to rods. This transition is, however, not visible to the unaided eye. The phases L1, L*1 , L*3 , and L3 are optically isotropic while the phases Lal and Lah are birefringent (the Lal phase is only weekly birefringent in most cases). The general features of this diagram are typical for such ternary systems and are practically independent of the chain lengths of the surfactant and the cosurfactant. Figure 11.1 200
11.2 From globular micelles towards bilayers
400 2Φ L3
6
cC OH mM
300 2Φ
200
Lαh Lαl-h L3 *
100 2Φ 0
0
L1
L*1 50
cC
Lαl
100 14
D MAO
/ mM
150
200
Figure 11.1: Section of the phase diagram of the ternary system C14DMAO/C6OH/H2O in the waterrich corner at 25 8C. For details see Section 11.4.1 and Fig. 11.11.
shows that the phases with lamellar structures are already formed at total surfactant concentrations around 1 wt%. These phases, except the Lal and Lah phase, which probably do not develop a phase boundary, are separated from each other by two-phase regions. Three phase regions can also exist in the phase diagram although all phases consist of about 99 wt% water. Therefore and because of the very similar densities of the different phases the phase separation takes a long time and hence the determination of such phase diagrams is dificult. Furthermore the refractive indices of the phases are almost the same, which makes it difficult to distinguish one and two-phase regions unambiguously as the single phases are often slightly turbid. Inspite of that most two-phase systems separate macroscopically into two phases after a sufficiently long time. The phase diagram of Fig. 11.1 serves as an example of what happens if the radius of curvature varies continuously with the mixing ratio Xc at a constant surfactant concentration. For the classic non-ionic alkyl polyglycolethers (C12E5) this can be done by varying the temperature [10]. For ionic surfactants this can be done by increasing the salinity [11], by mixing a cationic with an anionic surfactant, or vice versa [12]. In all these different situations we can expect to observe the same sequence of phases. The phase diagram in Fig. 11.1 was established on the basis of visual observation of samples. To check for birefringence the samples were viewed between crossed polarizers. Before we come to the microstructures there are several facts that are worth emphasizing. Note that the phase boundaries between the different phases are given by more or less straight lines, which means that the mixing ratio for the micelles is constant at the phase boundaries and does not depend on the concentration. Notice also that the equimolar mixtures of surfactant and cosurfactant are in the middle of the wide La region. The combination therefore acts like a real double-chain surfactant. The phases from L*1 to Lah are actually subphases of La, as has been demonstrated by freeze fracture electron micrographs. They consist of bilayer-type structures [13] but the topology of the bilayers is quite different.
201
11
11.3
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
Viscoelastic solutions with entangled rods
11.3.1 General behaviour Surfactant solutions with globular micelles are generally Newtonian liquids with a low viscosity which increases linearly with the volume fraction F of the particles according to Einstein’s law s
1 2:5 F :
1
Here Zs is the viscosity of the pure solvent. F is an effective volume fraction which also takes into account the hydration of the molecules. It can be up to three times the true volume fraction. But also in this case the viscosity of a 10 wt% surfactant solution is only about twice as high as the solvent viscosity. The same is true if anisometric micelles are present in the solutions as long as their rotational volumes do not overlap. On the other hand, many surfactant solutions are highly viscous even at low concentrations in the range of 1 wt%. From this observation it can be concluded that the micellar aggregates in these solutions must organize themselves into some kind of a supermolecular network. The viscosity of such systems strongly depends on parameters like surfactant concentration, ionic strength, temperature, or concentration of additives. The solutions usually also have elastic properties because the zero shear viscosity Z0 is the result of a transient network of entangled rods that is characterized by a shear modulus G0 and a structural relaxation time t according to 0 G0 t :
2
In this case the shear modulus is determined by the particle density n of entanglement points G0 n kT :
3
The networks of entangled cylindrical micelles could be made visible by cryo-electron microscopy by Talmon et al. [14]. These pictures clearly show the shape and the persistence length of the rods but they do not reveal their dynamic behaviour. According to Eq. 2 the viscosity is the result of structure and dynamic behaviour, i. e. the structural relaxation time constant t which strongly depends on many parameters and can vary by many orders of magnitude for the same surfactant, if for instance the counterion concentration is changed [15]. This is shown in Fig. 11.2 where Z0 for several cetylpyridiniumchloride (CPyCl) concentrations is plotted vs. the sodiumsalicylate (NaSal) concentrations. With increasing amounts of NaSal the viscosity passes a maximum, then a minimum, and finally a second maximum. This behaviour is due to a corresponding dependence of t on the NaSal concentration while G0 is independent of this parameter for a constant CPyCl concentration. Simi202
11.3
Viscoelastic solutions with entangled rods
106 30mM CPyCl 60mM CPyCl 100mM CPyCl
ηo / mPas
105 104 103 102 101 101
102
103
cN a S a l / mM
Figure 11.2: Double logarithmic plot of the zero shear viscosity Z0 for three different CPyCl concentrations vs. the concentration of added NaSal at 25 8C.
lar results have been observed for many systems by various groups. The viscosity of a 1 wt% surfactant solution varies from the solvent viscosity and up to a 106 times higher value. Figure 11.3 shows a schematic sketch of a network of rod-like micelles. It is generally assumed that the crosslinks of the network that cause the elastic behaviour are entanglements [16]. However, this is not always true. Adhesive contacts between the micelles or a transient branching point, like a many armed disc-like micelle, can act as crosslinks [17, 18]. Some experimental evidence for both possibilities have recently been observed. The entangled thread or worm-like micelles have typical persistence lengths between some 100 to 1000 Å and they may, or may not, be fused together at the entanglement points.
Figure 11.3: Schematic drawing of an entanglement network of long cylindrical micelles. Note the different length scales: k is the mesh size, l the mean distance between two knots, and m the contour length between two knots.
The cylindrical micelles have an equilibrium network conformation. They constantly undergo translational and rotational diffusion processes. They also break and reform. If the network is deformed or the equilibrium conditions are suddenly changed it will take some time until the system reaches equilibrium again. If a shear stress p21 is applied to a network 203
11
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
solution in a much shorter time than the equilibration time, the solution behaves like a soft material that obeys Hooke’s law, p21 G0 g ;
4
with the spring constant G0 and the deformation g. On the other hand, if the stress is applied for a longer time the system flows like a Newtonian liquid, p21 0 g_ ;
5
with the zero shear viscosity Z0 and the shear rate g_ . A mechanical model for a viscoelastic fluid is the so-called Maxwell model which consists of a spring with the constant G0 and a dashpot with the viscosity Z0. The zero shear viscosity of such a system can be expressed by the product of G0 and t according to Eq. 2. Both quantities can be determined by oscillating rheological measurements [19]. Many viscoelastic surfactant solutions can be described in a large frequency range by the Maxwell model with a single shear modulus G0 and a single structural relaxation time constant t [20]. This is shown in Fig. 11.4 a. However, there are surfactant solutions which behave in a completely different manner as can be seen from Fig. 11.4 b. The systems do not show a frequency-independent plateau value of the modulus and the viscosity cannot be expressed by a single G0 or a single t value. In such situations the shear stress after a rapid deformation relaxes exponentially [21], p21 p^ 21 exp
t a : t
6
102
103
101
102
100
101 G´ G´´ |η*|
10-1 10-2 -3 10
10-2
10-1 f/Hz
|η*| / P as
G´,G´´ / P a
Figure 11.4 a shows that the loss modulus G@ increases again with the frequency f. This increase can be related to Rouse modes of the cylindrical micelles. On the basis of a
100 100
10-1 101
Figure 11.4 a: Double logarithmic plot of storage modulus G', loss modulus G@, and complex viscosity |Z*| vs. frequency f for a solution with 100 mM CPyCl and 60 mM NaSal at 25 8C. The solution behaves like a Maxwell fluid with a single shear modulus G0 and a single structural relaxation time t.
204
11.3
Viscoelastic solutions with entangled rods
102
101
101
100
10-1 -3 10
G´ G´´ |η*| 10-2
10-1 f / Hz
100
100
101
|η*| / Pas
G´, G´´ / Pa
102
10-1
Figure 11.4 b: The same plot for a solution with 80 mM C14DMAO, 20 mM SDS , and 55 mM C6OH. The solution does not behave like a Maxwell fluid. Note the differences to Fig. 11.4 a: G@ does not pass over a maximum, G' does not show a plateau value but increases with f after the intersection with G@ with a constant slope of 0.25.
theoretical model [22] the minimum value of G@ can be expressed by the storage modulus G', the entanglement length le and the contour length lk of the cylindrical micelles according to G00min G0
le : lk
7
11.3.2 Viscoelastic systems Rod-like micelles from ionic surfactants are usually formed at high ionic strengths with strongly binding, or hydrophobic counterions, or with large hydrophobic groups, like double-chain or perfluoro surfactants [4]. Solutions of such surfactants become highly viscous and viscoelastic with increasing concentration. In Fig. 11.5 some results are shown as double logarithmic plots of viscosity vs. concentration [23–25] in spite of the different chemistry of the surfactants. All these surfactants show a concentration region where the slopes are the same and in the range of 8.5, which is very high. The viscosity starts to rise abruptly at an overlap concentration c* and follows the scaling law within a small transition concentration 0 /
c x ; c
8
where x is about 8.5 ± 0.5. The exponent must therefore be controlled by the electrostatic interaction in the solutions and is much bigger than the value of 4.5 ± 0.5 expected for large polymer molecules, which do not change their size with increasing concentration. From the large exponent it can be concluded that the rod-like micelles continue to grow with concentration above c*. This was postulated by McKintosh et al. [26]. 205
11
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions 106
Lec / C14 DMAO / SDS C8 F17 SO3 NEt4 C16 TMASal C16 PyCl + NaSal C16 C8 DMABr
105 ηο / mPas
104 103 102 101 100 100
101
102
c / mM
Figure 11.5: Double logarithmic plot of zero shear viscosity Z0 vs. concentration c for several solutions of charged surfactants. Note that all different systems show the same power law exponent within a limited concentration range above c*.
Figure 11.6 shows logarithmic plots of Z0 vs. c for the system CPyCl + NaSal [27]. It has been shown that this system can be described by the Maxwell model with one G0 and one t value above the first viscosity maximum. In this range the complex behaviour is due to the concentration dependence of t, while G0 steadily increases with concentration. Furthermore, it was found that t is controlled by the kinetics of breaking and reforming of the micelles [28]. These processes are faster than the reptation of the rods under the given experimental conditions. As already mentioned, the same is true for the system with a constant CPyCl concentration with increasing amounts of NaSal, which is shown in Fig. 11.2. The dependence of the viscosity on the counterion concentration is controlled by a corresponding behaviour of t, while G0 is independent of the NaSal concentration. The viscosity is therefore a result of the dynamics of the system and not of its structure. This can be proved by cryo-electron microscopy [14]. The electron micrographs show no differences between the structure of the 105
CPyCl +NaSal CPyCl +0.3 MNaSal uncharged
4
ηο / mPas
10
103 102 101 0 10
101
c / mM
102
103
Figure 11.6: Double logarithmic plot of zero shear viscosity Z0 vs. concentration c of CPyCl + NaSal for solutions at the first viscosity maximum (o), at the second maximum (p), and at the minimum (_) at 25 8C (Fig. 11.2).
206
11.3
Viscoelastic solutions with entangled rods
micelles for all four concentration regions. This is very remarkable because the micelles are differently charged in the concentration regions. Below the first maximum of the viscosity they are highly and positively charged, at the minimum they are completely neutral, and at the second maximum they carry a negative charge. The power law behaviour in the different concentration regions is also completely different, as can be seen from Fig. 11.6. The exponent at the first maximum is 8, at the minimum 1.3 and at the second maximum 2.5. No theoretical explanation is available for the extremely low exponent of 1.3 which has also been observed for systems which completely differ in chemistry and conditions. Therefore the exponent seems to represent a general behaviour determined by fundamental physics. Figure 11.7 shows logarithmic plots of Z0 vs. c for zwitterionic alkyldimethylaminoxide surfactants (CxDMAO) [17]. The data again show a power law behaviour over extended concentration regions. Some curves show a break indicating that also uncharged systems can switch the relaxation mechanism. At the lowest concentration region, where a power law behaviour is observed, the slope is the highest and almost the same as for the observed polymers. This is somewhat surprising because it could be expected that the length of the micellar rods increases with increasing concentration, which should lead to a higher exponent of the power law. It is therefore likely that the dynamics of the systems is already influenced by kinetic processes under these conditions.
106 105
C14DMAO C16DMAO OleylDMAO
ηo / mPa s
104 103 102 101 100 100
101
102 c / mM
103
Figure 11.7: Double logarithmic plots of zero shear viscosity Z0 vs. concentration c for solutions of alkyldimethylaminoxide surfactants of various chain lengths at 25 8C.
For higher concentrations a smaller exponent is observed. Obviously a new mechanism is operating under these conditions which is more effective in reducing a stress than the mechanism in the low concentration region. Generally a mechanism can only become dominant with increasing concentrations if it is faster than the one at low concentrations. The moduli increase with the same exponent in the various concentration regions. Furthermore, Fig. 11.7 shows that the absolute value of Z0 for C16DMAO and ODMAO (Oleyl) differ by an order of magnitude even though the slope is the same in the high concentration region. This is due to the fact that in the kinetically controlled region the breaking of the micelles depends very much on the chain length of the surfactant. 207
11
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
According to the theory of micelle formation, addition of cosurfactants to surfactant solutions leads to a transition of spherical micelles to rods or to a growth of rod-like micelles [29]. As a consequence the viscosity of a surfactant solution increases with increasing cosurfactant concentration. This is shown in Fig. 11.8 for the system of 100 mM C14DMAO with various cosurfactants. The viscosity first increases and then passes a maximum. The situation is similar as shown in Fig. 11.2. It is likely that the micelles still grow steadily with increasing cosurfactant concentration but the system switches from one mechanism on the left side of the maximum to a faster mechanism on the right side. The reason for the switch is probably that the rods become more flexible with increasing cosurfactant concentration. The different mechanisms become obvious in Fig. 11.9 where log (Z0 ) is plotted vs. log (c) for C14DMAO with different amounts of decanol (C10OH), which is completely solubilized in the micelles due to its poor solubility in water. The plot shows that the slopes of mixtures at the left side of the maximum are the same while the mixture with the highest cosurfactant/surfactant ratio has the lowest slope of 1.3. This value is equal to the one for CPySal at the viscosity minimum. Systems with similar low slopes from the literature [30] are shown
103
hexanol octanol decanol lecithin
ηo / mPas
102
101
100
0
10
20
30 40 c / mM
50
60
Figure 11.8: Semilogarithmic plot of zero shear viscosity Z0 of a 100 mM solution of C14DMAO vs. the concentration c of added cosurfactants at 25 8C. Note that all curves pass a maximum.
104
ηo / mPas
103
TDMAO/C10OH=5:1 TDMAO/C10OH=6.6:1 TDMAO/C10OH=10:1 TDMAO/C10OH=20:1 TDMAO
102 101 100
101
102
103
C14DMAO/ mM
Figure 11.9: Double logarithmic plot of zero shear viscosity Z0 of a mixture of C14DMAO and C10OH with different molar ratios of cosurfactant/surfactant vs. the concentration of C14DMAO at 25 8C.
208
11.3
Viscoelastic solutions with entangled rods
ηo / mPa s
103
102 CPyCl+NaSal (T=20oC) C14DMAO/C10OH=5.1(T=25oC) C16EO7 (T=45oC) CTAB+NaClO3
101 101
102 c / mM
103
Figure 11.10: Double logarithmic plot of zero shear viscosity Z0 vs. the total surfactant concentration for several surfactant systems with the same power law exponent of 1.3.
in Fig. 11.10. The chemistry and also the viscosity values for these systems are completely different, yet the slope is the same. The shear moduli for these systems are very similar for given surfactant concentrations and they also scale with the same exponent. The low exponent for the viscosity therefore comes about by the structural relaxation time which scales with an exponent of –1 according to t/
c c
1
:
9
11.3.3 Mechanisms for the different scaling behaviour All studied surfactant systems show the same qualitative behaviour. The viscosity rises abruptly at a characteristic concentration c* which is the lower the longer the chain length of the surfactant is. At the concentration c* the rotational volumes of rod-like micelles start to overlap and form a network. This network can be an entanglement network, as in polymer solutions, or the micelles can be fused together, or can be held together by adhesive contacts. All these types have been proposed and it is conceivable that they can really exist [31]. Theoretical treatments assume that the cylindrical micelles are worm-like and flexible. This can be the case for some of the presented systems, but certainly not for the binary surfactant systems. For example, C16DMAO and ODMAO have very low c* values and if the micelles would be flexible they would have to be coiled below c*. But both, electric birefringence and dynamic light scattering determinations, show that at c 7 c* the lengths of the rods are comparable with the mean distances between them. Hence, the rods must be rather stiff with persistence lengths of some 1000 Å. Similar results have been obtained by electron micrographs. The abrupt increase of Z0 at c* is difficult to understand for stiff rods even taking into account further growth of the rods with increasing concentration. Many systems with rod-like particles show that the rotational time constant for the rods is very little affected around c* and the solutions do not become viscoelastic above c*. We therefore have to assume that other interactions than just hard core repulsion between the rods must exist which 209
11
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
are responsible for the formation of the network at c*. It is conceivable that the rods form adhesive bonds or that they actually form a connected network of fused rods, as has been proposed by Cates [32]. In such situations two different types of networks have to be distinguished, namely saturated and interpenetrating networks. In the first case the mean distance between the knots or entanglement points is equivalent to the meshsize but in the second case this distance can be much larger than the meshsize between neighbouring rods. The viscosities above c* increase abruptly following the power law (Eq. 8) with an exponent x > 3.5, while the exponent of the power law for the shear modulus is always about 2.3. This behaviour has been treated in detail by Cates et al. [33]. The structural relaxation times are affected by both, reptation and bond breaking processes. Cates treats three different kinetic mechanisms. The first consists of the break of a rod with the formation of two new end caps. In the recombination step the rods have to collide at the ends in order to fuse into a new rod. In the second mechanism the end cap of one rod collides with a second rod and in a three-armed transition state a new rod and a new end cap is formed. In the third mechanism two rods collide and form two new rods through a four armed transition state. It is obvious that stress can release all three mechanisms. These mechanisms lead to somewhat different power laws for the kinetic time constant t according to t/
c x : c
10
But in all cases x is between 1 and 2. Mechanism 3 is less likely in systems with low c* values and stiff rods. For this argument it is likely that mechanism 2 or 3 is effective in the more concentrated region of the pure CxDMAO solutions. For C16DMAO and ODMAO solutions the slope of the log (Z0 )-log (c) plots suddenly changes at a characteristic concentration c**. For both regions the same scaling law for the shear modulus with an exponent of about 2.3 is found while the power exponent for the relaxation times changes from 1 to zero. From the constant exponent for G0 it can be concluded that the structures in both concentration regions are the same. The change in the slope must therefore be due to a new mechanism which becomes effective above c**. The independence of t of the concentration makes it likely that in this region the dynamics are governed by a purely kinetically controlled mechanism and that a reptation process is no longer possible. This situation has not yet been treated theoretically. Cates mentioned, however, that there might be situations where the reptation loses its importance. The more effective mechanism in this range could be the bond interchange mechanism. For the C14DMAO solutions with C10OH and the CPySal system at the minimum viscosity the extremely low power law exponent of 1.3 for the viscosity and an exponent of –1 for the structural relaxation times are found. A detailed explanation for this behaviour which has also been described by other authors [30] has not yet been given. The explanation for this behaviour could be that the cylindrical micelles for systems with such low exponents are very flexible. In such a situation the persistence length would be much shorter than the contour length between two neighbouring entanglement points. Furthermore, the persistence length should be independent of the concentration. The diffusion of the rods can therefore be described by a constant diffusion coefficient D. For two arms to collide they have to diffuse a distance x and for two neighbouring rods to undergo a bond exchange process they have to diffuse at least the average distance x between two arms. The time constant tD for 210
11.4
Viscoelastic solutions with multilamellar vesicles
the diffusion should be proportional to x2/D. Since the meshsize x decreases with the square root of the concentration one obtains for the structural relaxation time t the observed law t ! 1/D 7 c, which is identical with Eq. 9. We therefore conclude that for systems with the low exponent 1.3 the viscosity is governed by a diffusion-controlled bond interchange mechanism. The absolute values of Z0 and t can still vary from system to system because the persistence length lp of the rods should depend on the particular conditions of the systems. With increasing chain length lp should decrease and D increase. For such situations we would expect to find the lowest activation energies for the viscosity. A similar mechanism could be based on the assumption of connected or fused threadlike micelles as crosslinks. These crosslinks could be visualized as disc-like micelles from which the rods extend. This means that the transient intermediate species in the various bond interchange mechanisms are now assumed to be stable. In this situation all end caps could be connected. The resulting network could be in the saturated or unsaturated state. The crosslink points could then slide along the thread-like micelles like a one-dimensional diffusion process with a concentration-independent diffusion coefficient. A knot can be dissolved if two network points meet on their random path. If the structural relaxation time is determined by this random movement a similar equation t ! 1/c can be derived. Both models can describe the low exponent of 1.3 for the scaling law for Z0 and for both models reptation is no longer necessary for the release of stress. The mechanisms could probably be distinguished by the concentration dependence of the self-diffusion coefficients Ds of the surfactant molecules. In a solution with a connected network a surfactant molecule should be in the same situation as in a L3 phase for which it has been shown that Ds is independent of the surfactant concentration [34]. As Kato et al. [35] showed that the Ds values increase with concentration for a corresponding system with rods a diffusion limited bond interchange mechanism is more likely for the explanation of the scaling law of the structural relaxation time than the assumption of connected networks of thread-like micelles.
11.4
Viscoelastic solutions with multilamellar vesicles
11.4.1 The conditions for the existence of vesicles Thermodynamically stable vesicles have been found recently in many solutions with various types of surfactants [36–39]. Such vesicles occur especially in ternary systems of zwitterionic or non-ionic surfactants, aliphatic alcohols as cosurfactants, and water. As explained in detail in Section 11.4.2, this is due to the spontaneous curvature of the micellar surface which is continuously lowered with increasing amounts of the cosurfactant because of its very small headgroup area [3]. The systems try to come as close as possible to the spontaneous mean curvature without causing much bending energy by adjusting the two main curvatures on the micellar aggregates. As a consequence the micellar system undergoes several phase transitions with increasing cosurfactant concentration. 211
11
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
The vesicle phase occurs within a wide range of the total surfactant and cosurfactant concentration but only within a small range of the molar ratio of cosurfactant/surfactant around 1 : 1. If the surfactant bilayers are charged by the addition of an ionic surfactant the phase diagram becomes somewhat simpler because some mesophases are suppressed by the charge. As can be seen from the Fig. 11.11, the vesicle phase is still found under these conditions but it is shifted towards higher cosurfactant concentrations. Thermodynamically stable vesicles have also been found in ternary systems of non-ionic alkylpolyglycol surfactants, cosurfactants, and water [10] and also in the binary system of the double-chain surfactant didodecyldimethylammoniumbromide (DDABr) and water [42].
600
500 Lα
400
300
L1 +L3 L3 L3 Lαh L1* Lαl
L*1 L1 +L*1
L1 +Lα 200 L1 +L*1 100
L1
L1 0,0
cC6OH / mM
L1 +L2
0,2
0,4 XC
0,6 14
0,8
0 1,0
TMA B r
Figure 11.11: Section of the phase diagram of the quaternary system 100 mM C14DMAO/C14TMABr/ C6OH/H2O at 25 8C. For details see Refs. [9, 13, 40, 41].
11.4.2 Freeze fracture electron microscopy The vesicles can be made visible by freeze fracture transmission electron microscopy (FFTEM). Figure 11.12 shows the vesicles in a system of 90 mM C14DMAO, 10 mM tetradecyltrimethylammoniumbromide (C14TMABr), 220 mM n-hexanol (C6OH), and water. The cationic surfactant can also be replaced by the anionic surfactant SDS without causing a change of the vesicles or of the rheological properties. From this electron micrograph it is possible to recognize some general features which are of relevance for the properties of the 212
11.4
Viscoelastic solutions with multilamellar vesicles
Figure 11.12: Electron micrograph of vesicles in the system of 90 mM C14DMAO, 10 mM C14TMABr, 220 mM C6OH, and water (the bar represents 1 mm).
systems. The vesicles have a rather high polydispersity; some seem to be rather small unilamellar vesicles, while others consist of about 10 bilayers. The interlamellar spacing is fairly uniform and is in the range of 800 Å. The vesicles are very densely packed and the whole volume of the system is completely filled with them. They have a spherical shape even though the outermost shell can have a radius of several 1000 Å. Some of them do not consist of concentric shells but have defects. The larger vesicles have typical sizes in the range of 1 mm and the wedges, which result from the dense packing, are completely filled with smaller vesicles. Thus, each vesicle is sitting in a cage from which they cannot escape by a simple diffusion process without deforming their shells. Therefore, the system must have viscoelastic properties.
11.4.3 Rheological properties In Fig. 11.13 the viscoelastic properties of a vesicle phase of 90 mM C14DMAO, 10 mM C14TMABr, 220 mM C6OH, and water are demonstrated by plots of the storage modulus G', the loss modulus G@, and the magnitude of the complex viscosity |Z*| as a function of the oscillation frequency f. G' is much larger than G@ and almost independent of f in the whole frequency range. The system behaves like a soft solid and must have a distinct yield stress. This can be seen from the plot (Fig. 11.14) of the shear rate g_ as a function of the applied shear stress s. As can be seen from Fig. 11.15 both, shear modulus G0, the frequency-independent value of G', and yield stress sy, increase with increasing total surfactant concentration. Both quantities disappear at total concentrations below 1 wt%. This means that the vesicles are no longer densely packed and they can move around each other easily under shear flow. For higher concentrated systems sy varies linearly with G0 and is about one tenth of G0. This means the vesicles must be deformed by about 10 % before they can pass each other under shear. Similar results are obtained for the vesicles in the binary system of DDABr and water. 213
11
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions 102
G´ G´´ |η* |
101
101
100
100 10-2
10-1
f / Hz
|η*| / Pas
G´, G´´ / Pa
102
10-1 101
100
Figure 11.13: Double logarithmic plot of storage modulus G', loss modulus G@, and the magnitude of the complex viscosity |Z*| vs. frequency f for a vesicle phase of 90 mM C14DMAO, 10 mM C14TMABr, 220 mM C6OH, and water at 25 8C. 4
σ / Pa
3 2 1 0 0,0
0,1
0,2
0,3 γ. / s-1
0,4
0,5
Figure 11.14: Plot of the shear rate g_ vs. applied shear stress s for the same vesicle phase as in Fig. 11.13, showing a distinct yield stress value sy at 1.1 Pa.
50
4 3
30
σy / Pa
G´ / Pa
40
5 σy G´
20
2
10
1
0 50
100
150 csurfactant / mM
200
0
Figure 11.15: Plot of shear modulus G0 and yield stress sy vs. surfactant concentration csurfactant for a vesicle phase of C14DMAO and C14TMABr with a molar ratio of 9 : 1 and C6OH at 25 8C.
Figures 11.16 a and 11.16 b demonstrate the influence of charge density on the bilayers on G0. The modulus increases with increasing amounts of ionic surfactant and saturates at about 10 mol% of the ionic compound (Fig. 11.16 a). On addition of electrolyte the modulus decreases again linearly with the square root of the ionic strength (Fig. 11.16 b). 214
11.4
Viscoelastic solutions with multilamellar vesicles 100 mM Surfactant (C1 4 DMAO + C1 4 TMABr), 220 mM C6 OH
25
G´ / Pa
20 15 10 5 0
0
5
cC
10 1 4
15
20
TMA B r
Figure 11.16 a: Plot of shear modulus G0 vs. the concentration of the ionic surfactant C14TMABr for a vesicle phase with a total concentration of 100 mM C14DMAO + C14TMABr, and 220 mM C6OH at 25 8C. 70
G´(0.1 Hz) / Pa
60 50 40 30 20 10 0 0,0
0,5
1,0
1,5 2,0 2,5 (c N a C l / mM)1 /2
3,0
3,5
4,0
Figure 11.16 b: Plot of shear modulus G0 vs. the square root of the concentration of added NaCl for a vesicle phase with 85 mM C14DMAO, 15 mM C14TMABr, and 300 mM C6OH at 25 8C.
The effect of the chain length of the surfactant compound can be seen from Fig. 11.17. For systems with the same concentrations the modulus increases with increasing chain length from C10 to C16 and decreases for C18. This means that the modulus is not only determined by the electrostatic repulsion between the bilayers but also depends on the thickness of the bilayers. The headgroup of the surfactant in the vesicle phase has only a moderate effect on rheological properties. For example, a vesicle phase of 90 mM C12E6, 10 mM SDS, and 250 mM C6OH shows a very similar behaviour as the corresponding phase of 90 mM C14DMAO, 10 mM C14TMABr, and 220 mM C6OH. The absolute values of G', G@, and |Z*| are also very similar for both systems. The same can be found for systems where the concentration of the cosurfactant (Fig. 11.18) or the chain length of the cosurfactant is changed (Fig. 11.19). The figures show that the moduli and the yield stress increase slightly with the cosurfactant concentration and the chain length of the cosurfactant. But this increase is small in comparison with the strong dependence of these values on the concentration and the chain length of the surfactant compounds. The temperature has only a very small effect on both, G0 and sy, between 10 8C and 60 8C. An interesting effect is shown in Fig. 11.20 where the shear viscosity Z as a function of the shear rate g_ and the magnitude of the complex viscosity |Z*| as a function of the angular 215
11
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
G´ / Pa
103
x = 10 x = 14 x = 18
x = 12 x = 16
102
101 10-2
10-1
f / Hz
100
101
Figure 11.17: Double logarithmic plot of storage modulus G' vs. frequency f for vesicle phases of 90 mM CxDMAO, 10 mM C14TMABr, and 220 mM C6OH at 25 8C with various chain lengths x of the zwitterionic surfactant.
Figure 11.18: Plot of storage modulus G', loss modulus G@, and yield stress sy against the cosurfactant concentration for a vesicle phase of 90 mM C14DMAO, 10 mM SDS, and varying amounts of C6OH at 25 8C.
Figure 11.19: Double logarithmic plot of storage modulus G' vs. frequency f for a vesicle phase of 90 mM C14DMAO, 10 mM C14TMABr, and 160 mM CnOH for cosurfactants with various chain lengths at 25 8C.
216
11.4
Viscoelastic solutions with multilamellar vesicles
|η*|(ω) η(γ ) 90 mM C1 2 E6 , 10 mM C1 4 TMABr, 280 mM C6 OH |η*|(ω) η(γ ) 90 mM C1 4 DMAO, 10 mM C1 4 TMABr, 220 mM C6 OH
|η*|, η/ Pas
102 101 100 10-1
100
101 102 γ / s , ω / rad s-1 -1
103
Figure 11.20: Double logarithmic plot of shear viscosity Z vs. shear rate g_ and of the magnitude of the complex viscosity |Z*| vs. angular frequency o for two different vesicle phases at 25 8C.
frequency o are compared for two vesicle phases. The diagram shows the important difference to viscoelastic solutions of thread-like micelles [43]. They do not always fulfill the CoxMerz rule, stating that for all values of g_ the shear viscosity Z is equal to |Z*| at the corresponding value of o in the shear thinning region [44]. At low shear rates or frequencies both viscosities have the same value, while at higher shear rates Z (_g) is larger than |Z*|(o = g_ ). The curve for the zwitterionic system shows two breaks at characteristic shear rates. For shear rates above these characteristic values it is likely that the multilamellar vesicles undergo transformations to new structures as has recently been proposed by Roux et al. [45].
11.4.4 Model for the shear modulus In previous publications the shear modulus for the multilamellar phases was considered to be the result of the interactions of hard sphere particles [46–48]. In this picture each charged multilamellar vesicle is treated as a hard sphere. The theoretical treatment of the samples would then be similar to latex systems. The modulus of the systems depends on the chainlength of the surfactants that are used for the preparation of the systems if all other parameters like charge density, salinity, and concentration of surfactants and cosurfactants are kept constant. It can be argued that the differences of the moduli result from a change of the particle density of the vesicles. But these values are not known exactly. Systems with different chainlengths have similar conductivities which suggests that the particle density is also similar and therefore not responsible for the different shear moduli. Furthermore the birefringence looks the same, too. If the particle density of the vesicles decreases and if the mean size of the vesicles increases then the birefringence should increase. But this is not the case. The different moduli must therefore have a different origin. We propose, consequently, a different model for the explanation of the magnitude of the shear moduli. For the treatment of multilamellar vesicle phases and La phases this model was proposed by E. v. d. 217
11
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
Linden [49]. To our knowledge the theory has not yet been applied to experimental results. E. v. d. Linden assumes that multilamellar vesicles (droplets) are deformed in shear flow from a spherical to an elliptical shape. Turning into the deformed state the energy of closed shells is shifted because their curvature as well as their interlamellar distance D are changed. Due to the interaction of the bilayers, expressed by the bulk compression modulus B, the inner shells are deformed and the total deformation energy E of the lamellar droplet gets minimized. Assuming that the volume of a droplet is not modified by the deformation, the surface A must increase. One can define an effective surface tension seff = E/DA. E. v. d. Linden obtains: ef f
1 1
KB =2 ; 2
11
where K is the bulk rigidity which is correlated to the bilayer’s bending constant k by K = k/D. We can relate this effective surface tension to the shear modulus G of a vesicle with radius R. Using the identity G = 2seff /R yields: G
1
KB =2
R
k B D R
1=
2
:
12
Both, bulk compression modulus and bending constant, depend on the charge density of the bilayers and the shielding of the charges with excess salt. This means the theory of E. v. d. Linden results in a calculation of the geometrical average of the compression EB and bending energy EK per unit volume. The expression (Eq. 12) can be squared to G2
k B D R2
:
13
With n = R/D which denotes the number of bilayers in a vesicle we obtain: G2
nk B Ek EB : R3
14
Now we can try to find adequate expressions and values for the quantities B and K by other theories. For the bending constant as a function of the charge density we can use the expression (Eq. 15) that has been given by H. Lekkerkerker [50], Kel with q 218
kB T
q 1
q 2 2pQk
q 1q
p 2pQjsj p2 1 and p . ke
15
11.4
Viscoelastic solutions with multilamellar vesicles
kB is Boltzmann’s constant, T the absolute temperature, Q the Bjerrum length, k the reciprocal Debye length, and s the surface charge density. For the bulk compression modulus we can use the expression that is often used to describe the interaction between two charged particles K fa fs np k2 d 2 V
d ;
16
where V(d) is the energy of interaction between a pair of spherical particles V
d
z2 e 2 4p"
exp
ka
1 ka
2
exp
kd ; d
17
d denotes the separation of particles and a their radius. fa and fs are numerical factors [51]. There is a further possibility to get an expression for the compression modulus B. This quantity may be simply given by the osmotic pressure between the bilayers. According to a theory of Dubois et al. [52] we have calculated the osmotic pressure using the equation P cm kT :
18
where cm is the concentration of ion particles at the midplane between the bilayers, calculated by the Poisson-Boltzmann equation for the current conditions. In a previous paper we have discussed the possibility to identify the shear modulus with the osmotic pressure. This seems to be obvious because the osmotic pressure qualitatively increases like the shear modulus with increasing charge density of the bilayers and decreases on salt addition. But this attempt failed because the calculated values, which were in the order of several thousand pascals, were too large. In the current context, however, it looks reasonable to identify the osmotic pressure with the compression modulus B from Eqs. 11 and 12. Now there is a concept which is appropriate to reproduce the characteristic features of our experimental results. These are the influence of the charge density and the influence of salinity on the shear modulus. The growth of the shear modulus with the charge density can mainly be attributed to the increase of the bending constant and the osmotic pressure with the charge density. For small salinities the decrease with the salinity seems to come from the pair potential V (d) whereas the linear decrease at higher salinities seems to come from the bending constants. However, we should keep in mind that for both changes the vesicular structures do not remain constant. Therefore we cannot give a complete quantitative interpretation of the experimental results and the data cannot be fitted precisely to an exact theoretical model. At the end of this paragraph we check whether Eq. 12 gives reasonable results for our primary data. If we assume d = 80 nm for the interlamellar distance, R = 0.5 mm for the radius of a vesicle, 10 % for the charging degree (i. e. a surface charge density on the bilayers of e0 /500 H2, where e0 is the elementary charge), k = 0.56 kBT for the electrical contribution of the bending modulus (calculated according to Eq. 15) [47], and B = P = 3000 Pa for the compression modulus (calculated according to Eq. 18) [46] then Eq. 12 yields the shear modulus G & 18 Pa. This value is very close to the experimental one and demonstrates that a correct approach for a theoretical description of the rheological properties of the solutions seems to be found. 219
11
11.5
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
Ringing gels
11.5.1 Introduction For a large variety of amphiphilic compounds cubic phases are a particular class of surfactant systems normally observed in the more concentrated region of the phase diagram [53]. Such phases may occur in binary systems (surfactant and water) as well as in ternary or pseudo ternary systems, where usually a hydrocarbon is the third component. In addition they may contain a long alcohol chain as a fourth component (cosurfactant). Such a composition reminds one of that of microemulsions and for that reason they are also called microemulsion gels [54]. However, the denomination cubic phase is insofar more instructive since it relates directly to the structure of the corresponding systems. These phases are thermodynamically stable, optically isotropic, transparent, and highly viscous which distinguishes them easily from conventional L1 or L2 phases, i. e. micellar or reverse micellar phases. Often they possess a yield stress and exhibit elastic properties for not too large deformations [55]. In particular these elastic properties are responsible for the other name – ringing gels – which sometimes is used for these systems because samples of these phases usually show an acoustical resonance phenomenon (ringing sound) after being tapped with a soft object [56]. This effect is not necessarily associated with a cubic phase but quite frequently observed within this class of surfactant systems. The ringing phenomenon is observed for all the cubic phases studied by us. It might be mentioned here that such phases have been used already for pharmaceutical and cosmetical applications without having detailed structural information regarding the corresponding systems at that time [57]. More recently it has been claimed that cubic phases may play a key role in the fusion process of biological membranes [58, 59] and they are frequently formed by lipids, obtained from membrane extracts. The main perspective of our investigations was first to determine the detailed microstructure of the corresponding systems and then to relate them to the observed macroscopic properties. For that purpose two surfactant systems were chosen. Both have a cubic phase but at different locations in the phase diagram. In the first system – tetradecyldimethylaminoxide (C14DMAO)/hydrocarbon/H2O – the cubic phase is located between the isotropic L1 phase and the hexagonal phase at a constant surfactant/hydrocarbon ratio (Fig. 11.21 a) [55]. In the second system – bis-(2-ethylhexyl)sulfosuccinate (AOT)/1-octanol/H2O – it is situated between the lamellar phase, at lower octanol content, and an isotropic L2 phase, at higher octanol and lower water content, or a reverse hexagonal phase at higher AOT content (Fig. 11.21 b) [60]. These systems are insofar similar as they both are next to an isotropic phase. Moreover a binary system containing triblock copolymers of the polyethyleneoxide/polypropyleneoxide/polyethyleneoxide (PEO/PPO/PEO) type has been studied for which we also have found a formation of a cubic phase with location in the phase diagram similar to the aminoxide case.
220
11.5
Ringing gels
Decane
20
80
5
T=25°C 3
50
80 L1
0
20
H1
G Nc
50
2 + S
2
1
H2O
4
40
S
Lα
60
20
C14DMAO
80
100
Figure 11.21 a: Phase diagram of the ternary system C14DMAO/decane/H2O at 25 8C. Isotropic water continuous phase (L1), nematic phase (Nc), cubic phase (G), hexagonal phase (H1), lamellar phase (La), crystals (S), other phase regions (1, 2, 3, 4, 5).
Octanol 80
20
50
50 L2
80
H2O
20
F
I2 D
0 L1
20
40
60
80
AOT
100
Figure 11.21 b: Phase diagram of the ternary system AOT/1-octanol/H2O at 25 8C (in wt%). Isotropic water continuous phase (L1), lamellar phase (D), (bicontinuous) cubic phase (I2), isotropic oil continuous phase (L2), reverse hexagonal phase (F), other phase regions without symbols.
11.5.2 The aminoxide system For the aminoxide system investigations were done along a line of constant C14DMAO/decane ratio. Upon going along such a line close to the solubilization capacity of the surfactant one can cross from the L1 phase into the cubic phase by increasing the surfactant concentration. Samples located on such a dilution line contain spherical aggregates of constant size. This was shown by means of static and dynamic light-scattering experiments as well as by SANS measurements [55, 61]. The interactions are very well described by a hard sphere model. In Fig. 11.22 ex221
11
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions 3000
I(q) in a.u.
2000
1 2
1000
0 0,00
3 4 5 6 0,05
0,10
0,15 qin1/Å
0,20
0,25
Figure 11.22: SANS intensity curves as a function of the magnitude of the scattering vector q for samples of constant C14DMAO : decane weight ratio of 5.4 : 1 and different total concentrations of C14DMAO plus decane were 8 wt% for 1; 16 wt% for 2; 24 wt% for 3, 28 wt% for 4, 32 wt% for 5, 37.1 wt% for 6. Samples 1–5 are in the L1 phase whereas sample 6 is located in the cubic phase.
perimental SANS curves for various total concentrations (constant weight ratio of C14DMAO : decane = 5.4 : 1) are given which can be fitted curves in good agreement with the hard sphere model. Furthermore the SANS experiments show that the microemulsion droplets are fairly monodisperse, i. e. possess a polydispersity index of about 0.1 [62]. For the cubic phase the SANS experiment shows still the same type of aggregates but more concentrated. However, here the packing exceeds the critical volume fraction of 53 vol%, which is typical for a hard sphere crystallization [63, 64]. In the cubic phase the spherical aggregates are packed similar to metal atoms in a cubic lattice. From SANS investigations of samples in the cubic phase it seems that the packing is not primitive cubic but either face-centred cubic (fcc) or body-centred cubic (bcc). Between these two possibilities an experimental distinction was not feasible [65]. Samples of the cubic phase show a very distinct SANS behaviour when the scattering intensity is detected two-dimensionally. Then an isotropic pattern is no longer observed. One finds more or less pronounced spikes superimposed on a symmetric correlation ring. These spikes are distributed randomly on this ring. The occurrence of such scattering patterns is commonly observed with cubic phases as demonstrated by a typical example in Fig. 11.23 [61]. These relatively sharp peaks indicate a long-range ordering in the cubic phase. This scattering pattern can be explained by the presence of relatively large crystalline domains [66]. A similar series of samples as in the SANS experiments was studied in cooperation with the group of Prof. Wokaun by NMR self-diffusion experiments. The pulsed field gradient spin echo (PGSE) method [67, 68] allows the determination of the self-diffusion coefficient of each of the individual constituent components in particular water, surfactant, and hydrocarbon. Here, in order to obtain simpler NMR spectra the hydrocarbon was cyclohexane. The molar ratio of C14DMAO : cyclohexane was chosen to be 1 : 1.2, with three samples in the L1 phase and three samples in the cubic phase. The obtained self-diffusion coefficient of water shows a continuous, approximately linear decrease with increasing volume fraction of the micellar aggregates and with no dis222
11.5
Ringing gels
Figure 11.23: Three-dimensional plot of the scattered intensity in a SANS experiment for 32.2 wt% C14DMAO/6.0 wt% decane/61.8 wt% D2O.
continuity at the phase transition: L1 phase ? cubic phase. Such a behaviour is in good agreement with the model of a continuous aqueous phase where the diffusion of the water molecules is simply hindered by the steric restrictions imposed by the presence of micellar aggregates [69]. A much different picture is obtained for the diffusion of the surfactant and the hydrocarbon (Fig. 11.24). Within the L1 phase both diffuse with the same coefficient. This is in good agreement with the diffusion coefficient of the micellar aggregates calculated from their size, determined by scattering methods. Again some decrease of the diffusion coeffi100
Ds / 10 -12 m2/s
10
1
0,1
0,01 0,0
cubic phase
a) b) c) d) e)
0,1
L1 -phase
0,2
0,3
0,4
0,5
Φ
Figure 11.24: Self-diffusion coefficient Ds of surfactant and hydrocarbon in logarithmic representation as a function of the micellar volume fraction F for the system C14DMAO/cyclohexane/D2O. The molar ratio of C14DMAO : cyclohexane was always 1 : 1.2. a) Ds of (N(CH3)2); b) Ds of (CH2); c) Ds of (CH2) for samples with deuterated cyclohexane; d) Ds of H determined by means of a different instrument [69]; e) Ds of H for samples with deuterated cyclohexane [69].
223
11
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
cient with increasing micellar volume fraction occurs, as the movements of the aggregates get hindered increasingly. However, upon crossing into the cubic phase the situation is dramatically altered. The self-diffusion coefficient of surfactant and hydrocarbon drops suddenly by more than a factor of 20 for the cyclohexane and a factor of 200 for the surfactant. This is consistent with the picture from above, i. e. now the aggregates are frozen in and are no longer able to move in the cubic lattice which is formed. Hence the diffusion coefficient does not describe any more the motion of aggregates but is due to the diffusion of the individual molecules. This explains why the much smaller cyclohexane molecule now diffuses much faster than the larger C14DMAO molecule. Furthermore the very low diffusion coefficient excludes the possibility of a bicontinuous system since for such a structure a much faster diffusion would be expected [70]. In summary, this experiment clearly demonstrates that in the L1 phase and in the cubic phase the same water continuous structure is present. But while in the L1 phase the micellar aggregates are freely mobile their translational mobility is blocked in the cubic phase, i. e. the microemulsion droplets are condensed into a glass-like liquid crystalline state of high elasticity. In general, from the investigation of the more dilute L1 phase one may already conclude the microstructure of the cubic phase. This concept can be very useful for the explanation of macroscopic properties which are related to the aggregate size, like the shear modulus G0. By means of oscillatory rheological experiments one may determine G0 which is typically in the range of 105 –106 Pa for cubic phases. Experimentally one finds for aminoxide systems that G0 decreases with increasing size of the respective aggregates [71]. These experimental values may now be compared to theoretical calculations for hard sphere crystals [72]. In general, this ansatz will predict a proportionality of the elastic moduli to the particle density N of the aggregates. Furthermore, one can calculate for a given volume fraction the factor which relates the particle density to the elastic constants, like G0. The calculated values are in reasonable qualitative agreement with the experimental data [65]. This means that it is possible to deduce mechanical properties from the knowledge of the microstructure. Furthermore, this explains the strong dependence on the particle size since the particle density is proportional to 1/R3, i. e. the microstructure of the cubic phase directly determines the macroscopic elastic properties of the system.
11.5.3 The bis-(2-ethylhexyl)sulfosuccinate system As stated above, the cubic phase of the AOT system is located differently in the phase diagram than the aminoxide system, which is due to the fact that AOT as a double-chain surfactant has a tendency to form reverse phases. Again it was of great interest to investigate the relation between the isotropic L2 phase and the cubic phase. Furthermore, this cubic phase is interesting since here the surfactant concentration can be varied over a large range (30– 76 wt%). The transition from the L2 phase into the cubic phase has been studied by a variety of methods [73]. This can be done easily on a line of constant AOT content (in our case mostly 35 wt%). Then increasing the octanol/H2O ratio one crosses from the cubic phase into the 224
11.5
Ringing gels
L2 phase. Interestingly, measurements of the electric conductivity showed no discontinuity of the equivalent conductivity (specific conductivity/AOT concentration) upon crossing the phase boundary. Instead one observes a continuous increase of L with increasing H2O content. The value in the cubic phase is about 40 % of that of the free Na+ ions, which should mainly be responsible for the ionic conductivity (since the surfactant counterion will be largely immobilized being fixed in the amphiphilic film), which indicates that the structure must be water continuous. NMR PGSE self-diffusion studies on similar samples also showed a continuous increase of the water self-diffusion coefficient with rising water content. The value in the cubic phase is again about 40–50 % of the bulk water diffusivity, confirming the water continuous structure. For the alkyl chains of octanol and AOT a separation was not possible, but the values of (2–4)610–7 cm2/s are relatively large, more than one order of magnitude larger as in the case of the aminoxide, and show that the structure ought to be bicontinuous and similar to the neighbouring L2 phase [73]. SANS spectra of the cubic phase show the typical spikes on the isotropic diffraction ring (compare Fig. 11.22). From the position of the peaks and from the total scattering intensity, it can be concluded that the size of the structural units decreases with increasing AOT (water) content, at constant octanol/water (AOT/octanol) ratio. Measurements of the shear modulus G0 on a series of samples with constant ratio of octanol/water showed that G0 is proportional to c(AOT)3.1, i. e. again, as in the case of the aminoxide system, the elastic modulus increases with decreasing size of the structural units, since the SANS experiments show that the size of the structural units decreases with increasing AOT concentration. Correspondingly the particle density N of these units increases and, as stated above, G0 should be proportional to N. This means that the determination of the microstructure already allows the prediction of the elastic properties, as in the case of aminoxide. Upon heating the cubic phase will melt yielding a lowly viscous L2 phase. This phase transition was studied in some detail by differential scanning calorimetry (DSC). It was found that the higher the melting temperature the higher the surfactant content is, with melting temperatures in the range of 50 to 95 8C. The melting enthalpies DH show the same trend with typical values of 100–300 mJ/g. Similar values were also found for the aminoxide system. The maximum value, DH = 1034 mJ/g, was found for the cubic phase in the binary system of 75.8 wt% AOT and 24.2 wt% H2O at the melting temperature of 89.5 8C. For AOT concentrations close to the minimal value (around 28 wt%) necessary for forming the cubic phase, however, the situation becomes somewhat more interesting. Here, DH can be extremely small, below 0.5 mJ/g. This means that the macroscopically observable phase transition from the solid-like cubic phase to the lowly viscous L2 phase is associated with almost no enthalpy difference. Furthermore, for samples with c (AOT) < 37 wt% another very striking phenomenon occurs. Here a transition from the cubic phase to the L2 phase can be achieved by heating as well as by cooling! For the transition at lower and at higher temperature the transition enthalpy is very similar. Both cases are endothermic processes. Such a reverse melting process is quite unique for this type of cubic phase and has formerly only been reported for triblock copolymers of PEO/PPO/PEO-type (Section 11.5.4) [74–76] and PEO/PPO diblock copolymers [77]. However, for the block copolymers the situation is insofar different as the cubic phase is made up from normal micellar aggregates, which are transformed into a cubic phase upon heating since here the effective volume fraction of the 225
11
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
spherical aggregates increases [75]. In contrast to the cubic phase the AOT system is bicontinuous and transformed into a bicontinuous L2 phase by a different and still unknown mechanism.
11.5.4 PEO/PPO/PEO block copolymers PEO/PPO/PEO triblock copolymers exhibit properties similar to typical surfactants, i. e. they reduce surface and interfacial tension of aqueous solutions and form micellar aggregates above a critical micellar concentration [74, 78]. For some compounds of this type, like P104 (EO18PO58EO18), P123 (EO20PO70EO20), and F127 (EO106PO70EO106), a similarly located cubic phase like the one of the aminoxide systems has been found in the binary aqueous system [74, 79]. Structural investigations on these systems have shown that the block copolymers form spherical micellar aggregates in the L1 phase close to the cubic phase. Here the PPO block is the hydrophobic moiety. It forms the core of the micellar aggregate and is surrounded by the more hydrophilic PEO blocks which act as the hydrophilic part of the polymeric surfactant molecules. The temperature dependence of this system is interesting. At low temperatures the block copolymers dissolve in water as unimers. Upon increasing the temperature the PO groups are dehydrated rendering them more hydrophobic. This increased hydrophobicity is responsible for the aggregation of the monomers into micellar aggregates. As has been determined by DSC measurements this micellization process occurs over a fairly large temperature range of typically 15–30 K [79]. For this dehydration process the enthalpy changes are about 3 kJ/mol per PO group. This uncommon temperature behaviour is also responsible for the interesting, already mentioned, reverse melting transition, i. e. in a corresponding concentration range the lowly viscous L1 phase is transformed into the solid-like cubic phase upon heating. The simple explanation for this effect is that upon raising the temperature more and more monomers will aggregate in micelles. If the volume fraction F of the micellar aggregates exceeds the required one for the formation of a cubic phase (F = 0.53) then the gelification process occurs and a cubic phase, composed of individual micelles, is formed [64]. However, this phase transition is associated with an enthalpy about two orders of magnitude smaller than the heat of micellization – typically between 25 and 100 mJ/g, i. e. similar in extent as for the other cubic phases described above. Its nature is endothermic, i. e. the transition is associated with an increase of entropy. This transition can also be nicely monitored by SANS measurements. For the micellar solution an isotropic scattering ring is found where the typical spikes of the cubic phase can be observed if the transition temperature is crossed [74]. More recently it has been found that the formation of the cubic phase can be suppressed by the admixture of a simple surfactant, such as SDS [80]. This effect is due to cooperative binding of SDS molecules on the block copolymer molecules. By doing so the micelles are dissolved in favour of monomeric units until the volume fraction of the micelles gets to small to form a cubic phase, i. e. one can melt the cubic phase by adding surfactant. For example, for transforming the cubic phase of 25 wt% F127 a SDS concentration of 100 mM is required. 226
11.6
Lyotropic mesophases
Summarizing, cubic phases are a type of surfactant systems that can be observed for a large variety of different surfactants. Depending on the molecular structure of the surfactant they may be located in different places of the phase diagram with correspondingly different microstructures. They can be composed of an array of individual micellar aggregates or be of bicontinuous structure. However, even such a bicontinuous structure still will be characterized by a very well-defined typical size of the structural units with long-range ordering. At this point it is interesting to note that this size can already be obtained by studying isotropic phases close to the cubic phase. This is insofar advantageous since they are often much more amenable to experimental studies than the corresponding cubic phase itself. The knowledge of the size of these units (no matter whether they are individual aggregates or only structural repeat units of a bicontinuous structure) enables the prediction of the elastic moduli which will be proportional to the particle density of these units. For a given structural build-up theoretical relations can be used to calculate the moduli quantitatively. Therefore a detailed knowledge of the microstructure of the corresponding systems allows a quantitative understanding of macroscopic properties. Cubic phases are a nice example for a system where by now such structure property relations are well established.
11.6
Lyotropic mesophases
11.6.1 Introduction In general, at higher concentrations surfactant aggregates show a nearest neighbour order under the influence of the intermicellar interaction. Upon further increasing the concentration (mostly above 30–40 wt%) a long-range order between the micelles takes place and lyotropic mesophases are formed. Depending on the kind of micelles cubic phases are formed from globular micelles, rod-like micelles form hexagonal phases, and disc-like micelles form lamellar phases. At surfactant concentrations significantly above 50 wt% inverse hexagonal and cubic phases can also be built. Generally speaking, with increasing concentration the phase sequence: cubic > hexagonal > lamellar > inverse hexagonal > inverse cubic is found (Fig. 11.25) [81]. Figure 11.25 shows also schematically the structures of the different lyotropic mesophases. Figure 11.25 shows that between the hexagonal and the lamellar phases further isotropic phases, which also are cubic, can exist, although they do not consist of globular particles but show bicontinuous structures [73, 82]. Of course, not all these phases must be present in each surfactant solution.In most of the systems the first cubic phase is missing up to high surfactant concentrations because globular micelles are only present in few cases. Double-chain surfactants often have a lamellar phase as the first mesophase [83], but all systems show principally the given sequence of the mesophases. This is also valid for more-component systems, where according to the constitution of the further components, certain phases may be favoured or suppressed. Hydrocarbons, for example, favour globular micelles and hence cubic phases [55, 227
11
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
Figure 11.25: Schematic phase diagram of a binary system surfactant/water. Inverse hexagonal phase HII, lamellar phase La, hexagonal phase HI, isotropic (cubic) phases a, b, c, d.
61, 84]. But cosurfactants, like aliphatic alcohols, with an intermediate chain length support the formation of disc-like micelles and thus of lamellar phases [85]. The aggregates of the phases in Fig. 11.25 show a long-range order with respect to their orientation and to their centre of masses. Nevertheless, also lyotropic mesophases exist which are built up by rod-like or disc-like micelles, respectively. They show only a longrange order of their orientation, while their mass centres are statistically distributed in the solutions. These phases are called nematic calamitic (Nc) or nematic discotic (Nd) phases and exist at lower concentrations than the corresponding hexagonal or lamellar phases. Those nematic phases are very interesting because they can be uniformly oriented by weak external fields due to the lack of long-range order of their mass centres. This gives rise to the development of single crystals instead of the polycrystalline mesophases with differently oriented domains. On addition of chiral components the nematic phases can be transformed to lyotropic cholesteric phases with a helical twist of the orientation, as has been found for the corresponding thermotropic phases [86].
11.6.2 Nematic phases and their properties Lyotropic nematic phases have been discovered a long time after the finding of the lyotropic mesophases. In 1967 first evidence of such a phase was published [87] but it took more than 10 years to prove unambiguously the existence of these phases [88]. The first nematic 228
11.6
Lyotropic mesophases
phases have been found by fortune in complicated ternary and quaternary systems consisting of surfactants, cosurfactants, electrolytes, and water. Systematic studies on different surfactant systems have finally shown that lyotropic nematic phases do not occur so rarely as could be concluded from their late discovery. At the same time criteria could be established which allow a fairly precise prediction of the existence of a nematic phase and its position in the phase diagram [89]. This also allows the direct preparation of different types of the nematic phases and the study of their properties. Important conditions for the existence of nematic phases are the voluminous hydrophobic part and the headgroup area of the hydrophilic group which must be within certain values, for example 60–90 Å2 for double-chain ionic surfactants with a Nc phase under these conditions. The headgroup area can be lowered by adding cosurfactants (aliphatic alcohols with an intermediate chain length). This leads to the formation of a Nd phase. Both phases border to the corresponding hexagonal or lamellar phase from the side of lower concentrations. For smaller headgroup areas a lamellar phase is formed first while for larger headgroup areas a hexagonal or a cubic phase is the first mesophase [89]. For example, numerous nematic phases have been found in solutions of anionic perfluoro surfactants with their voluminous hydrophobic groups which mostly have been identified as Nd phases [90, 91]. Figure 11.26 shows a typical phase diagram of the binary system perfluoro-surfactant/water. It can be seen that the nematic phase exists only in a small concentration and temperature range which was probably the reason for the late detection of these phases. From Figure 11.26 the marked thermotropic behaviour of the nematic and the lamellar phases can also be seen. Thus it is possible by increasing the temperature to go through the phase sequence: lamellar > lamellar/nematic > nematic > nematic/ isotropic > isotropic. These thermotropic phase transitions are reversible. In most cases DSC measurements have shown that the phase transitions are of first order with very small heat transitions. Further experiments have shown that the nematic phases can be oriented uniformly in a magnetic field according to the anisotropy of the diamagnetism of the surfactant molecules. Isolated anisotropic aggregates cannot be oriented in micellar solutions because the energy of the magnetic field is much smaller than the thermal energy which achieves a random orientation of the micelles in this case. The lamellar phases, on the other hand, also cannot be oriented by the magnetic field because this would require the simultaneous orientation of a sheet of lamellae for which the energy of the magnetic field is not sufficient.
70 60
nematic isotropic
T/°C
50 40
lamellar
30 20 10
crystalline 0
10
20
30 wt%
40
50
Figure 11.26: Section of the phase diagram of the system C8F17CO2N(CH3)4/D2O.
229
11
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
But it is possible to orient the phases uniformly in the nematic region and to freeze the orientation by cooling the system to the lamellar region. This orientation remains without the magnetic field [92]. This allows a simple study of such phases and of the type of particles present in the phases. It is also possible to solubilize anisometric dye molecules in the aggregates and to orient them together with these micelles in the nematic phase by a magnetic field, leading to a dichroism of the systems [93]. The oriented nematic and lamellar phases show also a strong anisotropy of their electric conductivity, i. e. increase of the conductivity parallel and decrease of the conductivity perpendicular to the large axis of the aggregates. Nc phases have also been found in solutions of the cationic double-chain surfactants of the series CxCyN(CH3)2Br with x = 14 or 16 and y = 1–4 which fulfill the established conditions with respect to the headgroup area and the volume of the hydrophobic alkyl chain. On the other hand, surfactants with y = 6 do not form mesophases up to concentrations of 75 wt%, while surfactants with y > 8 show first a lamellar mesophase [89]. These mesophases also exist only in a small concentration range near the hexagonal phase, as can be seen, for example, in Fig. 11.27. Above a characteristic temperature they show a reversible phase transition of first order into an isotropic phase. Figure 11.27 also shows that a thermotropic transition hexagonal > nematic cannot be observed in these systems. Furthermore it could be shown for these phases that on addition of a cosurfactant a transition of the Nc into a Nd phase can take place. For one system it was found that both nematic phases can be present in equilibrium. From orientation experiments of these nematic phases in a magnetic field it could be concluded that the sign of the anisotropy of their diamagnetism is positive. If the Br counterion was substituted by benzene sulfonate, the nematic phases are kept but the sign of the anisotropy of the diamagnetism changes. Hence it is possible to prepare and to study all four possible nematic phases N+c, N–c, N+d, and N–d with these surfactants.
Figure 11.27: Section of the phase diagram of the system CTAB/H2O.
The identification of the Nc and Nd phases can be archived by polarization microscopy based on their different textures which are presented in Fig. 11.28 a and 11.28 b. 2H-NMR spectroscopic studies and orientation experiments on the nematic phases in a magnetic field are suitable for their detection, especially if SANS measurements are done on oriented samples. For these experiments the phases have to be prepared with D2O instead of H2O. This 230
11.6
Lyotropic mesophases
Figure 11.28 a: Texture of a calamitic nematic (Nc) phase in the binary system of 25 wt% hexadecyltrimethylammoniumbromide (CTAB)/water at 35 8C.
Figure 11.28 b: Texture of a discotic nematic (Nd) phase in the binary system of 47.5 wt% tetradecylpyridiniumheptanesulfonate (C14PyC7SO3)/water at 26 8C.
does not affect the nematic phases except a small shift of their existence region towards lower surfactant concentrations. Furthermore, SANS and SAXS measurements show a high degree of orientation of the nematic phases by the appearance of a third order scattering peak. Finally it shall be mentioned that until now no inverse nematic phases could be detected. Also the existence of nematic phases in non-polar solvents could not be proven for any studied system. Thus it is not yet possible to replace the thermotropic nematic phases in displays by lyotropic systems because lyotropic nematics cannot be orientated in an electric field with low voltages due to the rather high conductivity of the water solvent compared to non-polar organic compounds.
231
11
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
11.6.3 Cholesteric phases and their properties For thermotropic and also for lyotropic nematic phases it is known that the presence of chiral compounds leads to a transition of the nematic phase to a cholesteric one. This phase has also only a long-range order with respect to the orientation of the aggregates, but this orientation shows a twist within a domain, and the pitch depends on the concentration of the chiral compound. Cholesteric phases can be built up with chiral components (intrinsic cholesteric phases). The transition nematic > cholesteric can also take place by adding chiral samples (induced cholesteric phases) [86]. Intrinsic cholesteric phases require the use of chiral surfactants. For this purpose the group of the non-ionic sugar surfactants has been chosen. As the sugar head group is mostly very voluminous, it could be expected that only a few systems could fulfill the conditions for the formation of a nematic or cholesteric phase with respect to the headgroup area and the volume of the hydrophobic group. Further problems arose from the poor solubility of sugar surfactants with sufficiently small headgroups and because many of these surfactants are strongly sensitive to hydrolysis. Nevertheless, the first intrinsic cholesteric phase in a binary system surfactant/water could be detected for an alkylpolyglucoside with 12–14 C atoms and 1,1 glucose units on an average [94]. By orientation in a magnetic field this phase could be identified as a cholesteric discotic phase with negative anisotropy of its diamagnetism. On the other hand, it is no problem possible to transform Nc and Nd phases in a ternary system of the zwitterionic surfactant tetradecyldimethylaminoxide (C14DMAO), a cosurfactant (an aliphatic alcohol with 7–10 C atoms), and water into the corresponding induced cholesteric phases by adding chiral compounds [95]. This transition could be observed with a non-polar chiral additive (cholesterol), which can be solubilized only in the micellar aggregates, and with a polar chiral compound (tartaric acid), which remains dissolved in the aqueous phase. The cholesteric phases could be identified by their fingerprint textures in the polarization microscope, which are characteristic for non-oriented cholesteric phases. Uniformly oriented cholesteric phases show characteristic stripe textures. Both textures are presented in Figs. 11.29 a and 11.29 b. From the distance of the stripes the pitch of the phase can be cal-
Figure 11.29 a: Texture of a non-oriented cholesteric phase in the binary system of 60 wt% alkylpolyglucoside (APG 1)/water at 25 8C.
232
11.6
Lyotropic mesophases
Figure 11.29 b: Texture of a oriented cholesteric phase (aligned in a magnetic field of 2 T) in the binary system of 60 wt% alkylpolyglucoside (APG 1)/water at 25 8C.
culated. The experiments showed a linear relation between the reciprocal pitch and the concentration of the chiral additive. Until now, the reason for the pitch caused by the chiral additive is not understood. An interesting fact about these cholesteric phases is that above a certain temperature the pitch disappears and a nematic phase develops reversibly. This transition temperature is lower than the temperature for the transition nematic > isotropic. It could not be detected by DSC experiments, very likely due to its small enthalpy change.
11.6.4 Vesicle phases and L3 phases From Fig. 11.25 it can be seen that lyotropic lamellar phases normally exist at surfactant concentrations above 50 wt%. According to the theory of surfactant aggregation [4] surfactants with small headgroup areas and big hydrophobic groups are known to form lamellar phases already at concentrations far below 10 wt%. Such systems are perfluorosurfactants with strongly binding counterions [96] or double-chain surfactants [83]. In mixtures of surfactants with small headgroup areas like zwitterionic or non-ionic surfactants and cosurfactants – for example aliphatic n-alcohols with intermediate chain lengths – the geometrical conditions for the formation of lamellar phases in highly dilute systems are also fulfilled. Furthermore it is possible in these systems to change the natural curvature and the flexibility of surfactant lamellae simply by variation of the mixing ratio surfactant : cosurfactant. According to this, these ternary systems exhibit a typical phase behaviour with lamellar phases at the water rich corner, which has already been explained in detail in Section 11.4.1. The most powerful method for the identification of the different phases and for the determination of their microstructures is the transmission electron microscopy. This method requires a sample preparation using the freeze fracture and etching technique. With these measurements the microstructures of the various phases can be visualized directly. This is shown for three examples in Fig. 11.12 (Section 11.4.2) and in Figs. 11.30 a and 11.30 b [13, 41, 97]. 233
11
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
Figure 11.30 a: Electron micrograph of a lamellar phase in the system of 100 mM C14DMAO, 190 mM C7OH, and water (the bar represents 2.5 mm).
Figure 11.30 b: Electron micrograph of a L3 phase in the system of 100 mM C12DMAO, 220 mM C6OH, and water (the bar represents 140 nm).
With these technique it can also be demonstrated in the two phase regions that both phases with their different structures are really coexisting. This result can be supported by self-diffusion measurements with the pulsed NMR field gradient method from which the existence of bicontinuous phases and their topology can be concluded [73, 98]. The electron micrograph in Fig. 11.12 shows the existence of thermodynamically stable vesicles in the Lal phase which can be formed as small unilamellar vesicles besides large multilamellar vesicles (liposomes). Depending on the composition of the ternary system the vesicles can also show structural faults like holes (perforated vesicles) [37]. The vesicle phases show a significantly reduced electric conductivity because a part of the water phase together with the ions is included in the interior of the vesicles. With stopped-flow experiments and optical or conductivity readout it is thus possible to determine the permeability of the vesicle membranes for dissolved compounds [99]. Vesicle phases show often high viscosities and viscoelasticity due to the mutual hindrance of the vesicles in sheared solu234
11.6
Lyotropic mesophases
tions. Especially if the vesicle membranes are charged by incorporation of certain amounts of added ionic surfactants, the phases show with increasing charge of the surfactant film an increasing yield stress [40, 100]. The vesicle phases usually are not or only weakly birefringent and thus cannot be identified in the polarization microscope. From Fig. 11.30 a and for the Lah phases the presence of flat surfactant lamellae like in normal lamellar phases is evident. These phases are birefringent but they often do not develop their characteristic texture in the polarization microscope due to their low concentration. Until now it could not be proven whether there is a phase boundary between the La1 and the Lah phase or a continuous transition from Lal to Lah with increasing cosurfactant concentration. This problem is very difficult due to the high viscosity and turbidity of the phases. The bicontinuous structure of the L3 phase can also be seen directly from Fig. 11.30 b. This phase shows a low viscosity without elasticity and is optically isotropic but shows a strong flow birefringence. According to the similarity with normal micellar or reversed micellar solutions (L1 and L2 phases) these phases have been called L3 phases. The surfactant lamellae are arranged as branched tubes similar to a sponge. The low viscosity results from the very high flexibility of the lamellae which break and reform easily. As mentioned in Section 11.2, the phase sequence in Fig. 11.1 can be understood regarding the properties of the surfactant films. With low amounts of cosurfactant the natural curvature of the film is convex and the flexibility is usually low. Hence micellar aggregates are present in the L1 phase. With increasing amounts the curvature of the films decreases and their flexibility increases due to the small headgroup area of the cosurfactant. Thus transition into a vesicle phase and with more cosurfactant into a normal lamellar phase can take place. Further increase of the cosurfactant increases the flexibility of the films and permits the formation of the L3 phase. Adding even more cosurfactant finally leads to the separation of the cosurfactant phase. This concept also allows to understand the influence of further additives on the phase behaviour of the surfactant/cosurfactant/water systems. Adding an ionic surfactant to the L3 phase leads to a reduction of the flexibility and to a convex curvature of the surfactant films due to the electrostatic repulsion between the ionic headgroups. Thus a transition of the L3 phase to a vesicle phase is observed [38]. These vesicles have rather stiff membranes and the phase often shows a yield stress, as demonstrated in Section 11.4.3. But addion of an electrolyte to these systems shields the charge of the headgroups and gives rise to an increased flexibility and decreased curvature of the films. Hence the vesicle phase is transformed again into a L3 phase. Addition of an ionic surfactant to phospholipid vesicles, which normally must be prepared by sonification of the aqueous phospholipid dispersions [101], also leads to a destruction of the vesicles and to a transition to rod-like micelles above a certain concentration of the ionic surfactant [25]. Finally, it is worth mentioning that thermodynamically stable vesicles can also be found in the binary systems surfactant/water if the hydrophobic group is sufficiently voluminous. As already mentioned, this has been observed for the double-chain surfactant didodecyldimethylammoniumbromide DDABr in water [102]. Recently also a L3 phase has been found in the binary system of a polyoxyethylene-polyoxypropylene-polyoxyethylene triblock copolymer and water at high temperatures. This block copolymer has a composition EO5PO30EO5 and hence a very small hydrophilic headgroup. It forms a lamellar phase at room temperature which is transformed into the L3 phase above a certain temperature [103]. The reason for this transition is probably the increased flexibility of the block copolymer film at the high temperature. 235
11
11.7
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
Shear induced phenomena
11.7.1 General
pressure drop [hPa]
In this Section we discuss a remarkable and puzzling phenomenon that, because of its complicated nature, is not yet completely understood. The phenomenon has, however, considerable potential for technical applications where it is necessary to control the flow behaviour of aqueous solutions. In all applications where large amounts of water have to be circulated for cooling or heating purposes the energy expense for the pumping is a major economic factor. Usually one is interested in pumping as fast as feasible so that the flow in the water pipes is generally in the turbulent flow region. Under these conditions it is possible to reduce the friction coefficient by polymer additives or by drag-reducing surfactants (Fig. 11.31). In recycling operations polymers have the big disadvantage that they deteriorate under shear because the molecules break under shear forces. Surfactants do not have this disadvantage because the micellar structures which produce this effect are self-healing. Pilot operations in Europe have been running for months without loss of efficiency. The energy costs have been cut to less than a half.
360
12
300
10
240
8
180
6
120
4
60
2
0
0
1
2
3
4
0 5 0,0 0,2 flow velocity [m/s]
0,4
0,6
0,8
Figure 11.31: Plot of the pressure drop vs.the flow velocity in a capillary in the laminar and turbulent flow regions for water (solid line) and for a drag reducing surfactant solution (750 ppm C14TABr + NaSal at 27.5 8C, dashed line).
11.7.2 Under what conditions do we find drag-reducing surfactants? The phenomenon occurs often in surfactant solutions in which small rod-like micelles are formed that are charged and the charge is not screened by excess salt [104]. Many surfactant systems have been found where such conditions exist. Typically the length of the rods is 236
11.7
Shear induced phenomena
smaller than the mean distance between them. From this point of view the micellar solutions can be considered dilute even though there is repulsive interaction between the rods. Because of this repulsion the micelles try to be as far away from each other as possible and set up what is called a nearest neighbour order. The result of this order is a correlation peak in scattering experiments. In typical conditions the surfactant concentration is a few mM (about 0.1–0.2 wt%), the rods are a few hundred angstroms long and their mean separation is somewhat larger. Because there is no steric hindrance between the rods they can undergo Brownian rotations with rotation times of a few microseconds. Such conditions can easily be set up when charged surfactants are mixed with zwitterionic surfactants. Usually mixtures of ionic (10–30 mol%) and zwitterionic surfactants are favourable for the effect. These features of the micelles are the necessary prerequisites for the occurrence of the effect. They are, however, not the only ones, as will become clear. In shear measurements one expects the described solutions behave like normal Newtonian aqueous solutions. This is in fact the case for small shear rates (Fig. 11.32). In Fig. 11.32 the shear viscosity, which was measured in a capillary viscometer, is plotted vs.the shear rate. One observes a sudden rise of the viscosity at a characteristic shear rate g_ c and for g_ > g_ c the solutions show some shear thickening behaviour. Obviously something dramatic has happened to the micelles in the solutions. Some conclusions about what has happened can be drawn from flow birefringence measurements. Some typical results of flow measurements from a Couette system are shown in Fig. 11.33. We note a sudden increase of the flow birefringence at a critical shear rate. For g_ < g_ c no flow birefringence could be detected. In flow experiments, besides the birefringence it is also easy to measure the angle of extinction, which is the angle between the direction of flow and the mean orientation of the rods. In normal flow orientation this angle decreases smoothly from 458 to zero with increasing flow rate because the rods become more and more aligned. In the drag-reducing solution the situation is very different. In the Newtonian region for g_ < g_ c the solution remains isotropic and no preferential alignment can be detected. However, if g_ > g_ c the angle of extinction is close to zero, i. e. the new structures which are produced by the shear are completely aligned. Obviously the newly found structures must be much larger than the original small rods which were not aligned.
8
η[mPas]
6 4
60 mM 80 mM 100 mM 120 mM T= 25 °C
2 0 101
102
. γ[s-1 ]
103
Figure 11.32: The shear viscosity Z vs. the shear rate g_ for mixtures of C14DMAO : SDS = 6 : 4 with various total concentrations in a capillary viscometer at 25 8C.
237
11
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions 10 30 mM 60 mM 80 mM 100 mM
-∆n/ 10-6
8 6 4 2 0
0
100
200
300
400
.
500
600
700
800
900
γ / s-1
Figure 11.33: The flow birefringence Dn vs. the shear rate g_ for the same solutions as in Fig. 11.32.
It is believed now that the small rod-like micelles undergo collisions in the shear flow and they stick together for some time because of their interfacial properties. In this way long necklace-type structures are formed under shear and at the same time get aligned in the shear flow. This situation is schematically sketched in Fig. 11.34. These necklace-type structures act like high molecular weight polymers and give rise to drag reduction. These results show that shear is an important variable for micellar structures. We are aware that generally temperature, ionic strength, concentration, and cosurfactants can change micellar structures and hence the properties of surfactant solutions. We also should be aware that shear can change and influence micellar structures and even mesophases. It has been observed that micellar solutions can be transformed into liquid crystalline phases and single clear phase solutions become turbid as well as biphasic under shear. On the other hand, biphasic solutions can turn into a single phase under shear. A very striking example of a shear-produced transition is the transformation of a dilute L3 phase into a La phase with bright iridescent colours [105]. All these effects are based on the fact that the shape and size of the micelles depend to some degree on the intermicellar interaction energy, which itself depends on the mutual orientation of the micelles. When the interaction energy is changed the system responds with a change of structure and properties. These changes can be unexpected and large. They can, however, be used to our advantage.
Figure 11.34: Model for the explanation of the shear induced micellar structures. The small rod-like micelles can form long necklace-type structures under shear.
238
11.8
11.8
SANS measurements on micellar systems
SANS measurements on micellar systems
A large variety of our surfactant systems has been investigated by means of small angle neutron scattering (SANS) experiments mainly in cooperation with the group of Prof. Kalus (Universität Bayreuth). SANS is a method particularly suited for the study of self-aggregating colloids since its spatial resolution is typically in the range of 10–1000 Å, which is the size range of micellar aggregates. In the course of the investigation a shear apparatus was constructed in the group of Prof. Kalus which allows SANS experiments under shear. This has the advantage that one can align anisometric aggregates in the shear field and from the scattering curves of the aligned particles one can deduce more detailed information regarding their structure and their dynamic behaviour in the shear field. In addition, systems can be studied that exhibit shear induced structures, which are very interesting since they show drag-reducing behaviour. In the following we give just a short, exemplary overview over our large number of SANS experiments (some of them are discussed in the corresponding chapters, e. g. the work on cubic phases). The SANS is a method for the detailed determination of size and shape of the corresponding surfactant assemblies. For instance, from such experiments we found that tetramethylammoniumperfluoroctanesulfonate (TMAFOS) forms rod-like structures with a radius of 22 Å and a length of 200–300 Å [106]. For perfluorated surfactants the SANS method is only particularly advantageous because the refractive index of water and perfluorated compounds are very close. Therefore for these systems the powerful light scattering experiments for studying micellar systems will not work. Furthermore, detailed structural information might not be accessible for perfluorated surfactants, which are an interesting class of surfactants since normally they are even more surface-active than their hydrocarbon counterparts. However, elongated micelles are by no means restricted to perfluoro surfactants but also commonly found with conventional hydrocarbon alkyl surfactants. An interesting system is the mixed cationic/anionic surfactant tetradecylpyridinium-heptanesulfonate. Here SANS measurements have shown that below 160 mM charged rod-like aggregates are present which get shorter with increasing concentration. For these samples always a correlation peak is observed in the spectra but for more concentrated samples this peak will disappear. The reason is that with increasing concentration the degree of dissociation is decreasing. Therefore the charge density on the micelles decreases thus decreasing the electrostatic repulsion between the aggregates until they are effectively uncharged at high concentration [107]. Another interesting type of surfactant are double-chain amphiphiles. As an example of this surfactant-type hexadecyloctyldimethylammoniumbromide (C16C8DMABr) has been investigated. Here just above the cmc globular aggregates are formed whereas above a second transition concentration relatively short (150 Å) rod-like micelles are formed [108]. However, anisometry of the micellar aggregates is a necessary (yet not sufficient) prerequisite of liquid crystal formation. Indeed, at higher concentrations many perfluoro surfactants are known to form nematic, lyotropic, liquid, and crystalline phases [109]. One such system is the tetramethylammoniumperfluorononanoate (TMAPFN) which exhibits a nematic phase over a relatively large concentration range but only in a small temperature interval [91]. This system has been studied in detail by SANS [92]. The experiments showed that 239
11
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
the nematic phase is build up from disc-like micellar aggregates, i. e. the local structure of the amphiphilic film is planar. This phase can be oriented in an external magnetic field, thereby allowing for a more detailed analysis of the underlying micellar structures. The thickness of the disk was found to be 37 Å and from a rocking experiment the order parameter S & 0.85 was deduced. In addition, it is interesting to note that only the nematic phase can be oriented in an external magnetic field. But upon cooling the samples are transformed into the corresponding lamellar phase. This phase could not be oriented but it is highly ordered and also keeps its orientation without applying a magnetic field. As just discussed, lyotropic nematic phases can easily be aligned in an external magnetic field and it is of course of interest to study this alignment process as a function of time, i. e. to study the dynamics of the nematic phase. This has been done for a system made up from an aqueous solution of 10.2 wt% hexadecyltrimethylammoniumbromide (CTAB), 9.9 wt% hexadecyltrimethylammoniumbenzenesufonate, and 2.45 wt% decanol in D2O. This system forms a nematic phase that consists of disc-like micelles. Such a sample had been prealigned in a magnetic field of 7 T. The highly ordered phase gives a strongly anisotropic scattering pattern with two sharp peaks (Fig. 11.35). The preoriented sample which was in the neutron beam was then exposed to a magnetic field (1.4 T) oriented perpendicular to the axis of the originally employed magnetic field as well as to the neutron beam. Of course, this new magnetic field wants to reorient the texture of the sample and this reorientation process has been monitored by SANS. In Fig. 11.35 we see the time evolution of the scattering pattern: upon turning on the perpendicular field the two original peaks start to disintegrate into four smaller peaks which move on a ring to form finally again two narrow peaks which are then oriented perpendicular to the original peaks. The typical time for reorientation is about 60 min [110].
a)
b)
c)
d)
Figure 11.35: SANS curves of a prealigned nematic phase of a CTAB system (composition see text). the original aligned sample (a), the sample in a perpendicular magnetic field B1 after 20 (b), 50 (c), and 90 min (d). (B0 = 7 T, B1 = 1,4 T).
240
11.8
SANS measurements on micellar systems
Of course, anisometric micelles can also be oriented by shear and this again allows a more detailed study of the micellar structure as well as the dynamic alignment process. One such system has been cetylpyridiniumsalicylate in 20 mM NaCl D2O solution which at 20 mM concentration has been shown to contain rod-like micelles with a radius of 21.5 Å and a length of 500–750 Å [111]. Under shear an anisotropic scattering pattern is observed. It relaxes to the isotropic pattern after switching off the applied shear field. Time resolved measurements (time resolution of 250 ms) showed that the scattering curves during this relaxation process can be described by a single parameter, namely the rotational diffusion coefficient Drot. It was observed that Drot is time-dependent and decreases with time due to the interaction between the charged rod-like aggregates, i. e. at the beginning the aligned rods have the largest electrostatic repulsion giving a large driving force for the disorientation because for this aligned arrangement the electrostatic potential energy has its highest value. The less aligned the system the smaller becomes this driving force (since the electrostatic interaction becomes weaker) and correspondingly the rotational diffusion coefficient becomes smaller [112]. The experimental scattering curves can be explained by an orientation distribution function of the rods depending on the applied shear gradient [113]. Under the experimental conditions the product of shear gradient g_ and structural relaxation time t, which is increasing with the length of the aggregates, was always much larger than 1, i. e. a high orientational ordering was achieved because the ordering force becomes stronger than the diffusive force. Generally all anisometric aggregates can become oriented in the shear field but systems with shorter relaxation times t require higher shear gradients. This effect has been studied with the above mentioned C16C8DMABr at a concentration of 50 mM. Here the unsheared solution shows a correlation ring which becomes increasingly anisotropic with increasing shear rate. Figure 11.36 shows the scattering patterns for various shear gradients. In addition a higher order peak becomes visible. Finally for shear rates above g_ = 2000 s –1 the scattering pattern hardly changes any more. In this system not only the relatively short rodlike aggregates become weakly aligned but also a second type of micelle is formed in the shear field, where their relative concentration increases with rising shear rate. Therefore in this system a shear-induced transformation of micellar aggregates is observed [114]. A similar behaviour has also been observed for the system tetradecyltrimethylammoniumsalicylate (C14TMA-Sal) [115] and seems to be quite common for viscoelastic surfactant systems. A more detailed analysis of this system indicates that the sharp peaks that occur upon applying the shear are due to a hexagonal array of cylindrical micelles, i. e. again now 2 types of micelles are present: the originally present short rod-like micelles and the very long rod-like aggregates that are formed in the flow field [116]. This lattice-type array is formed within about 2 min as could be seen from time-resolved shear experiments [116]. This result agrees also well with flow birefringence experiments. At a given shear rate an equilibrium between the short rod-like micelles, which normally are only weakly aligned, and the long rodlike aggregates is present. This equilibrium depends on the shear rate. The higher the shear rate the more the equilibrium is shifted in favour of the long aggregates [117]. The growth process of the large micellar structures, which are strongly aligned, has been studied in more detail by transient SANS experiments. In these experiments the shear rate for the samples was raised stepwise from zero to a certain finite value. These experiments showed that the large micelles grow according to the Avrami law c
t cinf
1
exp
kt :
19 241
11
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions a)
b)
c)
d)
Figure 11.36: SANS patterns of C16C8DMABr in D2O (50 mM) for various shear gradients (the momentum transfer is given in units of nm –1): g_ = 0 s–1(a), g_ = 100 s–1(b), g_ = 400 s–1(c), g_ = 2000 s–1(d). The neutron beam is perpendicular to the direction of shear.
Originally this equation was used to describe nucleation and growth in metals and alloys. For the given system (C14TMA-Sal) the exponent n was found to be between 2 and 2.5 [118]. The exponent n should be i + 1 for i-dimensional growth, which means that the growth process observed here is close to a one-dimensional one as should be expected since the micelles grow in length without change of dimension. Such shear-induced structures (SIS) are an interesting phenomenon in particular for self-aggregating systems like micelles where the equilibrium structure often depends very subtly on small energetic changes. Of course, these structural changes have a profound influence on the properties of these systems, especially on their flow behaviour. For instance one may observe a shear-thickening behaviour that is coupled to drag-reducing properties of the system. Shear-induced transitions have been found in a variety of micellar systems [105]. Of particular interest in our studies have been systems that in the unsheared state contain small rod-like micelles and systems which show a strongly anisotropic behaviour beyond a critical threshold shear rate g_ c. Above this shear rate a strong increase of flow birefringence and viscosity together with a large anisotropy of the electric conductivity are observed. At the same time the scattering patterns of the SANS experiments exhibit also a strong anisotropy. This shear-induced effect will already occur at g_ trot P 1, where trot is the rotational time constant of the small type of micelles, i. e. in a range where the shear field should not be able to orient the small rod-like aggregates significantly. This means that the observed anisotropy is not due to the orientation of these originally present micelles but that larger oriented micellar aggregates have to be present in the solutions. So far the mechanism for formation of the SIS is not fully understood and several different mechanisms have been postulated [119–121]. 242
11.9
A new rheometer
The tetradecyldimethylaminoxide/sodiumdodecylsulfate (C14DMAO/SDS) system has been studied in much detail. This system shows a pronounced SIS formation around a molar mixing ratio of 7 : 3 for C14DMAO/SDS [122] – and it might also be noted here that for this composition the nematic phase, which is found for those systems, extends to the lowest surfactant concentration [123]. In order to find relations between the macroscopic behaviour of the system and the structure of the micellar aggregates, SANS study was performed [124]. Changing the contrast condition in the micellar aggregate by using both hydrogenated and deuterated SDS, yields detailed information regarding the structure of the micelles. The SANS experiments show that at the mixing ratios where the length of these aggregates reduces with increasing SDS content and where SIS is observed, elongated micelles are present which are best described by a three-axes prolate ellipsoid. From a contrast variation experiment using both, deuterated and hydrogenated SDS, it could be concluded that the buildup of these micelles is homogeneous and no internal segregation of the surfactant molecules within the aggregate could be deduced [124]. Such an internal segregation by having an enrichment of SDS at the end caps (here the relative area per molecule is necessarily larger because of the larger curvature and one could imagine that the SDS with its larger hydrophilic head group would preferentially be located at this position) would have been conceivable and, of course, such a build-up that would contain two more highly charged ends would have been an important factor to consider for the explanation of the SIS. However, this evidently is not the case and SIS formation has to be explained starting from originally short, homogeneously charged, elongated aggregates. If we summarize we find that SANS is a very powerful tool to investigate self-aggregating structures. It is perfectly adapted with the size range to be observed and in addition it allows a detailed observation of anisotropic samples which might be oriented by a magnetic field, shear field, order effects close to the wall of the cells, etc. Scattering patterns of such oriented samples contain even more information regarding the intra and interparticle structure. Furthermore, the contrast conditions in the samples can in principle be changed because they consist to a large degree of hydrogen and because the scattering properties of H and D nuclei differ largely. Therefore isotopic substitutions, that normally have only a very minor effect on the properties of the respective systems, will enable us to get much more structural information than would be accessible via other methods. This method is called the contrast variation. Those experiments have shed a lot of light on structural properties of micellar systems with respect to some of their dynamic properties.
11.9
A new rheometer
From electric birefringence measurements it is known that surfactant solutions, like those of tetraethylammoniumperfluorooctanesulfonate, show different time constants in the range from microseconds up to some seconds [24, 94, 125]. The shortest time t1 is attributed to the free rotation of rod-like micelles and amounts to 10 –5 to 10 –6 s. It can be found in almost the whole concentration range. The second time t2 is the birefringence which has a dif243
11
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
ferent sign and comes to 10–4 s. It can only be detected in the narrow concentration region where the rods begin to overlap (7 mM–15 mM). The longest time constant t4 (about some seconds) represents the structural relaxation time. It can be found in the electric birefringence and rheological measurements at sufficiently high concentrations (c > 30 mM) when a network is formed. The remaining time constant t3 (in the range of milliseconds) could not yet be detected by rheological measurements because the frequency range of commercial rheometers ends typically at frequencies of about 10 Hz. Therefore it is not possible to determine rheological time constants shorter than about 0.1 s. In order to overcome this problem, it was necessary to improve the frequency range for the dynamic rheological measurements. For this purpose a dynamic rheometer (HF rheometer) with a frequency range from 1 Hz up to 1 kHz was built. For samples with a high modulus (G & 1 kPa) the measurements can even be extended to 2 kHz. With this apparatus measurements on viscoelastic surfactant solutions were carried out. As expected, these solutions show short rheological time constants t3. They correspond well to those determined with the electric birefringence. The new HF rheometer is based on a prototype that was acquired from a group of the Universität Ulm, Abteilung Angewandte Physik, Prof. W. Pechhold. The sensitivity of the apparatus could be notably improved by numerous technical modifications. A schematic drawing of the mechanical part of the apparatus is shown in Fig. 11.37.
Figure 11.37: Schematic drawing of the mechanical part of the HF rheometer.
244
11.9
A new rheometer
The rheometer works with a concentric cylinder geometry. The gaps between the cylinders are 50 mm or 100 mm, respectively. The inner cylinder is driven by an electromechanical converter (shaker) and performs linear harmonic oscillations. The force is transferred by the sample to the outer cylinder and is detected by a very sensitive piezoelectric force transducer producing a signal which is amplified by a charge amplifier. The amplitude of the inner cylinder is determined by an inductive displacement transducer coupled with a carrier frequency measuring amplifier. A lock-in amplifier is used for the measurement of the force signal, the displacement signal, and the phase angle. In the present state measurements on viscoelastic samples with a modulus of 50 Pa, 100 Pa, and 1 kPa are possible up to 900 Hz (50 mM gap), 1.3 kHz and 2 kHz, respectively. The reliability of the apparatus was tested and the frequency range was experimentally determined by measurements on Newtonian liquids (silicon oils, glycerol/water mixtures). The viscosities of Newtonian liquids can be well detected down to 50 mPas. In this case the frequency range extends to 1.6 kHz. Solutions of tetraethylammoniumperfluorooctanesulfonate (C8F17SO3NEt4) were studied with the HF rheometer. A typical rheogram is shown in Fig. 11.38. The measurement was carried out on a 90 mM solution with the 50 mM gap in the frequency range above 10 Hz. Below 10 Hz a Bohlin CS 10 rheometer with a cone plate geometry was employed.
G´, G´´ / Pa, |η*| / Pas
102 101 100 10-1 G´ G´´ |η*|
10-2 10-3 10-4 10-2
10-1
100
f / Hz
101
102
103
Figure 11.38: Dynamic rheogram of a 90 mM solution of C8F17SO3NEt4 at T = 20 8C. Below 10 Hz the measurement was performed with a Bohlin CS 10 rheometer, above 10 Hz with the HF rheometer and the 50 mM gap (with 1% deformation).
In the lower frequency range the samples show Maxwell behaviour. G' rises with the slope 2, G@ with slope 1. At frequencies above the crossover of G' and G@, G' levels out and reaches a plateau value, whereas G@ first decreases and then increases again. At high frequencies both, G' and G@, increase. Even though a second plateau value of G' at high frequencies could not be found it was possible to fit the data by a Burger model (four parameter Maxwell model): G0
o G1
o2 t24 o2 t23 ot4 ot4 G and G0
o G1 G2 :
20 2 1 o2 t24 1 o2 t23 1 o2 t24 1 o2 t23 245
11
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
The structural relaxation time t4 (the indices have been chosen for correspondence with previous results) decrease with rising concentration from 25 ms (70 mM) to 0.2 ms (300 mM) and the short time constant t3 decrease from 0.35 ms (70 mM) to 0.1 ms (250 mM). Therefore, they correspond well with the time constants determined by dynamic electric birefringence measurements. For each concentration the minimum value of G@ is about a factor of 2.5 lower than the plateau value of G'. According to the theory of Granek and Cates [126] this yields a value of 2.5 for the ratio of the mean contour length of the micelles to the entanglement length (Eq. 7)). A decrease of the mean micellar length with rising concentration can be concluded on the basis of this value and from the decreasing structural relaxation times. Furthermore, mixtures of tetraethylammoniumperfluorooctanesulfonate and the pure perfluorooctanesulfonic acid (C8F17SO3H) were studied with the HF rheometer. In Fig. 11.39 a rheogram is shown of the 150 mM solution with the salt : acid ratio of 7 : 3.
G´, G´´ / Pa, |η*| / Pas
103 102 101 100 10-1
G´ G´´ |η*|
-2
10
10-3 10-2
10-1
100
101
102
103
f / Hz
Figure 11.39: Dynamic rheogram of a 150 mM solution of C8F17SO3NEt4/C8F17SO3H = 7 : 3 at T = 21 8C. The deformation was 1% again. The minimum of G@ is lower than in the case of the pure C8F17SO3NEt4.
From a qualitative point of view the rheogram does not differ too much from the one presented in Fig. 11.4 a. It can be noticed that after substitution of salt by acid the minimum of G@ is more pronounced. This means a larger ratio of the mean contour length to the entanglement length (here about 4), calculated by Eq. 7. Generally one observes that up to a mole fraction of 40 % of the acid the structural relaxation time t4 increases by a factor 10 (from 7 ms to 70 ms). From this result the growth of the micellar aggregates can be concluded. The short time constant t3 does not notably change and amounts to about 0.1 ms. At a molar ratio of more than 50 % of the acid the viscosity breaks down due to a decreasing length of the micelles. In this case it is not possible anymore to measure the samples with the HF rheometer. The high frequency increase of the moduli can be interpreted on basis of the following models: a) In addition to the continuous network there also exist unentangled shorter rods due to the equilibrium distribution of the micellar lengths. These shorter aggregates do not significantly influence the rheological behaviour of the samples at lower frequencies. At higher frequencies, however, they contribute to the moduli. 246
References b) At angular frequencies higher than the reciprocal value of the Rouse time the chain segment of the entanglement length cannot relax anymore by Rouse diffusion and a glass process starts causing an increase of the moduli.
References
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.
K.L. Mittal: ACS-Symp. Ser., 8, Am. Chem. Soc, Washington DC, (1975) C. Tanford: The Hydrophobic Effect, Wiley & Sons, New York, (1980) H. Hoffmann: Progr. Colloid Polym. Sci., 83, 16 (1990) J.N. Israelachvili, D.J. Mitchell, B.W. Ninham: J. Chem. Soc. Faraday Trans. II, 72, 1525 (1976) M. Borkovec: J. Chem. Phys., 91, 6268 (1989) K.S. Chan, D.O. Shah: J. Dispersion Sci. Technol., 1, 55 (1980) W.H. Wade, H. Seang, S.H. Yiv: J. Colloid Interface Sci., 92, 561 (1983) O. Abillon, P.B. Bings, D. Langevin, J. Meunier: Progr. Colloid Polym. Sci., 76, 84 (1988) G. Platz, C. Thunig, H. Hoffmann: Ber. Bunsenges. phys. Chem., 96, 667 (1992) M. Jonströmer, R. Strey: J. Phys. Chem., 96, 5994 (1992) O. Ghosh, C.A. Miller: J. Phys. Chem., 91, 4528 (1987) E.W. Kaler, A.K. Murthy, B.E. Rodriguez, J.A.N. Zasadzinski: Science, 245, 1371 (1989) H. Hoffmann, C. Thunig, U. Munkert, H.W. Meyer, W. Richter: Langmuir, 8, 2629 (1992) T.M. Clausen, P.K. Vinson, J.R. Minter, H.T. Davis, Y. Talmon, W.G. Miller: J. Phys. Chem., 96, 474 (1992) S. Toshiyuki, H. Hirata, T. Kodaka: Langmuir, 3, 1081 (1987) B.K. Mishra, S.D. Samant, P. Pradhan, S.B. Mishra, C. Manohar: Langmuir, 9, 894 (1993) S.J. Candau, E. Hirsch, R. Zana, M. Delsanti: Langmuir, 5, 1225 (1989) H. Hoffmann, A.K. Rauscher, M. Gradzielski, S.F. Schulz: Langmuir, 8, 2140 (1992) A. Khatory, F. Kern, F. Lequeux, J. Appell, G. Porte, N. Morie, A. Ott, W. Urbach: Langmuir, 9, 933 (1993) H.A. Barnes, J.F. Hutton, K. Walters: in: An Introduction to Rheology, Rheol. Ser., 3, Elsevier, New York, (1989) M.E. Cates: in: Proc. ACS-Symp. on Polymeric Microemulsions and Polymer-Microemulsion Interaction, New Orleans, (1987) A.K. Rauscher, H. Rehage, H. Hoffmann: Progr. Colloid Polym. Sci., 84, 99, (1991) A. Khatory, F. Lequeux, F. Kern, S.J. Candau: Langmuir, 9, 1456 (1993) H. Hoffmann, K. Reizlein, H. Rehage, H. Thurn: ACS-Symp. Ser., 27, 44 (1985) H. Hoffmann, U. Krämer, H. Thurn: J. Phys. Chem., 94, 2027 (1990) M. Angel, H. Hoffmann, U. Krämer, H. Thurn: Ber. Bunsenges. phys. Chem., 93, 184 (1989) H. Hoffmann, E. Schönfelder: Ber. Bunsenges. phys. Chem., 98, 842 (1994) F.C. McKintosh, S.A. Safran, P.A. Pincus: J. Phys. Condens. Matter, 2, 359 (1990) S.A. Safran, P.A. Pincus, M.E. Cates, F.C. McKintosh: J. Phys. France, 51, 503 (1990) H. Hoffmann, M. Löbl, H. Rehage, I. Wunderlich: Tenside Surf. Det., 22, 290 (1985) H. Thurn, M. Löbl, H. Hoffmann: J. Phys. Chem., 89, 517 (1985) H. Hoffmann, M. Löbl, H. Rehage: Suppl. Rheol. Acta, 26, 246 (1988) M. Valiente, C. Thunig, U. Munkert, U. Lenz, H. Hoffmann: J. Colloid Interface Sci., 160, 39 (1993) T. Imae, M. Sasaki, S. Ikeda: J. Colloid Interface Sci., 127, 511 (1989) S.J. Candau, A. Khatory, F. Lequeux, F. Kern: in: Proc. of the Workshop on Complex Liquid Systems, Polistena (1992)
247
11 31. 32. 33. 34. 35. 36.
37. 38. 39.
40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70.
248
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
J. Appell, G. Porte, A. Khatory, F. Kern, S.J. Candau: J. Phys. II, 2, 1045 (1992 T.J. Drye, M.E. Cates: J. Chem. Phys., 96, 1367 (1992) M.S. Turner, C. Marques, M.E. Cates: Langmuir, 9, 695 (1993) A. Ott, W. Urbach, D. Langevin, H. Hoffmann: Langmuir, 8, 1045 (1992) T. Kato, T. Terao, M. Tsukada, T. Seimiya: J. Phys. Chem., 97, 3910 (1993) E.W. Kaler, K.L. Herrington, A. Murthy, J.A.N. Zasadzinski: J. Phys. Chem., 96, 6698 (1992) K. Fontell, A. Ceglie, B. Lindman, B.W. Ninham: Acta Chem. Scand., A40, 247 (1986) T. Zemb, D. Gazeau, M. Dubois, T. Gulik-Krzywicki: Europhys. Lett., 21, 759 (1993) U. Munkert, H. Hoffmann, C. Thunig, H.W. Meyer, W. Richter: Progr. Colloid Polym. Sci., 93, 137 (1993) H. Hoffmann, U. Munkert, C. Thunig, M. Valiente: J. Colloid Interface Sci., 163, 217 (1994) Y. Okahata, T. Kunitake: J. Am. Chem. Soc., 101, 5231 (1979) B.W. Ninham, D.F. Evans, G.J. Wei: J. Phys. Chem., 87, 5020 (1983) D. Andersson, H. Wennerström, U. Olsson: J. Phys. Chem., 93, 4243 (1989) R. Strey, W. Jahn, G. Porte, P. Bassereau: Langmuir, 6, 1635 (1990) C.A. Miller, O. Ghosh, W. Benton: Colloids Surf., 19, 197 (1986) S. Szönyi, H.J. Watzke, A. Cambon: Progr. Colloid Polym. Sci., 89, 149 (1992) C. Thunig, G. Platz, H. Hoffmann: Springer Proc. in Physics, 66, 266 (1992) H. Hoffmann, C. Thunig, M. Valiente: Colloids Surf., 67, 223 (1992) M. Dubois, T. Zemb: Langmuir, 7, 1352 (1991) H. Rehage, H. Hoffmann: Mol. Phys., 74, 933 (1991) W.P. Cox, E.H. Merz: J. Polym. Sci., 28, 619 (1958) D. Roux, F. Nallet, O. Diat: Europhys. Lett., 24, 53 (1993) D. Roux, F. Nallet, O. Diat: J. Phys. II, 3, 1427 (1993) H. Hoffmann, C. Thunig, P. Schmiedel, U. Munkert: Langmuir, 10, 3972 (1994) H. Hoffmann, C. Thunig, P. Schmiedel, U. Munkert: Nuovo Cimento, 16D, 1373 (1994) H. Hoffmann, C. Thunig, P. Schmiedel, U. Munkert, W. Ulbricht: Tenside Surf. Det., 31, 389 (1994) E. van der Linden, J.H.M. Dröge: Physica A, 193, 439 (1993) H.N.W. Lekkerkerker: Physica A, 140, 319 (1989) T. Palberg, J. Kottal, F. Bitzer, R. Simon, M. Würth, P. Leiderer : J. Colloid Interface Sci., 169, 85 (1995) M. Dubois, T. Zemb, L. Belloni, A. Delville, P. Levitz, R. Setton: J. Chem. Phys., 96, 2278 (1992) K. Fontell: Colloid Polym. Sci., 268, 264 (1990) W. Jettka: PhD thesis, Universität Braunschweig (1984) G. Oetter, H. Hoffmann: Colloids Surf., 38, 225 (1989) H.N. Hoffmann, E.F. Paulus: Fette, Seifen, Anstrichmittel, 71, 399 (1969) E.S. Lower, W.H. Harding: Manuf.Chem., 37, 52 (1966) J.S. Patton: M.C. Carey, Science, 204, 145 (1979) D.P. Siegel: Biophys.J., 49, 1155 (1986) D.W. Osborne, A.J.I. Ward, K.J. O’Neill: in: D.W. Osborne and A.H. Amann (eds.): Topical Drug Delivery Formulations, Marcel Dekker, New York, (1990) M. Gradzielski, H. Hoffmann, G. Oetter: Colloid Polym. Sci., 268, 167 (1990) M. Gradzielski, H. Hoffmann: Adv. Colloid Interface Sci., 42, 149 (1992) P.N. Pusey, W. van Megen: Nature, 320, 340 (1986) K. Mortensen, J.S. Pedersen: Macromolecules, 26, 805 (1993) M. Gradzielski: PhD thesis, Universität Bayreuth (1992) H. Jousma, J.G.H. Joosten, G.S. Gooris, H.E. Junginger: Colloid Polym. Sci., 267, 353 (1989) E.O. Stejskal, J.E. Tanner: J. Chem. Phys., 42, 288 (1965) P. Stilbs: Progr. NMR Spectrosc., 19, 1 (1987) K.L. Walther, M. Gradzielski, H. Hoffmann, A. Wokaun, G. Fleischer: J. Colloid Interface Sci., 153, 272 (1992) A. Ceglie, K.P. Das, B. Lindman: Colloids Surf., 28, 29 (1987)
References 71. M. Gradzielski, H. Hoffmann, in: D.M. Bloor and E. Wyn-Jones (eds.): The Structure, Dynamics and Equilibrium Properties of Colloidal Systems, Kluwer Academic Publishers, Dordrecht, p. 427 (1990) 72. B.B. Laird: J. Chem. Phys., 97, 2699 (1992) 73. M. Gradzielski, H. Hoffmann, J.C. Panitz, A. Wokaun: J. Colloid Interface Sci., 169, 103 (1995) 74. G. Wanka, H. Hoffmann, W. Ulbricht: Colloid Polym. Sci., 268, 101 (1990) 75. K. Mortensen, W. Brown, B. Norden: Phys. Rev. Lett., 13, 2340 (1992) 76. W. Brown, K. Schillen, S. Hvidt: J. Phys. Chem., 96, 6038 (1992) 77. Y. Deng, G.E. Yu, C. Price, C. Booth: J. Chem. Soc. Faraday Trans. I, 88, 1441 (1992) 78. Z. Zhou, B. Chu: J. Colloid Interface Sci., 126, 171 (1988) 79. G. Wanka, H. Hoffmann, W. Ulbricht: Macromolecules, 27, 4145 (1994) 80. E. Hecht, K. Mortensen, M. Gradzielski, H. Hoffmann: J. Phys. Chem., 99, 4866 (1995) 81. M. Seddon: Biochim. Biophys. Acta, 1031, 1 (1990) 82. J. Charvolin, J.F. Sadoc: J. Phys. France, 47, 683 (1987) 83. J. Rogers, P.A. Winsor: J. Colloid Interface Sci., 30, 247 (1969) H. Kunieda, K. Shinoda: J. Phys. Chem., 82, 1710 (1978) 84. G. Oetter, H. Hoffmann, in: D.M. Bloor and E. Wyn-Jones (eds.): The Structure, Dynamics and Equilibrium Properties of Colloidal Systems, Kluwer Academic Publishers, Netherlands, p. 427 (1990) 85. P. Ekwall, L. Mandell: Acta Polytech. Scand., 74, 92 (1968) F.C. Larche, J. Appell, G. Porte, P. Bassereau, J. Marignan: Phys. Rev. Lett., 56, 700 (1986) 86. M. Boidart, A. Hochapfel, P. Peretti: Mol. Cryst. Liq. Cryst., 172, 147 (1989) 87. K.D. Lawson, T.J. Flautt: J. Am. Chem. Soc., 89, 5490 (1967) 88. T. Haven, K. Radley, A. Saupe: Mol. Cryst. Liq. Cryst., 75, 87 (1981) B.J. Forrest, L.W. Reeves: Chem. Rev., 81, 1 (1981) 89. G. Hertel: PhD thesis, Universität Bayreuth (1989) G. Hertel, H. Hoffmann: Progr. Colloid Polym. Sci., 76, 123 (1989) G. Hertel, H. Hoffmann: Liq. Cryst., 5, 1883 (1989) D. Tezak, G. Hertel, H. Hoffmann: Liq. Cryst., 10, 15 (1991) 90. K. Reizlein: PhD thesis, Universität Bayreuth (1983) 91. K. Reizlein, H. Hoffmann: Colloid Polym.Sci., 69, 83 (1984) 92. L. Herbst, H. Hoffmann, J. Kalus, K. Reizlein, U. Schmelzer, K. Ibel: Ber. Bunsenges. phys. Chem., 89, 1050 (1985) 93. M. Angel, H. Hoffmann, B. Schwandner, R. Weber: in: H. Sackmann (ed.): Proc. of the Liquid Crystal Conference in Halle, Kongreß- und Tagungsberichte der Martin-Luther-Universität HalleWittenberg, p. 120 (1985) 94. U. Krämer: PhD thesis, Universität Bayreuth (1990) 95. G. Bartusch, H.D. Dörfler, H. Hoffmann: Progr. Colloid Polym. Sci., 89, 307 (1992) 96. B. Lindman, K. Fontell: J. Phys. Chem., 87, 3289 (1983) 97. U. Munkert: PhD thesis, Universität Bayreuth (1994) 98. M. Gradzielski, H. Hoffmann, J.C. Panitz, A. Wokaun: J. Phys. Chem., 98, 6812 (1994) 99. S. Kaiser, H. Hoffmann: J. Colloid Interface Sci., 184, 1 (1996) 100. H. Hoffmann, A.K. Rauscher: Colloid Polym. Sci., 271, 390 (1993) 101. C. Huang: Biochemistry, 8, 344 (1969) 102. U. Lenz: PhD thesis, Universität Bayreuth (1991) 103. E. Hecht, K. Mortensen, H. Hoffmann: Macromolecules, 28, 5465 (1995) 104. D. Ohlendorf, W. Interthal, H. Hoffmann: Rheol. Acta, 25, 468 (1986) I. Wunderlich, H. Hoffmann, H. Rehage: Rheol. Acta, 26, 532 (1987) 105. H. Hoffmann, S. Hofmann, A.K. Rauscher, J. Kalus: Progr. Colloid Polym. Sci., 84, 24 (1991) 106. H. Hoffmann, J. Kalus K. Reizlein, W. Ulbricht, K. Ibel: Colloid Polym. Sci., 260, 435 (1982) 107. H. Hoffmann, J. Kalus, B. Schwandner: Ber. Bunsenges. phys. Chem., 91, 99 (1987) 108. G. Neubauer, H. Hoffmann, J. Kalus: B. Schwandner: Chem. Phys., 110, 247 (1986) 109. M. Boidart, A. Hochapfel, M. Laurent: Mol. Cryst. Liq. Cryst., 154, 61 (1988)
249
11
The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
110. J. Baumann, G. Hertel, H. Hoffmann, K. Ibel, V. Jindal, J. Kalus, P. Lindner, G. Neubauer, H. Pilsl, W. Ulbricht, U. Schmelzer: Prog. Colloid Polym. Sci., 81, 100 (1990) 111. H. Hoffmann, J. Kalus, H. Thurn, K. Ibel: Ber. Bunsenges. phys. Chem., 87, 1120 (1983) H. Thurn, J. Kalus, H. Hoffmann: J. Chem. Phys., 80, 3440 (1984) 112. L. Herbst, H. Hoffmann, J. Kalus, H. Thurn, K. Ibel, R.P. May: Chem. Phys., 103, 437 (1986) 113. J. Kalus, H. Hoffmann: J. Chem. Phys., 87, 714 (1987) 114. J. Kalus, H. Hoffmann, S.H. Chen, P. Lindner: J. Phys. Chem., 93, 4267 (1989) 115. J. Kalus, H. Hoffmann, K. Ibel: Colloid Polym. Sci., 267, 818 (1989) 116. V.K. Jindal, J. Kalus, H. Pilsl, H. Hoffmann, P. Lindner: J. Phys. Chem., 94, 3129 (1990) 117. C. Münch, H. Hoffmann, J. Kalus, K. Ibel, G. Neubauer, U. Schmelzer: J. Appl. Cryst., 24, 740 (1991) 118. C. Münch, H. Hoffmann, K. Ibel, J. Kalus, G. Neubauer, U. Schmelzer, J. Selbach: J. Phys. Chem., 97, 4514 (1993) 119. M.E. Cates, M.S. Turner: Europhys. Lett., 11, 681 (1990) 120. S.Q. Wang: J. Phys. Chem., 94, 8381 (1990) 121. R. Bruinsma, W.M. Gelbart, A. Ben-Shaul: J. Chem. Phys., 96, 7710 (1992) 122. S. Hofmann, A.K. Rauscher, H. Hoffmann: Ber. Bunsenges. phys. Chem., 95, 153 (1991) 123. H. Hoffmann, S. Hofmann, J.C. Illner: Prog. Colloid Polym. Sci., 97, 103 (1994) 124. H. Pilsl, H. Hoffmann, S. Hofmann, J. Kalus, A.W. Kencono, P. Lindner, W. Ulbricht: J. Phys. Chem., 97, 2745 (1993) 125. W. Schorr, H. Hoffmann: J. Phys. Chem., 85, 3160 (1981) 126. R. Granek, M.E. Cates: J. Chem. Phys., 96, 4758 (1992)
250
12
Photophysics of J Aggregates Herrmann Pschierer, Hauke Wendt, and Josef Friedrich
12.1
Introduction
Dye molecules of the pseudoisocyanine (PIC)-type form, under certain conditions, linear long chain aggregates [1, 2]. The conditions concern the solvent, temperature, and freezing procedure and depend, in addition, on the dye molecule and its counterion [3]. At sufficiently low concentration and quick freezing aggregate formation is hampered and one obtains the monomer. The spectral properties of the monomer show the usual features of dye molecules in frozen solution, namely a rather broad structureless inhomogeneously broadened absorption band. It is centred around 18850 cm –1 (Fig. 12.1). On the other hand, if aggregates are formed, the spectral properties change dramatically. The wavenumber is shifted from the monomer band by about 1450 cm –1 to the red and one or several extremely sharp bands appear (Fig.12.1), whose intensities depend on the formation procedure. These sharp bands are the so-called J bands [2]. They have exciton-like character with large coherence lengths. It is the large coherence length which determines – with respect to localized states – the unusual optical properties, namely the narrowing phenomena in the inhomogeneous band, the molecular superradiative properties [4–7], the characteristic temperature de-
Figure 12.1: Low temperature (4.2 K) absorption spectra of the pseudoisocyaninechloride (PIC-Cl) aggregate and monomer. Macromolecular Systems: Microscopic Interactions and Macroscopic Properties Deutsche Forschungsgemeinschaft (DFG) Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
251
12
Photophysics of J Aggregates
pendence of the homogeneous line width [7, 8], and non-linear optical features [9–12]. In this paper we focus on the narrowing phenomena. A physically transparent interpretation of this narrowing phenomenon is based on fast moving excitons which average the inhomogeneities of the host glass to a large degree. The magnitude of the band narrowing depends on the coherence length. pIf the number of coherently coupled monomers is Nc the narrowing is approximately 1/ Nc . From the ratio of the inhomogeneous band widths of monomer and aggregate, Nc is estimated to be of the order of 100. The problem with this estimation is that the inhomogeneous broadening cannot be determined accurately enough since phonons and molecular vibrations may be hidden below the inhomogeneous envelope. Another problem may arise from correlation effects in the disorder, see below. Even the nature of the inhomogeneity in the J band is not known exactly. There are various suggestions in the literature: chain length distribution [13], conformational inhomogeneity [14], and coherence length distribution [15, 16]. In this paper we present a comparative pressure and electric-field tuning hole burning study between PIC monomers and aggregates. It is our goal to gain information on the coherence length of the excitons on the aggregate chains from these comparative experiments.
12.2
Basic aspects of pressure and electric-field phenomena in hole burning spectroscopy of J aggregates
Pressure has a twofold influence on spectral holes: it shifts the hole and, in addition, broadens it. The pressure shift of spectral holes in aggregates is quite strong. This strong pressure shift is also related to the coherence length [17], because it scales with the polarizability of the exciton chain. In this paper, however, we focus on the pressure broadening which has a very direct relation to the coherence length. In small molecules, like the PIC monomer, pressure broadening of spectral holes arises because the local configuration of host molecules surrounding the probe is changed a little bit when the lattice is compressed. These configurational changes signal a lack of spatial correlation among the glass-forming molecules. The respective broadening is inhomogeneous in nature. Hence, it will be motionally narrowed by a fast moving exciton, very much in the same way as the inhomogeneous band is narrowed. As a consequence, by comparing the pressure broadening sM of the monomer with the respective sA of the aggregate, we directly get the number Nc of molecules within the coherence length [4–6, 11], sM sA
r 2
Nc 1 : 3
1
The advantage of a hole burning experiment is that the change in line width under pressure can be measured much more accurately than the inhomogeneous bands. No correc252
12.3 Experimental tion due to underlying phonons, or vibrations, or bandshape asymmetries is necessary. Hole burning in J aggregates was demonstrated first by de Boer et al. [18]. As to the influence of an electric field the situation is somewhat different: the monomer has a dipole moment which gives rise to a first order Stark effect. The result is a splitting and a broadening of the hole. In addition, the local environment determined by the host glass may induce a dipole moment which can make a significant contribution to the field broadening as well [19–21]. In the aggregate, a variety of things can happen: the structure of the chain may have an inversion center. As a consequence, the broadening will be reduced. In addition, similar to the pressure phenomena, the field broadening due to the environmentally induced dipole moments may be motionally narrowed. In this case motional narrowing can be interpreted in the sense that the influence of the matrix fields averages to zero within a scale given by the coherence length.
12.3
Experimental
So far, we performed comparative pressure experiments for two different PIC systems, namely PIC-Cl and PIC-I in an ethyleneglycol-water glass with a mixing volume ratio of 1 : 1. In the following, PIC stands for 1,1'-diethyl-2,2'-cyanine. Hole burning was performed with a ring dye laser pumped by an argon ion laser. The laser band width was about 1 MHz. The respective scan range covered 30 GHz. The holes were detected in transmission. In the J band, typical burning times were of the order of one minute. The respective power levels varied between 0.01 and 0.1 mW. The hole burning efficiency is very low in the monomers. Hence, burning times of the order of 20 minutes were used at power levels of about 100 mW. The aggregate spectra were measured at 4.2 K, whereas the monomer spectra were measured at 1.5 K. The reasons are the following: in the monomer, the holes are subject to thermal line broadening, hence are much broader at 4.2 K than at 1.5 K. This lowers the accuracy. As mentioned above, the photoreactive quantum yield in the monomer is extremely low. Hence, it is much harder to burn a hole at 4 K than at 1.5 K. However, at 1.5 K the pressure range is rather limited because He solidifies at a level of 2.4 MPa. The narrow pressure range at 1.5 K, on the other hand, limits the accuracy for the aggregates. In this case pressure broadening is so low that one needs a larger range to get significant results. Consequently, the respective experiments were performed at 4.2 K. We note that thermal line broadening between 1.5 K and 4.2 K is of no concern for the aggregate. Due to the molecular superradiance, the line width is not affected by phonons. Hence, there is no significant change with temperature in this range [8]. For the Stark experiments on J aggregates holes were burned with the ring dye laser but scanned with a pulsed dye laser system because of its larger scan range. The aggregate solution was dispersed on an indium tin oxide (ITO) coated glass substrate. Maximum field strengths were about 300 kV/cm. 253
12
Photophysics of J Aggregates
The monomers were investigated in a glass cuvette placed between two electrodes. In this case, maximum field strengths were about 10 kV/cm and both, burning and scanning of the holes, were performed with a pulsed dye laser system.
12.4
Results
In Fig. 12.1, we compare the inhomogeneous long wavelength absorption bands of PIC-Cl monomer and aggregate. Figure 12.2 shows for both cases as holes deform under isotropic pressure conditions.
Figure 12.2: Behaviour of spectral holes in the aggregate (left) and monomer (right) under different pressures. Burn-frequencies: 17334 cm –1 for the aggregate, 18727 cm –1 for the monomer. Temperature 4.2 K (aggregate) and 1.5 K (monomer). Lorentzian fit curves are indicated.
Figures 12.3 and 12.4 show the influence of pressure on the line width of monomer and aggregate. In the insets, it is demonstrated for the monomers on an expanded scale that, despite the small pressure range, line broadening is linear with pressure and that the respective slope can be determined with significant accuracy. For the monomer of PIC-Cl we measured (0.18 + 0.02) cm –1/MPa. For PIC-I a value of (0.21 + 0.01) cm –1/MPa was found. We stress that values of the same order of magnitude have been found for a series of similar sized molecules in organic glasses [22–24]. Going from the monomer to the aggregate a dramatic decrease in pressure broadening occurs. We measured (0.020 + 0.001) cm –1/MPa and (0.026 + 0.001) cm –1/MPa for PIC-Cl and PIC-I aggregates, respectively. The respective ratios in the broadening per pressure are (9 + 1) and (8 + 1). Figure 12.5 shows the broadening of a spectral hole for the PIC-I monomer (a) and aggregate (b) in an electric field. For the monomer the field-induced broadening is quite 254
12.4
Results
Figure 12.3: Pressure broadening of spectral holes for PIC-Cl monomer and aggregate. The inset shows the data for the monomer on a smaller scale.
Figure 12.4: Pressure broadening of spectral holes for PIC-I monomer and aggregate. The inset shows the data for the monomer on a smaller scale.
strong (about 1 GHz cm/kV) and linear with the applied Stark field ESt . In the aggregate the field-induced broadening is dramatically reduced. The fitted curve in Fig. 12.5 has a strong quadratic component but there is a linear contribution as well. Comparing the linear regime of the aggregate with the respective one of the monomer, it is obvious that the field-induced broadening is reduced by a factor of about 150.
255
12
Photophysics of J Aggregates
Figure 12.5: Electric field-induced broadening (Stark effect) of spectral holes for pseudoisocyanineiodide monomer (a) and aggregate (b). Note the different scales of the applied Stark field for monomer and aggregate.
12.5
Discussion
12.5.1 Pressure phenomena First, let us address broadening of the holes in the monomer band. The reason for pressure broadening is based on the fact that a large variety of microscopic environments in amorphous host materials can correspond with the same absorption energy of the probe molecule. This degeneracy is partly lifted under pressure and is reflected in pressure broadening. The magnitude of this broadening is determined through the magnitude of the probe solvent interaction and through the similarity of the microscopic probe-lattice interaction configurations with their respective change under pressure [25]. This similarity is expressed in a specific degree of correlation. In a rather perfectly ordered crystal, for instance, this degree of correlation is close to 1 and, correspondingly, the pressure broadening is close to zero as has been observed [26]. In amorphous solids, on the other hand, the respective correlation is low and broadening is rather large. Typical values are of the order of 0.1 to 0.2 cm –1/MPa [22–24]. As is obvious from the discussion above, pressure broadening in solids is inhomogeneous in nature. In the aggregates the wavefunction is delocalized over Nc monomer units. Then, we can consider two cases: a) the microenvironments of the individual monomers in the aggregate are statistically independent. In this case the site energy of molecule n is completely independent from molecule n + 1; b) there is a finite correlation length l0, i. e. within l0 the site energies of the molecules forming the aggregate are correlated to some degree. 256
12.5
Discussion
For case a), with statistically independent microenvironments, it was shown by several authors that the inhomogeneous line width of the monomers forming the aggregate is narq rowed according to Eq. 1, i. e. roughly by a factor N c 1. This narrowing is known as exchange narrowing and it can be interpreted in a way that a fast moving exciton on an aggregate chain averages over the local inhomogeneities. The amount of averaging is determined by the coherence length of the exciton. Hence, the coherence length can be determined from the ratio of the inhomogeneous band widths of aggregate and monomer. The exact determination of these band widths is a problem, because of hidden states and band asymmetries. However, we stressed above that pressure broadening of spectral holes is inhomogeneous in nature. It reflects in a specific way the disorder in the local environments and, hence, should be narrowed by a fast moving exciton in quite the same way as the inhomogeneous band. If so, we can determine the number Nc of monomers within the coherence length from Eq. 1, by comparing the pressure broadening in the monomer with the respective one in the aggregate. We found (120 + 32) for PIC-Cl and (101 + 24) for PIC-I. These numbers may slightly depend on the solvent and freezing procedure. Generally speaking they fit to what is known of J aggregates. For case b), with correlated site energies of the monomers in the chain, the situation becomes quite different. Equation 1 has to be substituted by sM sA
s 1 b 2
Nc 1 : 1b 3
2
Equation 2 is approximative and holds only for sufficiently small b values. Again, b is a degree of correlation between the site energies of the monomers in the chain. It can be written as [27] b exp
1 l0
3
with l0 being the correlation length, i. e. beyond l0 the site energies of the two monomers are statistically independent. As b goes to zero, we have case a) discussed above. As b goes to unity, sA approaches sM and there is no motional narrowing at all. This latter case cannot be directly derived from Eq. 2, since b = 1 is beyond the respective validity range. The main point is that, if there were correlation in the site energies, it would not be possible any more to determine Nc from a linear optical experiment alone because Eq. 2 would contain two unknown quantities. There are indications from non-linear experiments that l0 is finite [28]. Another way to estimate b is via the ratio of the radiative rates of monomer and aggregate. This ratio is also determined by the number Nc of coherently coupled molecules [7, 29] but is not influenced by correlation effects in the site energies, gA Nc gM ;
4
where gA is the radiative rate of the aggregate and gM the radiative rate of the monomer. In an earlier paper we measured the homogenous line width down to 0.300 K [8]. The mea257
12
Photophysics of J Aggregates
sured value corresponded with a lifetime of about 30 ps. Since the fluorescence quantum yield in the J band is supposed to be close to unity, the aggregate lifetime is governed by radiative processes. With g–1 M = 3.7 ns [30], we get from Eq. 4 an Nc value of about 123 which is close to what we found from our pressure tuning experiments. Hence, according to this estimation, b should be close to zero. We stress however, that the results for Nc in the literature, as obtained with various techniques, are subject to large variations. Clearly, more work must be done to achieve unambiguous results.
12.5.2 Electric field-induced phenomena The electric field-induced phenomena in the J band are much more complex than the pressure phenomena and not yet understood. Firstly we note that the reduction of the linear component in the Stark effect is in agreement with what we expected from motional narrowing effects. What was not expected at all, however, is the order of magnitude which is more than a factor of 10 larger as compared to the pressure experiments. We conclude that additional mechanisms must play a role. One such mechanism is the loss of the permanent dipole moment upon aggregate formation. The absence of a dipole moment has severe structural implications which are not clear at the present time. Our result is also at variance with experiments by Misawa et al. [31], who reported huge changes in the static dipole moment of J aggregates, which were attributed to their enhanced size. However, since these experiments were performed with spin-coated J aggregates, it may possibly be that geometric effects play an important role.
Acknowledgements
The authors thank the Deutsche Forschungsgemeinschaft (Sonderforschungsbereich 213, B15) and the Fonds der Chemischen Industrie for financial support.
258
References
References
1. G. Scheibe: Kolloidzeitschrift, 82, 1 (1938) 2. E. E. Jelley: Nature, 139, 631 (1937) 3. E. Daltrozzo, G. Scheibe, K. Gschwind, F. Haimerl: Photographic Science and Engineering, 18, 441 (1974) 4. E. W. Knapp: Chem. Phys., 85, 73 (1984) 5. E. W. Knapp: P. O. J. Scherer, S. F. Fischer, Chem. Phys. Lett., 111, 481 (1984) 6. F. C. Spano, S. Mukamel: Phys. Rev. A, 40, 5782 (1989) 7. H. Fidder, J. Knoester, D. A. Wiersma, Chem. Phys. Lett., 171, 529 (1990) 8. R. Hirschmann, J. Friedrich: J. Chem. Phys., 91, 7988 (1989) 9. S. Kobayashi, F. Sasaki: Nonlinear Optics, 4, 305 (1993) 10. F. C. Spano, S. Mukamel: Phys. Rev. Lett., 66, 1197 (1991) 11. F. C. Spano, S. Mukamel: J. Chem. Phys., 91, 683 (1989) 12. F. C. Spano, J. Knoester: in: Advances in Magnetic and Optical Resonance, Vol. 18, 117 (1994) 13. B. Kopainsky, W. Kaiser: Chem. Phys. Lett., 88, 357 (1982) 14. R. Hirschmann, W. Köhler, J. Friedrich, E. Daltrozzo: Chem. Phys. Lett., 151, 60 (1988) 15. H. Fidder, D. A. Wiersma: Phys. Rev. Lett., 66, 1501 (1991) 16. H. Fidder: Thesis, University of Groningen, (1993) 17. R. Hirschmann, J. Friedrich: JOSA B, 9, 813 (1992) 18. S. de Boer, K. J. Vink, D. A. Wiersma: Chem. Phys. Lett., 137, 91 (1987) 19. M. Maier: Appl. Phys. B, 41, 73 (1986) 20. A. J. Meixner, A. Renn, S. E. Bucher, U. P. Wild: J. Phys. Chem., 90, 6777 (1986) 21. L. Kador, D. Haarer, R. Personov: J. Chem. Phys., 86, 5300 (1987) 22. G. Gradl, J. Zollfrank, W. Breinl, J. Friedrich: J. Chem. Phys., 94, 7619 (1991) 23. J. Zollfrank, J. Friedrich: J. Phys. Chem., 96, 7887 (1992) 24. H. Pschierer, J. Friedrich, H. Falk, W. Schmitzberger: J. Phys. Chem., 97, 6902 (1993) 25. B. B. Laird, J. L. Skinner: J. Chem. Phys., 90, 3274 (1989) 26. P. Schellenberg, J. Friedrich, J. Kikas: J. Chem. Phys., 100, 5501 (1994) 27. J. Knoester: J. Chem. Phys., 99, 8466 (1993) J. Knoester: J. Lumin., 58, 107 (1994) 28. J. R. Durrant, J. Knoester, D. A. Wiersma: Chem. Phys. Lett., 222, 450 (1994) 29. J. Grad, G. Hernandez, S. Mukamel: Phys. Rev. A, 37, 3835 (1988) 30. H.-P. Dorn, A. Müller: Chem. Phys. Lett., 130, 426 (1986) 31. K. Misawa, K. Minoshima, H. Ono, T. Kobayashi: Chem. Phys. Lett., 220, 251 (1994)
259
13
Convection Instabilities in Nematic Liquid Crystals Lorenz Kramer and Werner Pesch
13.1
Introduction
Pattern formation in hydrodynamic instabilities has been studied intensely over the last decades [1, 2]. Although Rayleigh-Bénard convection (RBC) in simple fluids has been the prime example [3], the rich variety of scenarios found in nematic liquid crystals (LCs) has attracted increased attention. LCs are materials made up of highly anisotropic organic molecules in a phase that reflects this anisotropy. The class of nematic LCs (nematics) are fully liquid without longrange translational but with long-range uniaxial orientational ordering of the molecules. As ^ with a result of the coupling of the molecular alignment axis (described by the director n) the (mass) flow electric or thermal current the hydrodynamic equations involve numerous non-linearities (Section 13.2), which easily lead to instabilities when a state of non-equilibrium is maintained [4]. Convective flow can be driven electrically through space charges that naturally arise in an anisotropic conductor in the presence of spatial variations, the electrohydrodynamic convection (EHC), or thermally through buoyancy forces, the Rayleigh-Bénard convection (RBC). EHC has attracted more attention and will play a major role in this review. The study of EHC by Williams [5], Kapustin and Larinova [6] in 1963 initiated extensive experimental work in the typical thin-layer geometry shown in Fig. 13.1 a. The nematic is sandwiched between glass plates (separation d * 10–100 µm) with transparent electrodes. The surfaces are treated to provide (ideally) uniform anchoring of the director, in most cases along the x direction (planar or homogeneous alignment) but sometimes also in the z direction (homeotropic alignment) 1. Above an applied voltage Vc > 10 V (typically low-frequency ac) convection rolls appear with associated director distortions, which are easily detected optically. The spacing of the rolls is of order d except in the higher frequency dielectric range. Figure 13.1 b shows a typical pattern with the rolls in the y direction normal to the undistorted director.
1 We will concentrate here on the planar case. Some remarks about other alignments are made in Section 13.5.
260
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties Deutsche Forschungsgemeinschaft (DFG) Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
13.1 Introduction
v ^ E || z ^ z
^ z ^ y
rn
Directo
^ x
Figure 13.1: a) Cell geometry with section of a roll pattern for EHC (planar configuration). E = electric field, v = velocity; b) Normal roll pattern for EHC with a dislocation.
The mechanism for instability in EHC based mainly on space charges generated by preferential conduction along n^ (charge focusing, see Section 13.2) was suggested by Carr [7] and then incorporated into a first one-dimensional model by Helfrich [8]. Subsequently the linear theory, giving the onset of the instability and in principle the pattern up to degeneracy, was generalized to include the common case of ac driving [9, 10], a rigorous two-dimensional analysis [11], and finally a full three-dimensional treatment, signalizing the beginning of a renewed interest in the subject. For references see e. g. [12], for a comparison between the experimental and theoretical threshold see Fig. 13.2 a. Unfortunately already the linear theory is a numerical problem, but useful analytic approximations are possible [12–15]. The need for the full three-dimensional theory became particularly evident from the experimental work of Ribotta and co-workers [16, 17], where three-dimensional structures (oblique rolls) were observed at threshold, and from systematic measurements by Kai and co-workers [18, 19] under well-defined conditions. Figure 13.2 b shows the oblique rolls,
Threshold Voltage [V]
100
80
60
40
20
0
0
10
20 30 Driving Frequency f [Hz]
40
Figure 13.2: a) Threshold curve for EHC as function of frequency. Experimental points from [20]; b) Zigzag pattern after increasing the voltage in Fig. 13.1 b.
261
13
Convection Instabilities in Nematic Liquid Crystals
which in this case bifurcated from normal rolls after an increase of V. Note the appearance of grain boundaries separating domains with the two symmetry degenerate directions (zig and zag). At threshold the boundary would become less sharp. As the theory progressed expanding into a weakly non-linear analysis providing the Ginzburg-Landau amplitude equation description [12, 14] and eventually also including mean flow effects that allow to capture the transition from ordered periodic to weakly turbulent patterns [21–24] there was also a revival of experimental activity, For a phenomenological treatment, see Refs. [25, 26]. Some milestones were the identification of the Eckhaus instability (Section 13.3) that gives limits of the stable wave number band [27–29], the characterization of the instability that may lead to pattern turbulence [36, 37] (Fig. 13.3 b), and the structure and dynamics of single dislocations and their interaction [30–33]. For a comparison of the experiments [32, 34] with the results obtained from the Ginzburg-Landau theory [35] see Fig. 13.3 a. 8 7
exp. exp. ξ1, ξ2, τ
6
theo. ξ1, ξ2, τ
5 4 3 2 1
0 -1 -0.01
0.01
0.00
0.02
0.03
0.04
∆q [µm-1]
Figure 13.3: a) Climb velocity of a single defect vs. wave number mismatch. Parameters of the GLE are either from experiments [32] (solid curve) or from hydrodynamical calculations [12] (broken line); b) Snapshot of a defect turbulent pattern.
Further highlights were the identification of thermal noise slightly below threshold by Rehberg et al. [38–40] and finally the clear identification of a Hopf bifurcation leading to travelling rolls or waves in sufficiently thin layers, below about 50 µm and clean material (low conductivity) by Refs. [18, 41–43]. It is not possible to explain the Hopf bifurcation within the conventional theoretical framework the standard model (SM), see Section 13.2, where the LC is treated as an anisotropic ohmic conductor. Indeed some of the theoretical effort, in particular the inclusion of the rather complicated flexoelectric terms [14, 44–46], was aimed primarily at resolving this problem. The situation is further complicated by the fact that the bifurcation is often observed to be slightly subcritical, i. e. with a very small hysteresis [38–40, 47], whereas the theory predicts a supercritical bifurcation. Very recently an extension of the standard theory, the weak-electrolyte model (WEM), has been worked out where electric transport in the nematic is described in terms of two mobile ion species of opposite charge which are coupled via a slow dissociation-recombination reaction and whose densities are treated as dynamic variables [48]. Some results of the 262
13.1 Introduction WEM will be discussed in Section 13.5 together with experiments in the material I52 [49]. Whether the WEM can also capture the subcritical bifurcation remains unclear. In almost all the measurements the standard reference material 4-methoxybenzylidene-4'-n-butyl-aniline (MBBA) or a mixture, Merck Phase V, have been used, sometimes doped with an ionic substance. MBBA is the only room-temperature nematic with dielectric anisotropy ea < 0 where all the material parameters have been measured. For tabulated values see e. g. Ref. [12]. Unfortunately, it is a Schiff base and rather unstable when exposed to moisture. Therefore, it is difficult to control the long-time conductivity in situ. Thus the recent successful introduction of the very stable material 4-ethyl-2-fluoro-4'-[2-(trans-4-npentylcyclohexyl)-ethyl]-biphenyl (I52) doped with iodine is very promising [50, 51]. This material exhibits at low external frequencies strongly oblique travelling rolls which bifurcate supercritically, leading to a particularly interesting scenario. 2 In RBC the traditional instability mechanism by buoyancy forces is enhanced considerably by preferential conduction of heat parallel to the director leading to a heat focusing effect, as was first shown by Dubois-Violette in 1971 [52]. After some amount of experimental and theoretical work, for a summary see Ref. [53], Feng et al. [54] recently presented a fully three-dimensional treatment including the weakly non-linear analysis. Subsequent experiments in thick layers (some millimetres) with an additional stabilizing magnetic field Hx in the x direction (Fig. 13.1 a) have substantiated several predictions [55]. In Fig.13.3 a a zigzag pattern is shown which arises with increasing magnetic field after normal rolls (n is horizontal). Further increase of Hx leads to a larger angle of obliqueness with superposition of the two roll systems (Fig. 13.3 b) and eventually to rolls which are parallel to n. The very common nematic 4-n-pentyl-4'-cyanobiphenyl (5CB) was used, whose relevant material parameters have all been measured. 3
Figure 13.4: a) Oblique roll pattern for RBC in the planar configuration [55]; b) Superposition of zig and zag for RBC [55]. 2 In the note added at the end of this Chapter some very recent developments in EHC have been summarized. 3 Some very recent progress is summerized at the end of Section 13.4.
263
13
Convection Instabilities in Nematic Liquid Crystals
In this review we will address mostly the developments of the past twelve years, characterized by substantial progress in qualitative as well as quantitative understanding of the various instabilities. After introducing and explaining the basic equations (Section 13.2) the theoretical concepts of the linear and weakly non-linear analysis will be presented (Section 13.3). In Section 13.4 and Section 13.5 the results for the two systems, introduced above, are discussed in particular in the light of recent experiments. Finally, we will list topics which are omitted due to space limitations and we comment on some perspectives for future work (Section 13.6). For a classical review of convective instabilities in LCs see Ref. [56] and for EHC one may consult the books of Blinov [57], Pikin [58], and recent review articles [14, 15, 59–63]. RBC has been reviewed by Barrat [53] and very recently by Ahlers [64].
13.2
Basic equations and instability mechanisms
13.2.1 The director equation The macroscopic nematodynamic equations describe the dynamics of the slowly relaxing variables, which usually are either connected with conservation laws or with the Goldstone modes of the spontaneously broken symmetries. To formulate them we will follow the traditional approach [65–67] rather than the one based more directly on the principles of hydrodynamics and irreversible thermodynamics [68]. In the nematic state isotropy is spontaneously broken and the averaged molecular alignment singles out an axis whose orientation ^ i. e. an object that has the properties of a unit vector with n^ = – n. ^ defines the director n, The static properties are conveniently expressed in terms of a free energy density whose orientational elastic part is given by [69] felast
1 1 1 ^ 2 k22
n^
r n ^ 2 k33
n^
r n ^ 2: k11
r n 2 2 2
1
The elastic constants k11, k22, and k33 pertain to the three basic deformations splay, twist, and bend, respectively. For typical nematics with prolate molecules one has k33 > k11 > k22 and kii * 10–11 N. Electric and magnetic fields E and H exert torques on n^ that can be derived from an additional contribution to the free energy fem
1 m w
n^ H2 2 0 a
1 e0 ea
n^ E2 2
E Pf lexo :
2
Here wa = wk – wk and ea = ek – ek are the relative diamagnetic and dielectric anisotropies, so that the uniaxial susceptibility and dielectric tensors can be written in Cartesian coordinates in the form 264
13.2
Basic equations and instability mechanisms
wij w? dij wa ni nj ;
eij e? ij ea ni nj
3
and the flexoelectric polarization is [70] ^ n ^ e3
n^ r n^ : Pf lexo e1 n
r
4
R Static director configurations are obtained by minimizing the total free energy F = dV ( felast + fem) under the condition nj nj = 1 with suitable boundary conditions, which results in equating to zero the torque density G on the director. The generalization to dynamic situations is usually done by defining a vector S Srate Scons Sdiss
5
and requiring the balance of torques [65, 71, 72] n^ S 0 :
6
S has the form Srate
g1 N;
Scons
dF=dn^ ;
Sdissji
g2 Aij nj ;
^ N dn=dt
Aij
1 2
1
r v n^ ; 2
uvi uvj uxj uxi
7
8
:
9
Here d/dt = ut + v 7 r denotes the usual material derivative and in N the rigid rotation part x = 12
r v of the fluid has been subtracted out. A notation like uxj (ni) = uj ni = ni,j is used freely. g1 and g2 are called rotational viscosities, which couple the flow field v to the director. To confirm that Eq. 6 is indeed only a rate equation for the director one may take the vector product of Eq. 6 with n^ leading to ^ g1 dn=dt
1=2
r v n^
1
n^ n^ T
Scons Sdiss :
10
The typical relaxation time of the director in the thin-layer geometry of EHC is easily seen to be td g1 d 2 =
p2 k11 ;
11
where we chose the splay elastic constant as a representative; obviously g1 > 0 has to hold. td is typically in the order of 1 s. It can be seen that Eq. 6 and Eq. 10 have only two independent components which can be made explicit by transforming into one of the two local coordinate systems ^ x^ n; ^ n^
^x n ^ ; n;
12 265
13
Convection Instabilities in Nematic Liquid Crystals
^ z^ n; ^ n^
^z n ^ : n;
13
The first one becomes singular when n^ is parallel to x^ and is thus not suitable for planar alignment. Similarly, the other one is not practical for homeotropic alignment.
13.2.2 The velocity field The generalized Navier-Stokes equation for the velocity field v follows from momentum balance rm
dv f r T; dt
14
where rm is the mass density, f the bulk force to be discussed below, and T the stress tensor with components Tij
p dij
uF nk;j tij ; unk;i E0
15
and p is pressure. The viscous stress tensor for an incompressible nematic contains the six Leslie shear viscosity coefficients ai [72], tij a1 nk nm Akm ni nj a2 ni Nj a3 nj Ni a4 Aij a5 ni nk Akj a6 nj nk Aki :
16
It is instructive to consider the three simple geometries for plane parallel shear flow, the Miesovicz geometries [73], corresponding to the orientations of the director relative to the flow axis and the shear gradient. Choosing v = u (z) x^ one has the effective shear viscosities 4 1) for n^ = z^ along the shear gradient ? Z1 = (a4 + a5 – a2)/2, Gy = a2 uz ; 2) for n^ = x^ along the flow axis ? Z2 = (a3 + a4 – a6)/2, Gy = a3 uz ; 3) for n^ = y^ along the shear gradient ? Z3 = a4 /2, G = 0. The value of the non-vanishing component of the torque G on the director is also given. Another positive effective viscosity is Z0 = a1 + a4 + a5 + a6 . All shear viscosities are typical of order 10 –1kg m –1 s–1. From the Onsager reciprocity relations one finds [74] a6
a5 a2 a3
17
4 Unfortunately the definitions of Z1 and Z2 are not universally accepted. Our definition is adopted by Blinov [57] whereas the definitions in the review articles [67] and [73] are in the reverse order.
266
13.2
Basic equations and instability mechanisms
and also g1 a3
a2 ;
g2 a3 a2 :
18
Note that the flow-alignment parameter l = –g2 /g1 is a reversible quantity. For l > 1 the director can align in the flow plane at a fixed angle b with respect to the velocity (xaxis), where tan2b = (l–1)/(l + 1) = a3 /a2 . For l < 1 there is tumbling [73]. The pressure is to be determined from the incompressibility condition r v 0:
19
An efficient way to implement this relation is to represent the velocity field in terms of a toroidal and a poloidal potential g and f, respectively [75] v
x; y; z r z^ g r
r z^ f eg df
20
with eT
@y ; @x ; 0 ;
dT
@2xy ; @2yz ; @2xx
@2yy ;
21
where z corresponds to the coordinate perpendicular to the plane of the layer. Applying e and d on Eq. 14 gives two equations for g and f eliminating the pressure. The typical relaxation time for a velocity field in a thin-layer geometry is tvisc = rm d2/(p2 Z1). This time is usually of the order of 10 –5 s, much shorter than the others. Thus the velocity field can usually be treated adiabatically (neglect of inertial terms).
13.2.3 Electroconvection 13.2.3.1 The standard model Now we come to the additional equations that are specific to the processes driving the instability. In EHC the bulk force in the Navier-Stokes Equation 14 is derived from the Maxwell stress tensor, which here reduces to f rel E
P r E;
PD
E:
22
The equation determining the charge density rel is obtained from charge conservation and the Poisson law drel r jel 0; dt
jel s E ; (23)
rel r D;
D e E Pf lexo : 267
13
Convection Instabilities in Nematic Liquid Crystals
In the SM the usual assumption of an anisotropic but fixed ohmic conductivity is made. The conductivity tensor r has the same form as the other material tensors with sa = sk – sk, see Eq. 3. Equations 23 are easily seen to lead to charge relaxation with the time scale tq e0 e? =s?
24
which is typically of the order of 10 –3 s. Sometimes a dopant is added to the LC to obtain sufficient or well controlled conductivity (not much is needed). Now we are in a position to discuss the basic driving mechanism for EHC. The important point is that in almost all nematics sa is substantially positive, typically sa /sk& 0.3–1. Choosing materials with negative or only slightly positive dielectric anisotropy ea , here the materials show great diversity, one easily sees that in the presence of an applied field E0 and with a small spatial variation (fluctuation) of the director n a space charge rel results. For rel = 0 there is no solution of Eqs. 23. Roughly speaking, the charges are focused at locations where the director bends. The bulk force in the Navier-Stokes equation may then overcome viscous stresses and drive a velocity field v. Via the viscous coupling this may enhance the spatial variation of the director and thus generate an instability. The threshold voltage is for low frequency and for materials with not too large dielectric anisotropy of order V2c & p2k11 /(satq) and the introduction of the reduced control parameter R = V2 (satq)/(p2k11) is often useful.
13.2.3.2 The weak electrolyte model The SM fails in particular to describe the Hopf bifurcation leading to travelling rolls, which is observed quite frequently. For a Hopf bifurcation two processes that compete on a comparable time scale are necessary. In our case, however, the director relaxation is much slower than the other processes and thus determines the dynamics. Charge relaxation can usually also be treated adiabatically. In addition, director relaxation and charge relaxation do not compete but rather support each other usually, which also excludes a Hopf bifurcation. Thus another slow process appears to be operating. A recently proposed model assumes that this process is a relaxation of the mobile ion densities n+ and n– on the time scale trec, which may result from a dissociation-recombination reaction. One then gets if singlecharged ions are assumed rel e
n
n ;
sij s s0ij ;
25
where s e
m ? n m? n ;
s0ij dij
sa ni nj ; s?
26
+ Here m+ k, mk are the ionic mobilities perpendicular and parallel to the director, respectively. For simplicity the anisotropies were assumed to be the same for both types of + ions so that sa /sk = m+ k /mk. So now s is an additional variable. From the balance equations + – for n and n one easily recovers Eqs. 23 which now read
268
13.3
Theoretical analysis
ds r m ? m? s m? m? rel dt " # s m? rel s m? rel seq 1 1 s trec 2trec s2eq
m seq ?
2
m?
rel ;
27
+ – + mk ) neq contains the equilibrium ion density neq . The last expression is where seq = e (mk obtained by linearization in the quantities n+ – neq and n– – neq . Thus, in this model ion accumulation effects are included whereas ionic diffusion is neglected as in the SM. The charge accumulation counteracts the standard (Helfrich) mechanism of generation of space charges. If trec is sufficiently slow one can find an oscillatory behaviour of the system at threshold, i. e. a Hopf bifurcation (Section 13.5).
13.2.4 Rayleigh-Bénard convection In RBC the bulk force in the Navier-Stokes equation is f = r g. In the spirit of the Boussinesq approximation one has for the mass density r = rm [1 – a (T – T0)], where g is the gravitational acceleration and a the thermal expansion coefficient. One needs in addition the heat conduction equation dT r jT 0; dt
jT krT ;
28
with kij = kkdij + ka ni nj, ka = k|| – kk (k * 10 –7 m2 s–1). As first pointed out by DuboisViolette [52, 76] the conventional instability mechanism operative in isotropic fluids is here enhanced considerably by preferential conduction of heat parallel to the director (ka > 0) leading to a focusing effect in the presence of out-of-plane fluctuations of the director. For anisotropies of order one the reduction of the threshold is of order F = td /ttherm & 103, where ttherm = d2/(p2 kk) is the vertical thermal diffusion time. Other important dimensionless quantities besides F are the Prandtl number Pr = ttherm /tvisc & 103 and the Rayleigh number R = a g tvisc /ttherm (T/d), which is the traditional control parameter.
13.3
Theoretical analysis
We here present the general methods used to extract the relevant information from the basic equations. It is convenient to introduce the notation V = (n,v, …) for the collection of all field variables involved in the specific problem. We choose the fields in such a way that 269
13
Convection Instabilities in Nematic Liquid Crystals
V = 0 corresponds to the non-convecting basic and primary state. Then the set of macroscopic equations, as presented in the previous Section, can be written in the following symbolic form: LV N2
VjV N3
VjVjV
B0 B1
V B2
VjV
uV : ut
29
The vector operators N2, N3 … denote quadratic, cubic … operators in V and its spatial derivatives, whereas the operators L and Bi represent matrix differential operators of the indicated order on V. Computer algebra can be used to perform the expansion. A direct simulation of the coupled system of the partial differential equations, Eq. 29, with appropriate boundary conditions (v = 0, n prescribed at the confining plates, etc.) is at the limits of the supercomputers of today. It will turn out that in the liquid crystal systems a rich scenario of patterns, including spatio temporal chaos, develops already near threshold so that perturbational calculations are useful. The onset of the instability is obtained from a standard linear stability analysis of the basic (primary) state. The problem can be diagonalized with respect to horizontal coordinates x = (x, y) associated with the directions of idealized infinite extent by a Fourier transform, V
x; z; t
R
d2 qe
iq x
Uq
z el
qt ;
with q = (qx, qy). From Eq. 29 with boundary conditions at z = ± d/2 one arrives at an eigenvalue problem, lB0
iq; uz ; RUq
z L
iq; uz ; RUq
z ;
30
where R = R, S, … are the control parameters of the system. 5 In the typical scenario the real part s of one of the eigenvalues l (q, R) = s (q, R) ± io (q, R) crosses zero upon increase of the main control parameter R at fixed q beyond a value R0 (q) while the real parts of all other eigenvalues remain negative. Thus the neutral surface R = R0 (q), which separates the unstable (R > R0 (q)) from the stable (R < R0 (q)) modes, is given by the condition of vanishing growth rate s(q, R) = 0. Minimizing R0 (q) with respect to q gives the threshold Rc = R0 (qc) with the critical wave-vector qc = (qc, pc) and the critical frequency oc = o (qc), which is zero for a stationary bifurcation, which is the more common case, while non-zero for a Hopf (oscillatory) bifurcation. We will encounter situations where different minima of R0 (q) coincide. This multicritical behaviour is either accidentally or caused by some symmetry and calls for a special treatment. In isotropic systems qc is even continuously degenerate on a circle. In planarly aligned nematics, i. e. in an axially anisotropic system, one can distinguish two fundamental cases.
5 We will treat R as the main control parameter whose increase carries the system across the instability, i. e. the squared voltage in the case of EHC and DT in the case of RBC (both non-dimensionalized).
270
13.3
Theoretical analysis
a) If qc is parallel to one of the symmetry directions, pc = 0 or qc = 0, one speaks of normal (Fig. 13.1 b) or parallel rolls, respectively. b) If qc is at an oblique angle, one speaks of oblique rolls (Figs. 13.2 b and 13.4 a). Clearly, we get the two symmetry-degenerate directions zig and zag, which may superpose to give rectangles (Fig. 13.4 b). In the case of a Hopf bifurcation one has degeneracy between travelling waves in opposite directions, which may also superpose to give standing waves. Oblique rolls arising via a Hopf bifurcation give four degenerate modes. In EHC, for the usual case of driving with a pure ac field of frequency o = 2/p f, the eigenvector Uq of Eq. 30 inherits the additional periodic time dependence and the eigenvalue l becomes a Floquet coefficient. Then there is an additional discrete symmetry (z, t) ? (– z, t + 1/(2f )) and each component of Uq has a definite parity. Generally the conductive mode (for even parity the out-of-plane component of director and vz ; for odd parity the in-plane components of velocity) destabilizes first at low frequencies f. For materials with negative dielectric anisotropy, ea < 0, there exists a cut-off frequency fc so that the dielectric mode with the other parity destabilizes first for f > fc, where fc ? ? for ea ? 0. The existence of these two regimes was first pointed out by Orsay’s group [9, 10]. For further details see Refs. [14, 61]. The linear problem (Eq. 30) with realistic boundary conditions has to be solved numerically, often by treating the z-dependence by a Galerkin expansion with a suitable cutoff. In particular for stationary bifurcations a cut-off at lowest non-trivial order, i. e. one trial function for each component of Uq gives approximate analytic expressions for the neutral surface and the growth rate, from which the interplay of the many material parameters becomes transparent [12–15, 48, 54, 77]. The basic idea of the weakly non-linear analysis [1, 78–80] in its rather general form [54, 81, 82] is to reduce the phase-space dimension of the system by choosing an appropriate basis of states, characterized as the dynamically active ones [80]. We demonstrate the method only for stationary bifurcations, where the relevant time scale becomes slow near threshold but a generalization to Hopf bifurcations is straightforward. One expands V in Eq. 29 at lowest order as a wave packet of the eigenmodes Uq of Eq. 30 at the neutral surface R0 (q) and with the growth rate s = 0, V V1
R D
dq A
qUq
zeiqx c:c: ;
31
where A (q) denotes the amplitude order parameter, which vanishes at threshold and c.c. is the complex conjugate of the preceeding expression. The integration domain consists of small areas D± centred at ±qc which need not be specified precisely at this point. We will arrive at an order parameter equation for A (q) by expanding Eq. 29 up to order A3. In an intermediate step, one needs the contributions V2 * A2, generated by quadratic interaction from V1. They are determined from the relation LV2+ N2 (V1|V1) = 0 derived from Eq. 29. Here terms from the right-hand side can be neglected, since they contain in addition a (slow) time derivative. The solution V2 contains separate contributions with wave-vectors near ±2qc and 0. In the sector near q = 0 the so called mean flow or mean drift can be isolated systematically [24, 83]. The mean flow arises from long wavelength variations – on a scale much larger 271
13
Convection Instabilities in Nematic Liquid Crystals
than the spacing of the rolls – along the roll axis (undulations) leading to a lateral pressure gradient, whose spatial average across the cell is non-zero [1, 2]. A second amplitude B is introduced to describe the resulting Hagen-Poisseulle-like shear flow, also characterized by the non-zero vertical vorticity (curl v)z . The equations can be closed at order A3 by inserting V2 in Eq. 29 and projecting onto the subspace spanned by the linear modes V1. One arrives at two coupled integral equations for the amplitudes A (q) and B (s): a1
Z Z dA
q; t a2 A
q dq0 dq00 a3 A
q0 A
q00 A
q dt R dsb1 B
sA
q s ;
R c1 s2x c2 s2y B
s dqb2 A
qA
s
q0
q00
32
q :
33
The q-integrations are confined to the regions D = D+ & D– and s is near zero. The coefficient functions ai (q), bi (q, s) and the constants ci depend on the material parameters. They involve z-integrations and have to be calculated numerically. The above procedure guaranties that all coefficients are smooth functions of the wavevectors. Since the field B satisfies an anisotropic Poisson equation, transformed to Fourier space, its long-range character is evident. In nematics its effect turns out to enhance transverse modulations in distinct contrast to isotropic fluids in most cases. As will become clear below, the mean flow contributions can be neglected in the immediate vicinity of threshold. Stationary periodic roll solutions with wave-vector q0 can be calculated by the ansatz Ar (q) = cr d (q – q0) + c*r d (q + q0). Then the double integral on the right hand side of (Eq. 32) becomes trivial and one finds that B : 0. One also easily sees that |cr|2 is proportional to (R – R0 (q0))/R0 (q0), the reduced distance from the neutral surface. A subcritical bifurcation is signalized by a negative proportionality factor. Then higher powers in A would have to be included. The stability analysis of the periodic roll solution [54] is performed by introducing a small perturbation dA (q, t) of the amplitude Ar with a modulation wave-vector s in Eqs. 32 and 33, dA
q; t
c1
q; sd
q
q0
s c2
q; sd
q q0
s eLt ;
34
corresponding to a perturbation dV1 in Eq. 31. Roll solutions with wave-vector q0 are unstable if the maximum of Re (L (q0, s)) with respect to s is positive. Well-known long wavelength destabilization mechanisms involve: a) local dilation and compression of the roll pattern, the Eckhaus process (E) [84], i. e. s || q0, b) undulations along the roll axis, the zigzag process (ZZ) [86], i. e. skq0, c) combinations of both processes, the skewed varicose process (SV) [86]. In the last case mean flow is decisive. In addition one has short wavelength instabilities where |s| is of the order of qc. They are well-known in systems that are isotropic in the plane, where they can lead from rolls to squares, hexagons or bimodal structures. A different 272
13.3
Theoretical analysis
version is found in our anisotropic systems (planar director alignment) when oblique rolls become unstable with respect to a superposition of zig and zag (Fig. 13.4 b). Many features become more transparent when formulated in real (position) space in terms of amplitude (envelope) or Ginzburg-Landau equations (GLE). Then one sees that the important information is really condensed in a few parameters and the universal aspects of the systems become apparent. By model calculations, which can often be performed analytically, stability boundaries and secondary bifurcation scenarios are traced out. The real space formulation is essential when it comes to the description of more complex spatio-temporal patterns with disorder and defects, which have been studied extensively in EHC slightly above threshold (Figs. 13.1 b, 13.3 b). One introduces a modulation amplitude A(x) defined as A
x
R D
dqA
qei
q
qc x
:
35
Near threshold one expects that only a small region D+ (i. e. |q-qc| P qc) is relevant in Eq. 35 and that correspondingly the amplitude A(x) varies on a slow scale. The various coefficients of the order parameter equations (Eqs. 32 and 33) can now be expanded into Taylor series around qc with respect to q and around zero with respect to s. Since all non-analyticities have been absorbed in the amplitude B the expansion is smooth and powers of the components of (q-qc) and of s can obviously be identified with spatial derivatives of A(x) and B(x), which is constructed in analogy to Eq. 35. If the q-dependence is taken into account only in the linear coefficient a2 of Eq. 32 – all other coefficients taken at q = qc and s = 0, where b1 = b2 = 0 – one ends up with the famous, slightly generalized Ginzburg Landau equation [87] ut A l
qc
ir; eA
gjAj2 A ;
36
where l is the linear growth rate (Eq. 30). Clearly l is zero at threshold e = (R – Rc)/Rc = 0 and r = 0 and should be expanded in both arguments. For our purposes at lowest order it is sufficient to keep the following terms l
qc
ir; e e x21 u2x 2ax1 x2 ux uy x22 Wu2y
iZx1 x22 ux u2y
x42 u4y :
37
The reason for keeping higher powers in uy than in ux will become clear shortly. On this level the expansion can be cast into an overall expansion scheme in terms of e1/2, or equivalently A. In the anisotropic case, where there is no continuous degeneracy of the critical mode(s), one may in general assume e * A2* ut * u2x * u2y, so that the higher order terms *ux u2y and u4y drop out. Then Eq. 36 becomes uniformly of order e3/2. The remaining mixed derivative term in Eq. 37, that vanishes anyway for normal as well as for parallel rolls, can always be transformed away by rotating the coordinate system. Moreover, by rescaling x and y the differential operator becomes proportional to the Laplacian, so that Eq. 36 finally reduces to tut A eA x2 DA
gjAj2 A :
38 273
13
Convection Instabilities in Nematic Liquid Crystals
This constitutes the simplest GLE [88, 89]. Until now we have not specified explicitly the new spatial scaling. By appropriate scaling the time, length, and A all parameters can be scaled away. Note that the conventional way to derive the GLE starts from a multi-scale analysis in space and time, expanding systematically in powers of e1/2. Going back to Eqs. 36 and 37 with a = 0 it is easy to see that changing W from positive to negative – by changing some secondary control parameter like the frequency in EHC – describes a normal pitch fork bifurcation from normal to oblique rolls. 6 For W < 0 the maximum growth rate of plane wave solutions of Eq. 36 occurs at wave-vectors with nonzero y component. Details of this transition, which is the analogue of an LP in the theory of equilibrium phase transitions, have been discussed elsewhere [12, 21, 89, 90]. The corresponding uniform scaling *e3/2 as in Eq. 38 is recovered with W * e1/2 * ux and now uy * e1/4. This corresponds to the scaling adopted in isotropic media and in fact the wellknown Newell-Whitehead-Segel amplitude equation for isotropic systems [91, 92] can now be obtained as the special case W = 0 and Z = ±2 in Eq. 37. The general form of the amplitude equations is known a priori from symmetry, translation in space and time as well as rotation and reflexion symmetry within the plane of the layer, which manifests itself in the linear growth rate function. The coefficients for a specific problem have to be determined only for quantitative comparison with experiments. So far we concentrated on the simplest version of a stationary bifurcation where only a single mode becomes critical. One may have (near) degeneracy due to symmetry or accidentally by tuning of a secondary control parameter. If there is a n-fold degeneracy of marginal modes one is led to a coupled system of n amplitudes. Here degeneracy due to symmetry occurs in the oblique roll case (n = 2). From the ratio of the two non-linear coefficients for self and cross coupling one then deduces if rolls zig or zag with the possibility of domain boundaries (Fig. 13.2 b) or if their superposition (Fig. 13.4 b) is stable. In the case of a Hopf bifurcation, as observed in EHC, one has degeneracy due to reflection symmetry with two modes, corresponding to left and right travelling waves (rolls). Also, the coefficients of Eq. 38 become complex and from the linear dispersion relation one gains a term ± itvg 7 rA (the signs ± pertain to left and right travelling waves), which describes a group velocity vg. From a phenomenological point of view changes of complex coefficients and group velocity stem from the absence of reflection symmetry in the solutions. Destroying reflection symmetry by an external perturbation has a similar effect. Depending on the ratio of the real parts of the non-linear coefficients there are again two possibilities: either travelling waves with the possibility of domain boundaries (sources and sinks) or a superposition of the two wave systems leading to standing (oscillating) rolls. In the first case one is essentially left with the celebrated complex Ginzburg-Landau equation (CGLE), which exhibits transitions, e. g. at the Benjamin-Feir instability, to various forms of spatiotemporal chaos and is presently studied intensely. For general reviews see Refs. [2, 80]. The CGLE is applicable also to systems that show a Hopf bifurcation leading to a spatially homogeneous state (qc = 0), which is mainly found in oscillatory chemical reactions. The point defects of CGLE correspond to the famous spirals. In our system there is also the possibility of oblique travelling waves, which have in fact been observed in Merck Phase V [47] and in I52 [49, 51] and are known to lead to an interesting type of four-wave interaction [93]. 6 Now the higher powers in uy become essential; if we wanted to describe the transition between parallel and oblique rolls, higher powers in ux would have to be retained.
274
13.4
Rayleigh-Bénard convection
For the description of modulated roll patterns away from threshold (possibly only slightly) the amplitude B must be included, now also transformed to real space. A uniform scaling in e is then no more possible and coupling terms like AuyB appear in Eq. 36 as well as derivative terms in the cubic non-linearities. The gradient expansion starting from Eqs. 32 and 33 is systematically truncated in such a way that all O (s2) contributions to the growth rate (Eq. 34) are included [24, 83]. The additional equation for B is of the form
c1 u2x c2 u2y B q1 ux uy jAj2 q4 uy
iA u2y A c:c: . . .
39
The terms occurring are those allowed by symmetry. Due to the anisotropy more terms appear than in isotropic systems [94]. The clue for the characteristic appearance of the zigzag instability as a secondary bifurcation is that q4 is typically negative in nematics [23, 24] leading, in contrast to isotropic fluids, to amplification of transverse modulations of roll patterns. Model calculations that include this feature [25, 26] were quite successful in describing qualitatively the secondary instability and the behaviour beyond.
13.4
Rayleigh-Bénard convection
In the following two Sections we will discuss and compare theoretical and experimental studies on RBC and EHC. We often use non-dimensionalized units. Thus we write wave numbers as qi = qi'p/d, where the prime is sometimes omitted, and magnetic fields as Hi = hx Hf with the splay Freédericksz transition field Hf = (p/d) [k11/(m0 wa)]1/2. Other quantities have been introduced before. Note that for the Cartesian components of the wave-vector we use two symbols, q = (qx , qy) = (q, p). In the case of RBC we will mainly present an analysis of recent experimental investigations [55] in comparison with theoretical results [54]. In Fig. 13.5 a the critical Rayleigh number Rc (continuous curve) normalized to the value Rc0 = 1707.37 for isotropic fluids is shown as a function of a stabilizing magnetic field hx = Hx /Hf (Rc /Rc0 ? 1 for h2x /F p 1). The symbols are experimental results for the nematic 5CB. The linear theory predicts for increasing field a lower Lifshitz point (LP), where the roll orientation changes from normal ( pc = 0) to oblique and at higher field an upper LP with a transition from oblique to parallel rolls (qc = 0). In Fig. 13.5 b the experimental results for the squared wave numbers (qc, pc) together with |qc|2 are plotted and compared with the prediction of the theory (solid curves). The agreement is quite remarkable considering that there is no adjustable parameter. Weakly non-linear theory predicts two tricritical points (TP) with a subcritical bifurcation in the field range 5 < hx < 26. Note that the upper TP at hx = 26 is slightly above the lower LP. One may expect rather complex non-linear behaviour in that range, which has not been worked out in detail for 5CB so far. The situation should be simpler for MBBA where the TPs (hx = 4.15 and 31) are below and well-separated from the Lifshitz points (hx = 36 and 62). 275
13
Convection Instabilities in Nematic Liquid Crystals
Figure 13.5: a) Rayleigh number as a function of the stabilizing field hx in the planar configuration. b) The squared components of qc and |qc|2 as a function of hx.
In Figs. 13.6 a, b stability diagrams of MBBA in the e-q plane (normal rolls, p = 0) and in the e-p plane (oblique rolls, q = qc) are shown for a magnetic field hx = 34 between the upper TP and the lower LP (e = (R – Rc)/Rc). Rolls exist inside the neutral curves (NC) and are stable in a region bounded by the E, SV, and ZZ lines. The lowest-order theory (GLE) would only give the Eckhaus instability. Including the higher order terms with mean flow produces the skewed varicose instability SV, which however is hard to distinguish from the Eckhaus instability E. More important is that the stability regime for normal rolls is now limited from above by a ZZ line. When the ZZ line for normal rolls is extended on the left it joins the neutral curve at a point L which one may call a LP on the neutral curve. To the left of that point the growth rate for oblique rolls becomes larger than for normal rolls, or equivalently, the curvature of the neutral surface in the p-direction becomes negative. With 0.02
0.02
0.01
E
NC
ZZ
SV
ε=(R-Rc)/Rc
ε=(R-Rc)/Rc
P SV 0.01
ZZ
NC
L 0.00
–0.2 –0.1
0.0
0.1
q-qc
0.2
0.3
0.00 0.0
E 0.5
1.0
1.5
p
Figure 13.6: a) Stability diagram for normal rolls slightly below the lower LP; b) Stability diagram in the transverse (oblique) direction for q = qc.
276
13.4
Rayleigh-Bénard convection
increasing hx the point L moves down along the neutral curve until it reaches qc, e = 0 at hx = 36. This is the lower LP beyond which oblique rolls appear at threshold. In the vicinity of the LP the scenario can be described by the extended GLE as discussed in the last Section. The parameter W in Eq. 37 changes sign when the LP is crossed. From this equation without higher order terms one obtains a ZZ line that goes vertically up [89]. Coupling to the mean flow enhances the ZZ instability (Section 13.6) and consequently tilts the curve to the right. By contrast, in simple fluids, where the ZZ line always emanates from qc, mean flow effects, which are important for small to medium Prandtl numbers, turn the line to the left. Thus anisotropy is responsible for moving the LP away from qc along the neutral curve and mean flow tilts the line to the right. This seems to be a general scenario which also applies to EHC. Let us turn attention to the point P on the neutral curve NC in Fig. 13.6 a, where several stability limits merge. P corresponds to a tricritical point on the neutral curve at qtri . For q 6 qtri the solution bifurcates subcritically from the neutral curve, i. e. a small amplitude solution exists only outside the neutral curve. With decreasing magnetic field P moves down along the neutral curve and at hx = 31 it reaches e = 0 and q = qc , signalizing the change to a subcritical bifurcation with further decrease of hx . Interestingly, at small magnetic fields, where the bifurcation becomes supercritical again, the tricritical point moves upwards on the left branch of the neutral curve. Figure 13.6 b shows that normal rolls can escape the ZZ instability by undergoing a secondary transition to oblique rolls. Their stability range is bounded by two roughly parabolic curves. This transition has indeed been observed [55] and is quite well-known in EHC. There exists in addition a tertiary bifurcation (dotted lines), where the oblique rolls become unstable against a short wavelength mode which appears to lead to oblique rolls with p roughly reversed. At present it cannot be predicted into which state the system evolves. It could be a superposition of zig and zag rolls, as observed in the experiments, or a complex dynamic state where the system oscillates between the two states separated by grain boundaries (and maybe other defects) generated persistently. Also, in the light of the results for EHC (see below), one may expect that under some conditions in large aspect-ratio systems spatio-temporal chaos develops when the stability limit of normal rolls is exceeded. The non-linear scenarios have not been systematically studied in experiments so far. Actually investigations at small fields (hx ^ lower TP) would be very interesting. According to the theory the upper limitation of the stability regime is now of the SV-type, beyond which the system cannot evade into stable oblique rolls and complex behaviour seems unavoidable. Very recently a generalized weakly non-linear analysis including homogeneous twist of the director has been worked out [171]. Then the above mentioned SV instability changes into a ZZ instability in agreement with experiments [167]. In the non-linear regime one has a transition to abnormal rolls and bimodal convection as is also found in EHC, see note added at the end.
277
13
13.5
Convection Instabilities in Nematic Liquid Crystals
Electrohydrodynamic convection
The theoretical results are obtained with the SM (Section 13.2) unless otherwise stated.
13.5.1 Linear theory and type of bifurcation As pointed out before, most experiments were done with the material MBBA, sometimes doped with an ionizing substance. Typical material parameters can be found in the literature [12] which look fairly reliable except the flexoelectric coefficients. Unfortunately in most experiments the conductivity, which may vary strongly, has not been measured. One therefore has to determine the charge relaxation time tq, e. g. by fitting the cut-off frequency fc where the crossover between the low-frequency p conductive and the higher frequency dielectric regime occurs. The formula 2 p fc tq = C applies, where C is a function of s|| /sk, s|| /sk, and the viscosity ratio [12]. For MBBA C is about 6.3. In Fig. 13.2 a we show the threshold curve for a rather thick (d = 100 µm) and clean cell showing quite good agreement between theory and experiment. In the calculations the flexoelectric coefficients were taken considerably smaller than those estimated from some measurements. Otherwise one would find in contrast to the observations oblique rolls at a threshold in the dielectric regime. The conductivity was chosen as sk = 0.28610 –8 O–1m –1 and s|| /sk = 1.65. For more details see Ref. [14]. The theory correctly describes many qualitative features like the occurrence of oblique rolls at threshold at sufficiently low frequencies for materials with not too negative ea. MBBA (ea & –0.5) is a border-line case [14], Merck Phase V (ea & –0.3) has well developed oblique rolls and the newly introduced material I52 (ea & 0) exhibits strongly oblique rolls [50, 51]. For materials with ea < 0 the above behaviour with the conductive threshold apparently diverging at fc is a reasonable approximation if the charge relaxation time tq is much smaller than the director relaxation time td. Actually the conductive threshold curve always becomes vertical at a frequency fm ^ fc and bends back at higher voltages giving rise to restabilization of the conductive mode at high voltages. The effect becomes apparent in thinner and cleaner specimens. The fact that the conductive mode becomes ineffective for high voltages as well as for large ac frequencies is a consequence of the competition of the (destabilizing) electrohydrodynamic torques and the dielectric torques, which are stabilizing when ea < 0. For ea > 0 the homogeneous (qc = 0) Freédericksz transition competes with EHC [12] and in fact always preempts the latter at sufficiently high frequencies. For a recent investigation see [95]. Actually the quantitative agreement between theory and experiment over a large frequency range including the dielectric regime, suggested from Fig. 13.2 a, is somewhat deceptive. For the more common thinner specimens we are not always able to fit experimental curves satisfactorily over the whole frequency range by adjusting the conductivity, the flexocoefficients, and (to a lesser extent) the ratio s|| /sk. When adjusting the conductivity so that fc (and thereby essentially the whole conductive range) is described correctly, the dielectric threshold tends to come out too low. One could increase the conductivity to fit the dielectric threshold, which is roughly proportional to skd2, but then fc becomes too large. 278
13.5
Electrohydrodynamic convection
This discrepancy is in line with other observed quantitative discrepancies such as in the threshold behaviour when a stochastic component is present in the applied voltage [96]. As already mentioned, the most drastic non-standard behaviour, which is not understood from the SM, is the observed Hopf bifurcation in sufficiently thin and clean specimens [18, 41–43, 49–51] and the very small hysteresis sometimes observed at threshold [38–40, 47]. In fact a little further above threshold (but still near it) the amplitude of the pattern does appear to coincide reasonably with the results of the weakly non-linear theory [59]. Since the measurements on RBC exhibit such beautiful quantitative agreement with theory one has concluded that the electrodynamic part of the basic equations needs improvement. This was the motivation for introducing the WEM (Section 13.2). The linear stability calculations for the conductive mode have been carried out within this model [48, 49] using the same approximations which led to the analytic threshold formulas within the SM [12– 15]. It is found that there is an upward shift in the threshold, which may be quite small, and, more importantly, a Hopf bifurcation with critical frequency Rc a~ C oH td 2pfH td 1 o 02
s
1 o 02 ls td 2 1 ; Rc a~ C
40 s t
if the expression under the square root is positive. Here R = V 2 p2ak 11q is the reduced control para+ – meter, a~ 2 = mk mk g1 p2/(sa d2) is proportional to the geometric means of the mobilities, and
s =s q02 p02 1 o' = o tq b, with b =
ejjjj =ekk q02 p02 1 , is a reduced frequency. –1 a2 b/(1 + o'2)] < 0 is the damping rate of the (new) WEM Moreover, ls = – [t–1 rec + t d R~ mode. Its dominant contribution is usually just determined by the ion recombination rate 1/trec. Thus for the Hopf bifurcation to occur the quantity td /(~atrec ) must not be too large. This requires that the recombination of ions is sufficiently slow and that the layer is sufficiently thin and clean. Note, the Hopf condition is always satisfied near the cut-off for materials with negative dielectric anisotropy, where Rc diverges at the cut-off frequency (in the approximation used). But the Hopf frequency, which is then just given by the prefactor of the square root from Eq. 40, becomes large there. This appears to be consistent with the experiments [18, 41, 42]. Moreover, the prediction of the theory that for materials with vanishing dielectric anisotropy the Hopf condition and Hopf frequency become essentially independent of the external frequency has been verified experimentally using the material I52 [49]. I52 has the property that ea changes from negative to positive when the temperature passes through T & 60 8C.
13.5.2 Results of Ginzburg-Landau equation The large aspect-ratios, which can be achieved comparatively easily in EHC, make this system particularly well suited to test predictions of GLEs. Experimental results for the existence and stability of normal roll patterns in the q-e plane (Busse balloon) are shown in Fig. 13.7 [28, 36]. Similar experiments were reported by Ref. [29]. The symbols give the experimental points and the curves represent parabolic fits for the neutral curve (N) and the 279
13
Convection Instabilities in Nematic Liquid Crystals
Defect Turbulence
0.2
E
II
ε
ZZ Pattern
0.1 Normal Rolls
N 0.0 –0.2
I –0.1
0.0
q-qc
0.1
0.2
Figure 13.7: Experimental stability diagram (see text).
Eckhaus stability limit (E). The secondary and tertiary instabilities that limit the regions from above will be discussed further below. The wave number q can be controlled within certain limits on the small wave number side by the so-called frequency-jump technique [32]. Since qc increases with external frequency one can prepare first a state for the desired wave number by choosing the appropriate frequency and then by jumping to the frequency and e values. In the course of the experiment e can still be varied. Outside of the neutral curve the rolls decay whereas inside they grow up to their non-linear saturation. The neutral point is then determined by extrapolation. In this way also the GL relaxation time t was determined. The coherence length x1 in Eq. 37 can be obtained from the curvatures of the neutral surface at the minimum. x1 pertains to the direction parallel to the wave-vector of the pattern and is therefore often denoted as x||. Both, x1 and x2 (x2 = xk for normal rolls) have been measured by imaging the core of defects (see below) [32]. In the same paper comprehensive measurements on MBBA are presented for all parameters of the GLE as a function of external frequency in the conductive range. The comparison with theory shows quite good agreement. Another elegant method for measuring the linear parameters of the GLE makes use of thermal fluctuations very slightly below threshold [38, 39, 97, 98]. Then the dynamic structure function is exploited. The agreement with theoretical results [12] is generally, maybe except some cases of thin layers, satisfactory including the predicted variations with ac frequency. Slightly above onset the pattern should be dominated by the critical mode at q = qc and according to the p weakly non-linear analysis (Section 13.3) the pattern amplitude should grow proportional to e. This behaviour has been confirmed in experiments, where the optical contrast of the roll pattern was monitored as function of e [59] using the shadowgraph method [34, 38, 99]. The proportionality factor, which is determined by the non-linear (cubic) coefficient g in the GLE, agreed satisfactorily with theoretical results [12]. In a next step the Eckhaus instability boundaries (curves E in Fig. 13.7), which probe non-linear aspects of the system, could be identified by observing the destabilization of the pattern via longitudinal modulations after a frequency jump [29, 36, 37] 7. Subsequently the 7 The forcing of a pattern into a wave number state with q ( qc can also be done by the use of digitized electrodes and led to the first detailed investigation of the Eckhaus process in a remarkable experiment [27].
280
13.5
Electrohydrodynamic convection
pattern evolves into a state with wave number in the stable range by spontaneous creation of defects either at the surface or in pairs. p The prediction of the GLE that the Eckhaus curve is narrower by the universal factor 1/ 3 than the neutral curve, both in parabolic approximation, has been confirmed. Actually also the full evolution process from an unstable to a stable wave number can be analyzed with the GLE [100]. A particularly beautiful example for the usefulness of the GLE is the description of the structure and dynamics of point defects (dislocations), see Fig. 13.1 b. A defect is characterized by a zero of the amplitude A (x) where it behaves as |x| exp (± i f), where f is the polar angle and the topological charge is ±1 in the isotropic scaling. A stationary, isolated defect solution exists only for a pattern with the background wave-vector q = qc , i. e. with modulation wave-vector Q = q – qc = 0 in the GL description. Otherwise the defect moves with constant velocity V perpendicularly to Q. Each defect crossing the system carries away one periodic unit and brings the system nearer to its globally stable state Q = 0 (or q = qc) by an amount 2 p/L, where L is the system length. Since the GLE has a minimizing potential one can speak in such terms and the force on the defect is in fact the analogue of the PeachKohler force in solids [101, 102]. Due to the two-dimensional nature of the problem V (Q) is non-analytic at V = Q = 0. One has V log (V t/(3.29 x)) = 2 (x 2/t) Q for small Q [88, 103]. The full universal curve V (Q) can be obtained numerically [14] and is compared with detailed experimental results in EHC for motion of defects along the rolls [32] (Fig. 13.3 a). Also the interaction of defects, which is repulsive for equal topological charge and attractive otherwise, was investigated and compared with experiments [33, 104]. The theoretical results discussed above pertain to the normal-roll regime with wave number changes confined to the longitudinal component, i. e. the rolls remain normal. Some rather qualitative experiments with transverse wave number changes, showed that defects moved perpendicular to the rolls, as theoretically expected [30]. One also gets a neutral curve and a (generalized) Eckhaus stability limit roughly as expected [19]. In the latter experiments a magnetic field, applied in the plane of the layer at an oblique angle, was used to influence qc.
13.5.3 Beyond the Ginzburg-Landau equation From the foregoing discussion we know that the weakly non-linear behaviour of EHC in the range of validity of the GLE is understood fairly well. The situation is not quite as satisfactory when it comes to secondary bifurcations, which confine the q- range and which are captured in the theory only if corrections to GLE are included (Section 13.3) or a full evaluation is done to the hydrodynamic equations.
13.5.3.1 Experimental results From the most recent experiments it is quite clear that, beginning with normal rolls at threshold and increasing e, the ZZ instability, where long wavelength modulations with wave-vector s = (0, sy) perpendicular to q = (q, 0) start growing, provides the most impor281
13
Convection Instabilities in Nematic Liquid Crystals
tant destabilization mechanism, which sets in at quite low e (^ 0.2). This has been observed for MBBA (Fig. 13.7) [29, 36, 37, 105], where there are usually normal rolls at threshold for all frequencies. Furthermore, in Merck Phase V, where one has an LP at moderately low frequencies [105], and in I52 too, where oblique rolls at threshold extend to higher frequencies [50, 51]. Actually in the first clear observation of oblique rolls [106] they appeared after a secondary bifurcation. It also seems fairly clear now that under sufficiently ideal conditions – very homogeneous large aspect-ratio system, slow increase of e – beyond the ZZ instability the system can settle down in an (ideally) stationary oblique roll state. Usually zig and zag domains seem to persist. For MBBA the angle of obliqueness remains small, below about 108. The necessity to change sufficiently slowly has been stressed in particular by Ribotta [167]. Otherwise the system tends to remain in a dynamic defect turbulent state, which was called fluctuating Williams domains by Kai and co-workers [19]. It is characterized by continuous generation and annihilation of dislocation pairs modulating the (normal) roll pattern, see e. g. the snapshot shown in Fig. 13.3 b. Thus one seems to have two coexisting attractors. The transition to normal roll defect turbulence takes place at e above ezz (diamonds in Fig.13.7) [36, 37]. At even higher values of e a transition to a rectangular or grid pattern is typically observed before the onset of the strong turbulence types with creation and annihilation of disclinations in the director field [17, 62]. According to some measurements of (doped) MBBA the SV instability also plays a significant role, particularly in the higher frequency part of the conductive range, where it appeared to replace the ZZ instability as the first roll destabilization mechanism [29, 37]. When increasing e slowly beyond the SV instability the system settled down at first in an interesting regular defect-lattice structure [37, 107, 108]. At higher values of e a transition to defect turbulence again sets in. For an ac driving o around the crossover frequency ocr between the two types of behaviour a (fairly) direct transition from normal rolls to defect turbulence is observed. Below ocr the SV instability manifested itself as transients in experiments where e was increased suddenly, starting out from below the ZZ line. Whereas immediately above the ZZ line the first destabilization took place via the ZZ instability as expected. It changed over to a SV destabilization at higher values of e thus defining a SV line in the q-e plane. With increasing o the SV line moved closer to the ZZ line and apparently joined it at ocr . Although defect-lattice structures of the above type have apparently been observed by other researchers [168], they did not characterize quantitatively the existence range. Possibly their cells with undoped MBBA behaved somewhat differently. We only mention in passing that various undulated structures have been reported by Ribotta and co-workers [17, 105] but could not always be reproduced by others [169]. Actually such structures can be obtained near the LP even within the GLE approximation, but they then turn out to be metastable since their (generalized) potential is higher than that of straight (usually oblique) rolls [14, 89, 90, 109, 110]. This may explain why they are difficult to observe.
13.5.3.2 Theoretical results and discussion The experimental scenario can be partly understood with the SM [24, 111, 112]. One finds that the general shape of the measured ZZ instability limit in the q-e plane, shown in 282
13.5
Electrohydrodynamic convection
Fig. 13.7 (triangles), is reproduced by the theory in a rather robust fashion, whereas the actual position depends crucially on the material parameters and is very sensitive to approximations. Thus using standard MBBA parameters the ZZ instability comes out too high within the order-parameter approach, which involves an expansion up to cubic order in the pattern amplitude (Section 13.3) [24, 111]. One needs the full Galerkin computations (Section 13.3), which lead to rather good quantitative agreement (triangles and broken line in Fig. 13.8 a) [112]. We conclude that higher order terms in the amplitude become important already at quite low values of e. In any case corrections to the GLE approximation in particular the coupling of the pattern amplitude to the mean flow [24] (Eq. 39), which is made explicit in the order parameter approach, has the effect of reinforcing the ZZ instability with increasing amplitude, in distinct contrast to the effect in isotropic fluids 8. This results in a ZZ line roughly horizontal in the q-e plane in the range where it determines the stability boundaries. A convenient characterization of the ZZ line is its position at q = qc, which is called ezz . On the low-q side one expects the ZZ line to extend to the neutral curve and join it at a point L, which is sometimes called an LP on the neutral curve, because below normal and above oblique rolls become critical on the neutral curve. This point can be calculated from the linear theory and is also shown in Fig. 13.8 a. Following the ZZ line through the Eckhaus unstable range on the low q side there is a delicate numerical problem and the curve may exhibit rather unexpected variations (Fig.13.8 a).
Figure 13.8: a) Stability diagram for normal rolls for a rather thick cell with MBBA at o tq = 0.2. Full Galerkin computation. For details see text; b) Stability diagram for rolls escaping in the oblique direction at fixed q = qc. The unstable bubble was tested by using Galerkin expansion with more terms (Z). For details see text.
In any case, if decreasing the external frequency the LP on the neutral curve typically moves down and may eventually cross the minimum at qc leading to oblique rolls at threshold. This is the case for the materials Merck Phase V and I52. Then, simultaneously ezz moves down to zero. Of course shifts of ezz and the LP on the neutral curve can also be effected by changing the material parameters. Figures 13.9 a,b give an impression of the de8 In technical terms it is the opposite sign of the coefficient q4 of Eq. 39 which is responsible for the different behaviour
283
13
Convection Instabilities in Nematic Liquid Crystals 0.070
0.08
εZZ
εZZ
ωτ0 = 0.5 εa = –0.53 d = 55µm
0.06
σa = 0.5 d = 55µm
0.065 0.060 0.055
0.04
0.050 0.02 0.045 0.00 0.1
0.3
0.5
σa
0.7
0.9
0.040 –1.0
–0.8
–0.6
εa
–0.4
–0.2
0.0
Figure 13.9: Onset of the ZZ instability at band centre as a function of (a) the anisotropy of the conductivity sa and (b) the anisotropy of the dielectric constant ea.
pendence of ezz on the anisotropies of the conductivity and the dielectric tensor. The trend shown in Fig. 13.9 b is in qualitative agreement with experiments when comparing MBBA to Merck Phase V and I52, which have ea & –0.5, –0.3, 0, respectively. 9 The mean flow also has the effect of transforming the Eckhaus instability on the large q side into a SV instability. This effect becomes noticeable only in the upper part as the ZZ line is approached because at smaller e the ratio sy /sx is very small, see the broken line in Fig. 13.8 a. In fact the SV instability then turns around and joins the ZZ line smoothly. This effect indicates that the SV instability may become relevant and could be a clue to the observations quoted above [29, 37, 108]. In order to show that the system can escape at ezz into an oblique roll state, as is actually observed, the stability of rolls with q, p & 0 was investigated. An example of the stability diagram for MBBA is shown in Fig. 13.8 b, where q is fixed at qc. The point ZZ on the axis corresponds to ezz. It can be seen that the smallest roll angle arctan ( p/qc), where rolls are stable, first increases with increasing e but saturates at about 88. This is in agreement with experiments [36, 37, 113]. With increasing e the minimal angle decreases again and, surprisingly, beyond the point denoted by SV normal rolls could restabilize. It is interesting to note that the order parameter approach gives the ZZ destabilization qualitatively correct but fails to reproduce the restabilization of normal rolls [24]. We conclude that at ezz a forward-type bifurcation to oblique rolls can occur as has been observed. From the curve CR in Fig. 13.8 b we see that at larger e a short wavelength instability comes into play, i. e. a roll system with a different wave-vector, particular in different orientation, starts growing. This may saturate the often-observed rectangular patterns or, for a non-symmetric superposition, lead to the sometimes-observed oblique modulated structures [17, 105]. In order to examine this possibility one would have to test the stability of such patterns by a suitable Galerkin procedure, which was done for normal rolls. However, since there are other possibilities, in particular turbulent states, this approach is not exhaustive and has to be complemented by simulations of the dynamics. 9 Recently most material parameters of Merck Phase V have been measured, see note added at the end. The material parameters of I52 have either been measured directly or fitted to EHC measurements [49].
284
13.5
Electrohydrodynamic convection
Unfortunately, solving the full hydrodynamic equations is impossible. Therefore the order parameter approach, applied here, becomes useful. As mentioned before, this approach [24] (Section 13.3) captures all features of the secondary bifurcation scenarios found in rigorous calculations, though the ZZ destabilization line comes out too high for realistic material parameters. This may be corrected by using a larger value of sa /sk. When formulated in real space one is led to coupled amplitude equations which have been simulated numerically [21, 24] 10. Below the ZZ line the system approaches rapidly the stable normal roll attractor when starting from random initial conditions. Beyond the ZZ line the situation becomes complicated. Starting slightly above the ZZ line, from small random initial conditions, one could either end up in a stationary zigzag pattern (Fig. 13.10 a) for sa /sk, e = 0.1, and otq = 0.5) or in states with shorter and more sinusoidal undulations (Fig. 10 in Ref. [24]). Then a few slowly moving defects appeared to persist. There is no clear-cut separation between undulations and zigzag, analogue to the situation near the LP without the mean flow mentioned above. For larger e (e = 0.2), on the other hand, the pattern evolves into a state similar to the one shown in Fig. 13.10 b. This is a time-dependent state with alternating zigs and zags where defects are continuously generated in pairs and is reminiscent of some cases of the observed defect turbulence, although in the experiments the rolls often tend to be more aligned in the normal direction. This discrepancy could be an artefact of the order parameter approximation, where normal rolls do not restabilize at higher e (see above), so the roll orientation remains more oblique. The generation and annihilation of defects can be understood from an advection of the roll pattern by the mean flow, which amplifies small undulations. Because the anisotropy counteracts the bending of rolls the stress is released by straightening the rolls and dislocations are left behind. The situation is reminiscent of Rayleigh-Bénard convection in isotropic fluids where stable roll attractors apparently compete with complex patterns, spiral defect turbulence [115–117]. It has been shown very recently that if anisotropy is introduced into this system by inclining the convection cell, a normal roll pattern with dislocation defect turbulence occurs, which looks quite similar to patterns observed in EHC [118].
Figure 13.10: (a) Stationary zigzag pattern for e = 0.1. (b) Snapshot of a defect turbulent zigzag pattern at e = 0.2. 10 Similar results have been obtained from quite simple model equations if certain parameters are adjusted ad hoc [25, 114].
285
13
13.6
Convection Instabilities in Nematic Liquid Crystals
Concluding remarks
Clearly this review could not be comprehensive, so we first list here some of the omitted material. Recently some experiments in planarly aligned samples with an additional destabilizing magnetic field in the z-direction (parallel to the electric field) were performed [119, 120]. Some earlier work is discussed in [60]. For not too high magnetic fields the EHC threshold is merely reduced in accordance with the standard theory. For fields that are larger than a critical value, however, the Freédericksz transition comes first and then EHC sets in as a secondary instability, similarly to case F in homeotropic alignment, see below. The voltage threshold should then increase with increasing field, because destabilization of the Freédericksz distorted state becomes increasingly difficult, which is in qualitative agreement with experiments. Also the change to a subcritical bifurcation appears to be born out by theory [121]. In the experiments with I52 oblique travelling rolls were observed. In this four-wave scenario (left-right and zigzag) a number of superpositions are possible, depending on the non-linear coefficients [93]. The left and right travelling roll systems always appear to separate, whereas in some parameter range the zigs and zags make a transition from separation to coexistence [51]. However, the superposed state is disordered (weakly turbulent). One can show theoretically that just beyond this codim-2 point a Benjamin-Feir-type modulational instability (Section 10.3) becomes generic, depending only on the sign of coefficients and not on their magnitude [122]. Presumably the above experiments were done in that parameter range. EHC is particularly suited to apply space or time-modulated forcing. The effect of spatially periodic forcing of stationary patterns in a quasi one-dimensional situation was studied by using structured electrodes [27, 123]. In particular, commensurate-to-incommensurate transitions were observed as predicted by theory based on phenomenological amplitude equations [124, 125]. Additional two-dimensional effects in spatially forced isotropic systems were studied theoretically [126]. Experiments appear to be lacking at this time. The effect of resonant time-periodic forcing of travelling (normal) rolls was first discussed theoretically using normal-form equations [127, 128] and was then confirmed experimentally [43]. Interesting scenarios including a transition to standing, oscillating rolls can be induced in this way. Similar effects for travelling oblique rolls, where one has four-wave mixing, have also been considered [47, 129]. The effect of stochastic driving was investigated theoretically [96, 130] and experimentally [131, 132]. Some statistical properties of defect turbulent states in EHC were studied experimentally [42, 133, 134] and, as pointed out before, the role of thermal fluctuations slightly below threshold was measured and analyzed. Of the numerous investigations of far-off threshold effects we only mention the study of phase waves in the oscillatory bimodal state [135, 136] and the transition between strongly turbulent states [60]. 11 Besides the case of planar surface alignment of the director, considered here, one can have homeotropic alignment where by appropriate treatment of the confining plates the di11 Very recently a multifractal analysis of electroconvective turbulence has been performed [170].
286
13.6
Concluding remarks
rector is oriented perpendicular to the plane. Then the system is isotropic in the plane of the layer and therefore from a symmetry point of view more related to convection in simple fluids. Both, RBC and EHC, have recently been investigated theoretically using the methods described before [81, 137, 138]. In RBC, when heated from below, a subcritical Hopf bifurcation is predicted, which at high stabilizing magnetic fields first transforms into a stationary bifurcation and subsequently becomes supercritical. The results are consistent with experiments, mostly without visualization of the patterns, by Guyon et al. [139]. One also obtains convection when heating from above leading to stationary squares. Application of a weak planar magnetic field renders the system anisotropic and then one has rolls in a small range above threshold. Experiments were performed in [140]. In homeotropic EHC one has to distinguish two cases. For materials with positive or only very slightly negative dielectric anisotropy, like e. g. the newly introduced material I52 in an appropriate temperature range [49–51], theory predicts a direct transition to convection (case C) in the form of stationary squares, except at large ea where one finds rolls. The linear results can be described in good approximation by an analytic threshold formula [138]. For materials with more negative ea, like MBBA, one has first a bend Freédericksz transition (case F) leading to a planar component of the director, which (ideally) singles out spontaneously a direction in the plane. Subsequently one has a transition to EHC, which on the linear level exhibits similar scenarios as in the planar case. Some experimental results have been reported in [107]. The tendency towards oblique rolls is more pronounced in the homeotropic case, and in fact MBBA should show oblique rolls at low frequency. Experiments without [141] and with a stabilizing magnetic field [142] have verified these predictions. On a deeper (non-linear) level there is, however, a very essential difference that makes this system rather unique. The direction singled out by the Freédericksz transition in the plane of the layer is the result of a spontaneously broken symmetry in the absence of an applied planar magnetic field. This makes itself known in experiments, of course, by the fact that the planar component is not really uniform over the cell but varies slowly in space being pinned by inhomogeneities. For a description (under ideal conditions) one has to couple the EHC mode from the beginning on with the Freédericksz Goldstone mode. For normal rolls this situation may often not be very important, although it could lead to a destabilization of the pattern from the very beginning. But for oblique rolls there is certainly a drastic consequence. One easily understands that oblique rolls exert a mean torque on the director via the velocity field. This torque cannot be balanced. So the director has to react by a rotation which in turn acts back on the rolls reducing their amplitude. Clearly there cannot be a static state even under ideal conditions and indeed in the experiments a pattern with slow dynamics shows up [141]. The weakly non-linear theory is being worked out now. Hopefully there will be more experiments to study the transition between order and chaos as the magnetic field is decreased and a better characterization of the chaotic state. Besides the planar and homeotropic alignment there is also the possibility of hybrid anchoring, i. e. planar on one side, homeotropic on the other. Then, in the basic state the director bends by p/2 when going from one plane to the other and there are in fact two symmetry equivalent directions for the bend to occur. Usually the preparation procedure of the probe, i. e. the filling with the LC, will single out one direction. Thus the basic state is not reflection symmetric and the pattern is expected to drift. Recently, this has been shown theoretically [143]. Under dc driving drift can also be imposed by non-ideal planar alignment, 287
13
Convection Instabilities in Nematic Liquid Crystals
i. e. by a pretilt [144, 145]. Indeed a small pretilt is hard to avoid. There is some similarity to systems with an externally imposed drift that has been studied recently in Taylor vortex flow in simple fluids [146, 147]. Finally we point out that the uniaxial symmetry of planarly oriented cells can be perturbed in at least two ways: by applying a planar magnetic field at an oblique angle to the alignment direction, which has been utilized to some extent in [19], or by imposing a twist on the director by having different planar alignment directions on the two plates. In this case one also has to break reflection symmetry around the midplane of the layer, which may be done by applying a dc voltage (the flexoelectric effect is linear in the field). As a consequence one loses the sharp separation between normal and oblique rolls and in fact the roll direction should turn smoothly by changing any parameter, e. g. the voltage. This has indeed been demonstrated experimentally [148]. Actually, twisted cells have interesting properties also under ac excitation [149]. We point out that twisted cells are employed in the most common type of LC displays. But the materials used there have strongly positive dielectric anisotropy that leads to a Freédericksz transition instead of EHC. Although during the last 12 years much progress could be achieved, there still remain many open problems. For planar alignment there exist some quantitative discrepancies with the SM which are possibly not explained by the WEM. It is also not yet clear if that model will explain the (very weakly) subcritical bifurcation, observed under some conditions. But this should be resolved in the near future. A different source of space charges is operative in the very beautiful electroconvection experiments in thin freestanding smectic films [150, 151]. Here the free surface charges and the inhomogeneity of the applied field certainly play a crucial role. Very recently a linear and weakly non-linear theory has been worked out [152].
Acknowledgements
We have benefited from discussions with G. Ahlers, A. Buka, W. Decker, M. Dennin, A. Hertrich, S. Rasenat, I. Rehberg, A. Rossberg, H. Richter, M. Treiber, and W. Zimmermann. A. Buka, I. Rehberg, H. Richter, and A. Rossberg have also kindly provided graphs of their results. We are grateful to W. Decker, A. Hertrich, and F. Schmögner for help in preparing the manuscript. Financial support by Deutsche Forschungsgemeinschaft (Sonderforschungsbereich 213) is gratefully acknowledged. L. Kramer wishes to thank the Center Emile Borel, Institut Henry Poincare, and the ECM, Universidad de Barcelona, and W. Pesch the LASSP, Cornell University, where part of this work was performed, for their hospitalities.
288
Note added
Note added
We here list some recent developments in EHC. Firstly, on the level of the SM, the weakly non-linear theory for homeotropic surface alignment in the usual case F, where one first encounters a bend Freédericksz transition, has been worked out [153, 154]. One obtains in the normal roll regime one GLE for the patterning mode (two equations in the oblique roll range) coupled to a dynamic equation for the Goldstone mode that signifies the broken continuous symmetry. In the normal roll case the equations essentially scale uniformly in e. It turns out that in the absence of any external symmetry breaking (no stabilizing magnetic field) no stable periodic roll solutions exist and one seems to have a disordered state in all cases. Theory suggests that the state is always dynamic, and the type of spatio-temporal defect chaos has been termed soft-mode turbulence [155, 156], some experiments appear to show in the normal roll range for very small e static (frozen) disorder [141, 142]. The destabilization of normal rolls is brought about by the important fact that, although the growth rate is maximal for the rolls oriented perpendicular to the (undistorted) director, they exert an abnormal torque around the z-axis on the director that amplifies fluctuation-induced misalignments away from the normal orientation. The abnormal torque is a signature of the non-variational nature of the system, even at onset. In the presence of a (weak) stabilizing magnetic field H there are stable rolls up to ec * H2. For normal rolls the destabilization occurs in a frequency range oL < o < oAR at eZZ by a ZZ instability, and for higher frequencies by a spatially homogeneous transition (in the x-y plane) at eAR to abnormal rolls, where the director is rotated away from its normal position, either to the left or to the right. The term abnormal rolls has been introduced earlier in the context of experiments without magnetic field [166]. The rotation is again a manifestation of the abnormal torque. The abnormal rolls destabilize at 1.5 ec (in the weak-field limit) and then one has defect turbulence. Interestingly, at even higher values of e, the coupled GLEs describe in a substantial parameter range a new spatial superstructure where defects of equal topological charge tend to assemble along chains aligned parallel to the magnetic field. The chains form a periodic stripe pattern and the topological charge alternates from chain to chain. Between the chains the rolls (and the director) are rotated alternatively to the right and to the left. The roll angle is related to the defect density along a chain by a simple topological constraint. These structures are reminiscent of the chevrons, observed for more than 25 years, in the dielectric range of EHC in planarly aligned cells – for photographs see any of the standard textbooks [57, 65, 66] – and it appears that a similar mechanism is operative there. The formation of chevrons can be modelled as a Turing-like instability arising in the defect turbulent state [157]. Secondly, it has been realized recently that also in the conductive range of planarly aligned cells the transition to abnormal rolls occurs generically at e = eAR * 0.1 [158]. As in the homeotropic case the transition occurs at high frequencies in the stable range and can be preceded for lower frequencies (oL < o < oAR) by a ZZ instability. In contrast to the (nearly) isotropic homeotropic case one finds for o < oAR restabilization of abnormal rolls above a certain eARstab. Thus, in the oblique roll range and in the normal roll range below oAR normally oriented rolls reappear as abnormal rolls. The restabilization had been found before in the full Galerkin computations, described in Section 13.5.3 (Fig. 13.8 b), without understanding the physical basis. Although in planarly aligned systems abnormal rolls cannot be distin289
13
Convection Instabilities in Nematic Liquid Crystals
guished from normal rolls by the usual optical techniques, there exists indirect evidence for their occurrence [158]. Increasing e further, the abnormal rolls are destabilized by a short wavelength bimodal varicose instability that transforms at large frequencies into a long wavelength instability of the SV-type. Finally, we mention some recent progress in the application and analysis of the WEM model. The linear analysis was shown to describe quantitatively recent systematic measurements on Merck Phase V of Vc, qc, and the Hopf frequency oc as a function of o for cells of different thickness [159]. This agreement is particularly noteworthy because it is based on recent measurements of the viscosities a2 – a6 [160] as well as on independent measurements of the elastic constants. The only remaining SM parameter a1 was fitted to give the correct Lifshitz frequency. So it appears that the WEM can describe at least the three systematically studied materials MBBA [161], Merck Phase V, and I52 [49]. Meanwhile it has also become clear that the WEM model can explain the weakly subcritical nature of the bifurcation observed under some conditions in MBBA and Merck Phase V [38–40, 47] and, more recently, also in I52 [162–164]. Whereas the results for I52 show that the bifurcation is subcritical only in the stationary regime near the crossover to the Hopf bifurcation, in agreement with the predictions [165], one observes hysteretic behaviour in the other materials also in the Hopf range. This can be understood in terms of the subcritical stationary branch persisting in the non-linear range even in the region where the Hopf bifurcation precedes slightly the stationary one.
References
1. P. Manneville: in: Dissipative Structures and Weak Turbulence, Academic Press, New York, (1990) 2. M.C. Cross, P.C. Hohenberg: Rev. Mod. Phys., 65, 851 (1993). 3. F.H. Busse: Fundamentals of thermal convection, in: W.R. Peltier (ed.): Mantle convection, Plate Tectonics and Global Dynamics, Gordon and Breach, p. 23 (1989) 4. A. Buka, L. Kramer: in: Pattern formation in liquid crystals, Springer, New York, (1995) 5. R. Williams: J.Chem.Phys., 39, 384 (1963) 6. A. Kapustin, L. Larinova: Kristallografya, 9, 297 (1963) 7. E.F. Carr: Mol. Cryst. Liq. Cryst., 7, 253 (1969) 8. W. Helfrich: J. Chem. Phys., 51, 4092 (1969) 9. Orsay Liquid Crystal Group: Phys. Rev. Lett., 26, 1642 (1970) 10. E. Dubois-Violette, P.G. de Gennes, O.J. Parodi: J. Phys. (Paris), 32, 305 (1971) 11. P.A. Penz, G.W. Ford: Phys. Rev. A, 6, 414 (1972) 12. E. Bodenschatz, W. Zimmermann, L. Kramer: J. Phys. (Paris), 49, 1875 (1988) 13. W. Zimmermann, L. Kramer: Phys. Rev. Lett., 55, 402 (1985) 14. L. Kramer, E. Bodenschatz, W. Pesch, W. Thom, W. Zimmermann: Liquid Crystals, 5(2), 699 (1989) 15. L. Kramer, W. Pesch: Electrohydrodynamic instabilities in nematic liquid crystals, in: A. Buka, L. Kramer (eds): Pattern formation in liquid crystals, Springer, New York, (1995) 16. R. Ribotta, A. Joets: Defects and interactions with the structure in ehd convection in nematic liquid crystals, in: J. E. Wesfreid, S. Zaleski (eds): Cellular Structure in Instabilities, Springer, Berlin, p. 249 (1984)
290
References 17. 18. 19. 20. 21.
22.
23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.
38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50.
51. 52. 53. 54. 55. 56.
A. Joets, R. Ribotta: J. Phys. (Paris), 47, 595 (1986) S. Kai, K. Hirakawa: Prog. Theor. Phys. Suppl., 64, 212 (1978) S. Kai, N. Chizumi, M. Kohno: Phys. Rev. A, 40, 6554 (1989) S. Kai: unpublished (1988) E. Bodenschatz, M. Kaiser, L. Kramer, W. Pesch, A. Weber, W. Zimmermann: Patterns and defects in liquid crystals, in: P. Coullet, P. Huerre (eds): New Trends in Nonlinear Dynamics and Pattern Forming Phenomena: The Geometry of Nonequilibrium, Plenum Press, NATO ASI Series, (1990) W. Pesch, W. Decker, Q. Feng, M. Kaiser, L. Kramer, A. Weber: Weakly nonlinear analysis of pattern formation in nematic liquid crystals, in: J. M. Coron, F. Helein, J. M. Ghidaglia (eds): Nematics: Mathematical and Physical Aspects, Kluwer Academic Publishers, Dordrecht, NATO ASI Series, p. 291 (1991) M. Kaiser, W. Pesch, E. Bodenschatz: Physica D, 59, 320 (1992) M. Kaiser, W. Pesch: Phys. Rev. E, 48, 4510 (1993) S. Sasa: Prog. Theor. Phys., 83, 824 (1990) S. Sasa: Prog. Theor. Phys., 84, 1009 (1990) M. Lowe, J. Gollub: Phys. Rev. Lett., 55, 2575 (1985) S. Rasenat, E. Braun, V. Steinberg: Phys. Rev. A, 42, 5728 (1991) S. Nasuno, S. Kai: Europhys. Lett., 14(8), 779 (1991) S. Nasuno, S. Takeuchi, Y. Sawada: Phys. Rev. A, 40, 3457 (1989) G. Goren, I. Procaccia, S. Rasenat, V. Steinberg: Phys. Rev. Lett., 63, 1237 (1989) S. Rasenat, V. Steinberg, I. Rehberg: Phys. Rev. A, 42, 5998 (1990) E. Braun, V. Steinberg: Europhys. Lett., 15(2), 167 (1991) S. Rasenat: PhD thesis, Universität Bayreuth, (1991) L. Kramer, E. Bodenschatz, W. Pesch: Phys. Rev. Lett., 64, 2588 E. Braun, S. Rasenat, V. Steinberg: Europhys. Lett., 15(6), 597 (1991) S. Nasuno, O. Sasaki, S. Kai, W. Zimmermann: Stability diagram in electrohydrodynamic convection of nematics electrohydrodynamic convection in nematics: the homeotropic case, in: S. Kai (ed.): Pattern formation in complex dissipative systems and Global Dynamics, World scientific, p. 275 (1992) I. Rehberg, S. Rasenat, M. de la Torre Juarez, W. Schöpf, F. Hörner, G. Ahlers, H. Brand: Phys. Rev. Lett., 67, 596 (1991) I. Rehberg, F. Hörner, L.Chiran, H. Richter, B. Winkler: Phys.Rev. A, 44, R7885 (1991) I. Rehberg, F. Hörner, G. Hartung: J. Stat. Phys., 64, 1017 (1991) A. Joets, R. Ribotta: Phys. Rev. Letters, 60, 2164 (1988) I. Rehberg, S. Rasenat, V. Steinberg: Phys. Rev. Lett., 62, 756 (1989) I. Rehberg, S. Rasenat, M. de la Torre-Juarez, V. Steinberg: Phys. Rev. Lett., 61, 2448 (1988) N.V. Madhusudana, V.A. Raghunathan, K.R. Sumathy: Pramana J. Phys., 28, L311 (1987) V. Raghunathan, P.N. Madhusudana: Pramana J. Phys., 131, 163 (1988) W. Thom, W. Zimmermann, L. Kramer: Liq. Cryst., 4, 309 (1989) M. de la Torre Juarez, I. Rehberg: Phys.Rev. A, 42, 2096 (1990) M. Treiber, L. Kramer: Mol. Cryst. Liq. Cryst., 261, 951/311 (1995) M. Dennin, M. Treiber, L. Kramer, G. Ahlers, D. Cannell: Phys. Rev. Lett., 76, 319 (1995) M. Dennin, G. Ahlers, D. Cannell: Measurement of material parameters of the nematic crystal I52, in: P. Cladis, P. Palffy-Muhoray (eds): Spatio-temporal patterns in nonequilibrium complex systems, Santa Fe Institute Studies in the Sciences of Complexity XXI, Addison-Wesley, New York, (1994) M. Dennin, D. Cannell, G. Ahlers: Mol. Cryst. Liq. Cryst., 261, 977/337 (1995) E. Dubois-Violette: C. R. Acad. Sci., 21, 923 (1971) P.J. Barratt: Liquid Crystals, 4(3), 223 (1989) Q. Feng, W. Pesch, L. Kramer: Phys. Rev. A, 45, 7242 (1992) L.I. Berge, G. Ahlers, D. S. Cannell: Phys. Rev. E, 48, 3236 (1993) E. Dubois-Violette, G. Durand, E. Guyon, P. Manneville, P. Pieranski: Instabilities in nematic liquid crystals, in: L. Liebert (ed): Solid state physics, Suppl. 14, Academic Press, New York, p. 147 (1978)
291
13
Convection Instabilities in Nematic Liquid Crystals
57. L.M. Blinov: in: Electrooptical and Magnetooptical Properties of Liquid Crystals, John Wiley, New York, (1983) 58. A. Pikin: in: Structural Transformations in Liquid Crystals, Gordon and Breach Science Publishers, New York, (1991) 59. I. Rehberg, B.L. Winkler, M. de la Torre, S. Rasenat, W. Schöpf: Adv. Solid State Physics, 29, 35 (1989) 60. S. Kai, W. Zimmermann: Prog. Theor. Phys. Suppl., 99, 458 (1989) 61. W. Zimmermann: MRS Bulletin, XVI, 46 (1991) 62. S. Kai: Forma, 7, 189 (1992) 63. L. Kramer, W. Pesch: Annu. Rev. Fluid Mech., 27, 515 (1995) 64. G. Ahlers: Experiments on thermally driven convection, in: A. Buka, Kramer (eds.): Pattern formation in liquid crystals, Springer, New York, (1995) 65. P. de Gennes, J. Prost: in: The Physics of Liquid Crystals, Clarendon Press, Oxford, (1993) 66. S. Chandrasekhar: in: Liquid Crystals, University Press, Cambridge (1992) 67. M.J. Stephen, J. P. Straley: Rev. Mod. Phys., 46, 617 (1974) 68. H. Pleiner, H. Brandt: Hydrodynamics and electrohydrodynamics of liquid crystals, in: A. Buka, L. Kramer (eds.): Pattern formation in liquid crystals, Springer, New York, (1995) 69. F.C. Frank: Discuss. Faraday Soc., 25, 19 (1958) 70. R.B. Meyer: Phys. Rev. Lett., 22, 918 (1969) 71. J.L. Ericksen: Arch. Ration. Mech. Anal., 4, 231 (1960) 72. F.M. Leslie: Quart. J. Mech. Appl. Math., 19, 357 (1966) 73. E. Dubois-Violette, P. Manneville: Flow instabilities in nematics, in: A. Buka, L. Kramer (eds.): Pattern formation in liquid crystals, Springer, New York, (1995) 74. O. Parodi: J. Phys. (Paris), 31, 581 (1970) 75. F.H. Busse, E.W. Bolton: J. Fluid Mech., 146, 115 (1984) 76. E. Dubois-Violette: J. Physique, 33, 95 (1972) 77. A. Buka, L. Kramer: Theory of transient patterns in the splay freedericksz transition of nematics, in: S. Kai (ed.): Pattern formation in complex dissipative systems, World Scientific, Kitakyushu, Japan, (1991) 78. H. Haken: in: Synergetics, Springer, (1978) 79. M.C. Cross: Phys. Fluids, 23, 1727 (1980) 80. A.C. Newell, T. Passot, J. Lega: Annu. Rev. Fluid Mech., 25, 399 (1993) 81. Q. Feng, W. Decker, W. Pesch, L. Kramer: J. Phys. France, II 2, 1303 (1992) 82. W. Pesch, L. Kramer, General mathematical description of pattern forming instabilities, in: A. Buka, L. Kramer (eds.): Pattern formation in liquid crystals, Springer, New York, (1995) 83. W. Decker, W. Pesch: J.Phys. France, II 4, 493 (1994) 84. W. Eckhaus: in: Studies in Nonlinear Stability Theory, Springer, New York (1965) 85. F.H. Busse: Rep. Prog. Phys., 41, 1929 (1978) 86. F.H. Busse, R.M. Clever: J. Fluid Mech., 91, 319 (1979) 87. V. Ginzburg, L. Landau: Zh. Eksp. Teor. Fiz., 20, 1064 (1950) 88. E. Bodenschatz, W. Pesch, L. Kramer: Physica D, 32, 135 (1988) 89. W. Pesch, L.Kramer: Z.Phys. B, 63, 121 (1986) 90. L. Kramer, E. Bodenschatz, W. Pesch, W. Zimmermann: in: W. Güttinger, G. Dangelmayr (eds.): The Physics of Structure Formation, Springer, Berlin, (1987) 91. A.C. Newell, J. A. Whitehead: J. Fluid Mech., 38, 279 (1969) 92. L.A. Sege: J. Fluid Mech., 38, 203 (1969) 93. M. Silber, H. Riecke, L. Kramer: Physica D, 61, 260 (1992) 94. A. Zippelius, E. Siggia: Phys. Fluids, 26, 2905 (1983) 95. P. Kishore, T. Raj, A. Iqbal, S. Sastry, G. Satyanandam: Liquid Crystals, 14, 1319 (1993) 96. Müller, U. Behn: Z. Phys. B – Condensed Matter, 69, 185 (1987) 97. F. Hoerner, I. Rehberg: Using thermal noise to measure the correlation lengths of electroconvection, in: S. Kai (ed.): Pattern formation in complex dissipitative systems and Global Dynamics, World scientific, p. 429 (1992)
292
References 98. 99. 100. 101. 102. 103. 104. 105.
106. 107.
108. 109.
110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120. 121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138.
B. Winkler, W. Decker, H. Richter, I. Rehberg: Physica D, 61, 284 (1992) S. Rasenat, G. Hartung, B. Winkler, I. Rehberg: Exp. in Fluids, 7, 412 (1989) L. Kramer, H. Schober, W. Zimmermann: Physica D, 31, 212 (1988) G. Tesauro, M. C. Cross: Phys. Rev. A, 34, 1363 (1986) E.D. Siggia, A. Zippelius: Phys. Rev. A, 24, 1036 (1981) L.M. Pismen, J. D. Rodriguez: Phys. Rev. A, 42, 2471 (1990) E. Bodenschatz, A. Weber, L. Kramer: J. Stat. Phys., 64, 1007 (1991) A. Joets, R. Ribotta: Electro-hydrodynamical convective structures and transitions to chaos in liquid crystals, in: J. E. Wesfreid, S. Zaleski (eds.): Cellular Structure in Instabilities, Springer, Berlin, p. 294 (1984) C. Hilsum, F. Saunders: Mol. Cryst. Liq. Cryst., 64, 25 (1980) S. Kai, Y. Adachi, S. Nasuno: Stability diagram, defect turbulence, and new patterns in electroconvection in nematics, in: P.E. Cladis, P. Palffy-Muhoray (eds.): Spatio-Temporal Patterns in Nonequilibrium Complex Systems, SFI Studies in the Sciences of Complexity, Addison-Wesley, (1994). S. Nasuno, O. Sasaki, S. Kai, W. Zimmermann: Phys. Rev. A, 46, 4954 (1992) L Kramer, W. Zimmermann, E. Bodenschatz, W. Pesch, in: J.E. Wesfried, R. Brand, P. Manneville, G. Albinet, N. Boccara (eds.): Propagation in Systems far from Equilibrium, Springer, Berlin, (1988) E. Bodenschatz: PhD thesis, Universität Bayreuth, (1989) M. Kaiser: PhD thesis, Universität Bayreuth, (1992) W. Decker: PhD thesis, Universität Bayreuth, (1995) V. Steinberg: private communication. S. Sasa: Defect chaos in 2 d anisotropic systems, in: S. Kai (ed.): Pattern formation in complex dissipitative systems and Global Dynamics, World scientific, p. 336 (1992) S. Morris, E. Bodenschatz, D.S. Cannell, G. Ahlers: Phys. Rev. Lett., 71, 2026 (1993) M. Assenheimer, V. Steinberg: Phys. Rev. Lett., 70, 3888 (1993) W. Decker, W. Pesch, A. Weber: Phys. Rev. Lett., 73, 648 (1994) B.B. Plapp, E. Bodenschatz: Bull. APS., 40, 765 (1995) J. Gleeson, D. Todorovic-Marinic: Mol. Cryst. Liq. Cryst., 261, 327 (1995) A. Hertrich, W. Pesch, J. Gleeson: Europhys. Lett., 34, 417 (1996) A. Hertrich: PhD thesis, Universität Bayreuth, (1995) H. Riecke, L. Kramer: unpublished (1995) M. Low, B. Albert, J. Gollub: J. Fluid. Mech., 173, 253 (1986) P. Coullet: Phys. Rev. Lett., 56, 724 (1986) P. Coullet, P.Huerre: Physica D, 63, 123 (1986) W. Zimmermann, A. Ogawa, S. Kai, K. Kawasaki, T. Kawakatsu: Europhys. Lett., 24, 217 (1993) H. Riecke, J. Crawford, E. Knobloch: Phys. Rev. Lett., 61, 1942 (1988) D. Walgraef: Europhys.Lett., 7, 495 (1988) H. Riecke, M. Silber, L. Kramer: Phys.Rev., E 49, 4100 (1994) A. Lange, R. Müller, U. Behn: Z. Phys. B, 100, 477 (1996); U. Behn, A. Lange, T. John: preprint (1997) S. Kai, in: S. Moss, P. McClintock (eds.): Noise in Nonlinear Dynamical Systems, volume 3, Cambridge University Press, p. 22 (1989) U. Behn, A. Buka, N. Kloepper: unpublished (1995) L. Gil, J. Lega, J. Meunier: Phys. Rev. A, 41, 1138 (1990) S. Kai, M. Kohno, M. Andoh, M. Imizaki, W. Zimmermann: Mol. Cryst. Liq. Cryst., 198, 247 (1991) M. Sano, H. Kokubo, H. Janiaud, K. Sato: Prog. Theor. Phys., 90, 1 (1993) V. Delev, O. Scaldin, A. Chuvyrov: Liquid Crystals, 12, 441 (1992) A. Hertrich, W. Decker, W. Pesch, L. Kramer: J. Phys. (France), II 2, 1915 (1992) L. Kramer, A. Hertrich, W. Pesch: Electrohydrodynamic convection in nematics: the homeotropic case, in: S. Kai (ed.): Pattern formation in complex dissipitative systems and Global Dynamics, World scientific, p. 238 (1992)
293
13 139. 140. 141. 142. 143. 144. 145.
146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170. 171.
294
Convection Instabilities in Nematic Liquid Crystals
E. Guyon, P. Pieranski, J. Salan: J. Fluid Mech., 93, 65 (1979) J. Salan, E. Guyon: J. Fluid Mech., 126, 13 (1983) H. Richter, A. Buka, I. Rehberg: Phys. Rev. E, 51, 5886 (1995) H. Richter, N. Klöpper, A. Hertrich, A. Buka: Europhys. Lett., 30, 37 (1995) A. Hertrich, A.P. Krekhov, W. Pesch: J. Phys. (France), II 5, 773 (1995) V. Raghunathan, P.M. Murthy, N. Madhusudana: Mol. Cryst. Liq. Cryst. 199, 48 (1992) W. Zimmermann: On travelling waves in nematics, in: J.M. Coron, F. Helein, M. Ghidaglia (eds.): Nematics: Mathematical and Physical Aspects, Kluwer Academic Publishers, NATO ASI Series, Dordrecht, pages 401–426 (1991) K. Babcock, G. Ahlers, D. Cannell: Phys. Rev. Lett., 67, 3388 (1991) A. Tsameret, V. Steinberg: Europhys. Lett., 14, 331 (1991) S. Frunza, R. Moldovan, T. Beica, M. Giurgea, D. Stoenescu: Europhys. Lett., 20, 407 (1992) A. Hertrich, A. P. Krekhov, O. A. Scaldin: J. Phys. (France), II 4, 239 (1994) Z.A. Daya, S. Morris, J.R. de Bruyn, A. May: Phys. Rev. A, 44, 8146 (1991) S.S. Mao, J.R. de Bruyn, S.W. Morris: Phys. Rev. E, 54, R1048 (1996) Z.A. Daya, S.W. Morris, J.R. de Bruyn: Phys. Rev. E, 55, 2682 (1997); V.B. Deyirmenjian, Z.A. Daya, S. Morris: preprint (1997) A.G. Rossberg, A. Hertrich, L. Kramer, W. Pesch: Phys. Rev. Lett., 76, 4729 (1996) A.G. Rossberg, L. Kramer: Physica Scripta, T67, 121 (1996) S. Kai, K Hayashi, Y. Hidaka: J. Phys. Chem., 100, 19 007 (1996) Y. Hidaka, J.-H. Huh, K. Hayashi, M. Tribelski, and S. Kai, Phys. Rev. E56, R6256 (1997) A.G. Rossberg, L. Kramer: Physica D, 115, 19 (1998) E. Plaut, W. Decker, A. Rossberg, L. Kramer, W. Pesch: Phys. Rev. Lett., 79, 2367 (1997) M. Treiber, N. Eber, Á. Buka, L. Kramer: J. Phys. (France), II 7, 649 (1997) H.H. Graf, H. Kneppe, Schneider: Molecular Physics, 77, 521 (1992) M. Treiber: PhD thesis, Universität Bayreuth, (1996) M. Dennin, G. Ahlers, D.S. Cannell: Phys. Rev. Lett., 77, 2475 (1996) M. Dennin, G. Ahlers, D.S. Cannell: Science, 272, 388 (1997) M. Dennin, D.S. Cannell, G.Ahlers: Phys. Rev. E, 57, 638 (1998) M. Treiber, L. Kramer: Phys. Rev. E, 58, 1973 (1998) H. Richter, A. Buka, and I. Rehberg, in: P.E. Cladis and P. Palffy-Muhoray (eds.): Spatio-Temporal Patterns in nonequilibrium Complex Systems, Addison-Wesley Publishing Company, (1994) Ribotta, private communication S. Rasenat, private communication I. Rehberg, private communication V. Carbone, N. Scaramuzza, and C. Versace, Physica D, 106, 314 (1997) E. Plaut and W. Pesch
14
Preparation and Properties of Ionic and Surface Modified Micronetworks Michael Mirke, Ralf Grottenmüller, and Manfred Schmidt
14.1
Introduction
In recent years micronetworks have gained increasing interest in academic and industrial research. Like dendrimers microgels could principally serve as molecular containers in order to transport guest molecules (drug delivery) or as microscopic reaction sites for the formation of nanosized colloidal particles. In the present work we have a) investigated the preparation and surface modification of microgels via crosslinking reactions in microemulsion; b) utilized microgels as the core in core-shell and star-like structures; c) prepared and characterized ionic micronetworks.
14.2
Polymerization in normal microemulsion
14.2.1 Mechanism and size control Antonietti et al. [1] and Wu [2] have reported on the size control of the polymerized microemulsion in the system: styrene, di-isopropenylbenzene (cross-linker), cetyltrimethylammoniumchloride (CTMACl) or a similar surfactant, and water. According to a simple model developed by Wu [2] the resulting latex size is a function of the fleet ratio s = ms /mM with ms and mM, the masses of surfactant and monomer respectively, given by S
1
NA r a0 R C 3Ms
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties Deutsche Forschungsgemeinschaft (DFG) Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
1 295
14
Preparation and Properties of Ionic and Surface Modified Micronetworks
with NA the Avogadro number, Ms the molar mass of the surfactant, a0 the surface area per surfactant molecule, r the mean particle density, and the constant [3] C
NA r a0 2d 3Ms
1
2
with d the surfactant microgel interpenetration depth. Whereas Eq. 1 is almost perfectly confirmed by the experimental data of Antonietti et al. and Wu, the present data do not follow Eq. 1 and show much more scatter than literature results (Fig. 14.1). Variation of the ionic strength of the continuous phase, of the reaction temperature, of the total monomer content, and of the concentration of crosslinking agent has essentially no effect on the resulting microemulsion. 6 5
S-1
4 3 2 1 0 -1
0
5
10 15 20 25 30 35 40 45 50 55 60
Rh / nm
Figure 14.1: Inverse fleet ratio S –1 versus size Rh for AIBN initiated, polymerized latices: data from Ref. [1] (_), Ref. [2] (M), and present results (y).
As shown in Fig. 14.2 differences, however, were observed for different initiators, e. g. AIBN, redox system (K2S2O8 /K2S2O5), and dibenzylketone. Although much effort was taken to improve the reproducibility and size control we could never reproduce the literature data for AIBN initiation particularly at small S. We presently cannot offer a satisfactory explanation for this discrepancy. Concerning the mechanism of particle formation the polymerization starts in a two phase or in a monophasic region of the phase diagram depending on the total monomer concentration and on the fleet ratio [4]. The biphasic regime is easily detected by a bimodal decay of the time correlation function g1 (t) as recorded by a dynamic light scattering instrument whereas for the monophase an extremely fast, monomodal decay curve is observed. The hydrodynamic radius Rh, deducted from the measured diffusion coefficient, is less than 1 nm which is – within experimental error – identical to the radius of empty micelles formed by the pure surfactant in water at the respective concentration. It is tempting to interpret these results in terms of a bicontinuous monophase rather than in terms of monomer-filled microdroplets. Since similar investigations are currently performed in Hoffmann’s group [5], 296
14.2
Polymerization in normal microemulsion
6 5
S-1
4 3 2 1 0 -1
0
5
10 15 20 25 30 35 40 45 50 55 60
Rh / nm
–1
Figure 14.2: Inverse fleet ratio S versus size Rh for differently initiated polymerized lattices: AIBN (y), redox (o), and photo initiation by dibenzylketone ($).
we have not further persued the phase diagram of the unpolymerized emulsion but concentrated on the changes during polymerization. As shown in Fig. 14.3, the bimodal decay of the correlation function has merged into a single, almost monoexponential curve after 90 min reaction time, which does not change with monomer conversion any more. The final latex size is already reached after 90 minutes or at about 50 % monomer conversion! Again, this observation is not easily understood, because it means that polymerized particles do not grow beyond a certain size and new particles are formed even during the later course of reaction. Surface tension measurements as function of the reaction time and as function of the fleet ratio only show the well-known surface activity of styrene as compared to polystyrene, which is still significant at 50 % monomer conversion. At this point also the rate of conversion exhibits a maximum as function of time which is
1.0 0.5
g1 (t) 0.1
0
100
200
300
400 500 t/msec
600
700
800
900
Figure 14.3: Time correlation function g1 (t) for AIBN initiated latices at different polymerization times: 0 min (T), 30 min (y), 90 min (o), 150 min (M), and 2 days (6).
297
14
Preparation and Properties of Ionic and Surface Modified Micronetworks
usually interpreted in terms of a two stage mechanism. At the beginning of the polymerization more and more growing particles are formed until a certain number of particles is reached. Then the reaction rate decreases because the monomer concentration in the system decreases or the number of active polymerization sites is diminished. In view of the discussion above, the hypothesis of the constant number of growing particles appears questionable. Qualitatively similar results are obtained for emulsions which were initiated by light or redox initiators. Despite the extreme lack of understanding and reproducibility in most cases the resulting microemulsions were monodisperse and spherical. Therefore they are principally well suited as starting material for surface-functionalized spherical particles.
14.2.2 Surface functionalization of microgels A suitable functional group on the surface of the microgel could be an azo-moiety which can be utilized to start a grafting from reaction in order to synthesize core-shell or star-like structures. Here, we have taken advantage of the cosurfactant properties of the methacrylic acid ester (Fig. 14.4) which is simply added with DTMACl to the reaction mixture. During the polymerization the methacrylic group is bound into the microgel thus forming the pendent azo group. Since we wish to preserve the azo group during microgel formation the microemulsion was redox initiated. Redox initiation with the system K2S2O8 /K2S2O5 introduces ionic SO3– groups to each primary chain, thus creating an ionic micronetwork. In fact, increasing the initiator to monomer ratio results in an increased conductivity of the microgels dissolved in DMF. O O N
N
CN
CH3 CN
Figure 14.4: Cosurfactant containing the azo-moieties methacrylic acid [4-(1,1-dicyanoethyl)azo]benzylic ester.
The presence of ionic groups makes the particle characterization somewhat ambigious, since now interparticle interactions might seriously influence the interpretation of the light scattering data. One example is shown in Tab. 14.1, where the measured apparent radius of gyration Rg,app of the latex in water changes from negative values to about 17 nm with increasing salt concentration, whereas the hydrodynamic radius Rh remains approximately constant at about 8.5 nm. This large ratio of Rg /Rh = 2 is not compatible with the value of 0.775, expected for a spherical structure. Also, most of the redox-initiated structures do not dissolve in toluene after careful removal of the surfactant and the solution characterization was performed in DMF. Again, the experimental Rg /Rh values were larger than 1.5 indicating that no spherical structures were 298
14.3 Polymerization in inverse microemulsion Table 14.1: Influence of added salt on the light scattering results of redox polymerized latices. c (NaCl)/mol/l
Rg,app /nm
Rh,app /nm
Rg /Rh
0 4 610 –3 8 610 –3 2 610 –2 3.4610 –2 6 610 –2
–10.5 2680 > 51200 45.1 44.7
166 3180 2.80 2.78
[68] [68] unpublished unpublished
Ala-tRNAAla (E. coli) Ala-MinihelixAla (12 bp) Ala-MicrohelixAla (7 bp)
18.9 110 > 4040
5.82 214
unpublished unpublished unpublished
17.0 17.4 42.0 120 22.8 189 1230 1830
1.02 2.47 7.06 1.34 11.1 72.4 108
[72] unpublished unpublished unpublished unpublished unpublished unpublished unpublished
(AEDANS)Tyr-tRNATyr Tyr-tRNATyr Tyr-tRNATyr 2'dA Tyr-tRNATyr 3'dA Phe-tRNAPhe Phe-tRNAPhe 3'dA Phe-tRNAPhe 3'NH Phe-tRNAPhe oxi/red
Table 19.2 b: Variations of aminoacyl-tRNA using E. coli EF-Tu 7 GTP. When not indicated tRNAs originate from E. coli. Aminoacyl-tRNA Tyr-tRNATyr (AEDANS) Gln-tRNAGln Tyr-tRNATyr Phe-tRNAPhe AcPhe-tRNAPhe tRNAPhe tRNA (unfractionated) Ser-tRNASer (GCU) Ser-tRNASer (UGA) Ser-tRNASec Met-tRNAMet Met-tRNAfMet fMet-tRNAfMet Met-tRNAiMet (Yeast) Met-tRNAiMet (Yeast, ox) Thr-tRNAThr Trp-tRNATrp Glu-tRNAGlu Leu-tRNALeu (UAA) Leu-tRNALeu (GAG) Asp-tRNAAsp Leu-tRNALeu (CAA, E. coli) Val-tRNAVal (GAC) Val-tRNAVal (UAC) Leu-tRNALeu (CAG)
390
KD [10–10 M] 2.4 1.9 5.6 6.0 3800 26100 28200 7.0 7.2 500 7.3 173 1360 6000 110 7.7 13.5 17.0 22.3 29.7 31.2 33.8 36.2 47.1 64.1
Fold decrease
0.79 2.33 2.50 1580 10875 11750 2.92 3.0 208 3.04 72.1 567 2500 45.8 3.21 5.63 7.08 9.29 12.4 13.0 14.1 15.1 19.6 26.7
Reference [62] [64] [64] [62] [63] [63] [63] [64] [64] [65] [64] [63] [63] [66] [66] [64] [64] [64] [64] [64] [64] [64] [64] [64] [64]
19.3 Structure of elongation factor Tu arm does not disturb the interaction with EF-Tu [71]. Different aminoacyl-tRNA species interact with EF-Tu with different affinities. These sequence and aminoacyl-dependent variations are up to 64-fold (Tab. 19.2 b). Peptidyl-tRNAs and initiator Met-tRNAs interact only weakly with EF-Tu 7 GTP (Tab. 19.2 b).
19.3.6
H NMR of yeast Phe-tRNAPhe EF-Tu 7 GTP complex
1
For codon-specific binding of aminoacyl-tRNA to the ribosome the tRNA must form a complex with the elongation factor Tu. The complex formation manifests itself also in the imino resonance region of the 1H NMR spectrum of the tRNA. Since the molecular mass of tRNA (25,000 Da) is only about one third of the molecular mass of the aminoacyl-tRNA-EFTu 7 GTP complex (70,000 Da) the complex formation should become apparent by an increase of the line widths of the imino resonances. This was indeed observed with a complex of yeast Phe-tRNAPhe and EF-Tu 7 GTP from T. thermophilus (Fig. 19.14).
Figure 19.14: The imino proton resonances in the NMR spectrum of free Phe-tRNAPhe and in ternary complex with EF-Tu 7 GTP and EF-Tu 7 GDP [47]. a) 270 µM tRNAPhe, b)270 µM Phe-tRNAPhe EFTu 7 GTP ternary complex, c) the same complex as in (b) after GTP hydrolysis and deacylation of PhetRNAPhe. The dashed lines in (b) and (c) correspond to the 1H NMR imino proton spectrum of free tRNAPhe, as shown in (a). The numbers denote the imino proton resonances of the base pairs in the tRNAPhe acceptor stem as assigned by [74]. The numbers in circles indicate the signals of the first three imino protons in the acceptor stem, which disappear from the NMR spectrum after ternary complex formation.
391
19
Site-Directed Spectroscopy and Site-Directed Chemistry of Biopolymers
The spectra recorded immediately after complex formation, display resonance signals which are broadened by a factor of about 2.6 in comparison to free tRNA. In addition, it should be kept in mind that during the acquisition of the spectrum (about 1 hour), part of the GTP has already been cleaved to GDP [75]. In the GDP bound form, however, the affinity of EF-Tu for aminoacyl-tRNA is considerably lower than in the GTP bound form (Tab. 19.2 c).
Table 19.2 c: Variation of EF-Tu using Tyr-tRNATyr(AEDANS). EF-Tu (from T. thermophilus) EF-Tu.GTP EF-Tu.GDP EF-Tu.GTP (Thr62Ala) EF-Tu.GTP (Thr62Ser) EF-Tu.GTP (His67Ala) EF-Tu.GTP (His85Gln) EF-Tu.GTP (His85Leu) EF-Tu.GTP (Glu55Leu) EF-Tu.GTP (Glu56Ala) EF-Tu.GTP (Arg59Thr) EF-Tu 7 GTP 181–190 wild type (6His-tag) EF-Tu-Domain I EF-Tuf EF-Tu.GMPPCP EF-Tu.GMPPNP EF-Tuf.EF-Ts EF-Tu (from E. coli) EF-Tu.GTP EF-TuAR EF-TuBo Trypsin cleaved EF-Tu TPCK-treated EF-Tu
KD [10–10 M] 17.0 98000 618 31.7 60.9 260 54.6 16.3 49.2 36.4 24.0 82.3 >2120 35 82.0 98.9 50.6 KD [10–10 M] 2.4 37 56 54 2400
Fold decrease
Reference
5760 36.4 1.86 3.58 15.3 3.21 0.96 2.89 2.14 1.41 4.84 125 2.06 4.82 5.82 2.98
[72] [72] unpublished unpublished unpublished [54] [54] unpublished unpublished [47] unpublished unpublished unpublished [72] unpublished unpublished unpublished
Fold decrease
Reference
15.4 23.3 22.5 1000
[62] [73] [73] [73] [73]
The imino spectrum of the Phe-tRNAPhe-EF-Tu 7 GTP complex also reveals several distinct line shifts in comparison to the spectrum of the free tRNA. Moreover, some of the imino resonances, namely the ones of base pairs 1 through 3 of the acceptor stem, seem to be missing. However, it cannot be excluded that these lines overlap with other lines, due to larger shift changes. The assumption of vanishing imino resonances is also supported by the work of Heerschap et al. [76], who observed a disappearance of the imino resonances of the first acceptor stem base pairs upon complex formation of tRNAPhe with E. coli EF-Tu. As the ternary complex dissociates in the course of the GTP hydrolysis and the PhetRNAPhe deacylates, the line width of the imino resonances decreases again and the chemical shifts adopt the positions they had before in the free tRNA. However, even after com392
19.3 Structure of elongation factor Tu plete GTP hydrolysis and total deacylation the line widths do not fully recover the initial values of the spectra of free tRNAPhe, indicating a complex between tRNAPhe and EF-Tu 7 GDP (Fig. 19.14). Since the above-described NMR experiments have been undertaken at complex concentrations of about 0.3 mM, the still noticeable association of deacylated tRNA and EFTu 7 GDP represents no contradiction to the numerous biochemical studies which typically use concentrations below ca. 1 µM.
19.3.7
C NMR studies of the Val-tRNAVal EF-Tu 7 GTP ternary complex
13
E. coli Val-tRNAVal, 13C labelled at the carbonyl group, was used for 13C NMR analysis of the ternary complex formed by Val-tRNAVal and EF-Tu 7 GTP [47]. Labelling the carbonyl group appeared particularly favourable since it represents the carbon atom which functions as a linker between the ribose of the tRNA terminal adenosine76 and the amino acid residue. This linkage is the most important prerequisite for ternary complex formation (Tab. 19.2 b). Free Val-tRNA provides two peaks in the 13C NMR spectrum around 170 ppm corresponding to 2' and 3' isomers, with a slight preference for 3' derivative (Fig. 19.15 a). Complex formation of Val-tRNAVal with EF-Tu 7 GTP leads to a drastic upfield shift to a region around 64 ppm (Fig. 19.15 b). Such an extremely large upfield shift cannot be explained by conformational changes of the protein or isomerizations of Val-tRNA. Rather it requires the assumption of substantial alterations in the electronic environment of the carbonyl group. A change of the hybridization state from sp2 (carbonyl) to sp3 (orthoester) upon complex formation of Val-tRNAVal with EF-Tu 7 GTP is a conceivable explanation for the observed resonance shift.
Figure 19.15: Part of the proton-decoupled 125.7 MHz 13C NMR spectrum of a) E. coli [1-C-13C] ValtRNAVal. The resonances at 169.8 ppm and 169.5 ppm represent the 3' and 2' isomers of Val-tRNAVal, respectively; b) [1-C-13C] Val-tRNAVal EF-Tu 7 GDP/GTP. The resonances at 63.7 ppm and 63.5 ppm were suggested to belong to the GDP and GTP forms of the protein, respectively [75].
393
19
Site-Directed Spectroscopy and Site-Directed Chemistry of Biopolymers
Such an orthoester can be formed either with the oxygen atoms of both, the 2' and the 3'-hydroxyl groups of the A76 ribose (Fig. 19.16 a) or with a nucleophilic functional group of the protein (Fig. 19.16 b). The structure in Fig. 19.16 a seems to be reasonable as during the (relatively slow) transacylation reaction of the amino acid residue this intermediate state has to be passed [77]. The structure shown in Fig. 19.16 b would explain the observed chemical shift as well. In this case a nucleophile leading to an easily cleavable transient bond should be provided by the protein. A deprotonated carboxylate located in the interface between domains I and II of EF-Tu [1] is the most obvious candidate for such a function.
A
B
Figure 19.16: Orthoester acid intermediate structure of the aminoacyl residue in the Val-tRNAVal EFTu 7 GTP ternary complex. The function of nucleophile is fulfilled a) either by the 2'-OH group or b) by a functional group of the EF-Tu.
Evidently, EF-Tu stabilizes the orthoester intermediate possibly via electrostatic or hydrogen bond interactions. This could also be the reason for the strong upfield shift of this resonance, which is extraordinary as compared to the available model molecules. Moreover, a strong conformational constraint in the vicinity of the sp3 hybridized carbonyl group could contribute to this strong upfield shift. The relative intensities of the two lines at 63.7 ppm and 63.5 ppm are varying with time after complex formation. The 63.7 ppm line increases at the expense of the 63.5 ppm signal [47]. The disappearance of the 63.5 ppm resonance proceeds with the time constant of the GTP hydrolysis [3]. Thus, the two signals could be associated with two slightly differing conformations of the orthoester in the GTP and GDP forms of EF-Tu. This assumption is corroborated by the observation of only one single resonance at 63.7 ppm for a complex of [1-13C]ValtRNAVal with EF-Tu which has bound the slowly hydrolyzable GTP analogue GMPPNP [47]. Positional isomers (2' or 3') of aminoacyl-tRNA do not have a dramatic effect on the affinity of EF-Tu 7 GTP. However, the replacement of the ester bond (–O–CO–) for the amide bond (–NH–CO–) attachment of aminoacyl residue to tRNA strongly influences the affinity (Tab. 19.2 a). 394
19.3 Structure of elongation factor Tu
19.3.8 Role of EF-Tu in complex with aminoacyl-tRNA Domains II/III function as an effector promoting the binding of aminoacyl-tRNA. In the inactive GDP conformation these domains are tilted away from the domain I, forming a large cavity. This conformational change has considerable functional consequences, since EFTu 7 GDP interacts with aminoacyl-tRNA 5000 times less efficiently than EF-Tu 7 GTP. Thus it is conceivable to suggest that the binding site of the main determinant of the aminoacyltRNA binding, namely the aminoacyl-residue, is placed in the binding site which is formed by the EF-Tu 7 GDP to EF-Tu 7 GTP conformational change. Indeed it was demonstrated that the reactive e-bromoacetyl-lysyl-tRNALys labels specifically a histidine67 residue located near the interface of domain I and II in EF-Tu 7 GTP [78]. The X-ray structure determination of aminoacyl-tRNA 7 EF-Tu 7 GppNHp [79] and aminoacyl-AMP EF-Tu 7 GppNHp ternary complexes confirm the affinity labelling results and, in addition to histidine67, locate glutamate271 of domain II in the vicinity of aminoacyl-tRNA binding site. Aminoacyl residues 50–60 in the effector loop of EF-Tu Cys82 in domain I [64], arginine300 [54], and threonine393 [55] of domain III are other aminoacyl residues from the domain interfaces which are involved in aminoacyl-tRNA binding. Mutations in EF-Tu which affect GTPase activity (Thr62, His85) do not influence aminoacyl-tRNA binding (Tab. 19.2 c). The aminoacyl domain of tRNA composed of the CCA end, seven base pairs of the aminoacyl stem and five base pairs of the T stem, is sufficient to promote efficient binding to the EF-Tu 7 GTP [68]. About 10 base pairs of the aminoacyl domain are bound to domain III of EF-Tu [79]. This binding is probably governed by ionic interactions of the RNA-phosphate backbone. The sequence of nucleotides, however, also plays some role in this process. EF-Tu 7 GTP serves for aminoacyl-tRNA not only as a vehicle transporting it to the ribosome but in addition as a matrix setting the correct conformation of the L shaped molecule [80]. The precise distance between the anticodon loop and aminoacyl-residue is evidently required for aminoacyl-tRNA functioning during translation [50].
19.3.9 EF-Tu interaction with EF-Ts Elongation factor Ts forms stable complexes with EF-Tu and fulfills the role of the nucleotide exchange protein (NEP). It is not known how this protein binds to EF-Tu and how the acceleration of the GDP dissociation is achieved. However, it is clear that EF-Tu 7 EF-Ts interaction changes the structure of nucleotide binding domain. Such effects were identified especially in the L2 region of EF-Tu, which alters its protease accessibility upon EF-Tu 7 EFTs complex formation [81]. Replacement of the lysine in the NKXD-guanine binding region of E. coli EF-Tu for glutamate causes a dissociation of nucleotide and formation of stable EF-Tu 7 EF-Ts complexes. This effect can be reversed by a second mutation in EF-Tu which prevents EF-Tu 7 EF-Ts interaction. Such mutants were found by replacement of some amino acids in the C-terminal region of domain I [82] (residues 154–199). The binding of EF-Ts may dislocate the guanine binding SA consensus element (Fig. 19.2) leading to dissociation of the bound nucleotide. The interaction of transducin-b with rhodopsin takes place presum395
19
Site-Directed Spectroscopy and Site-Directed Chemistry of Biopolymers
ably also in the vicinity of the region of the GTP-binding domains [83]. In addition to domain I of EF-Tu, EF-Ts apparently interacts also with domains II or III. This is evident from investigation of EF-Tu species of domains I/II [84] and II/III [72].
19. 3. 10 Site-directed mutagenesis of EF-Tu An elegant way to achieve a site-directed replacement of one or more amino acids in the protein is provided by genetically induced mutations into particular genes followed by expression of the mutant protein. This method was extensively used in the investigations of T. thermophilus EF-Tu. A compilation of mutants is presented in Tab. 19.3. In many cases the replacement of invariant and functionally important amino acids leads to the loss of the
Table 19.3: Derivatives of EF-Tu, prepared by site-directed mutagenesis of the wild type T. thermophilus tufA gene variants, were overexpressed in E. coli, purified, and fully characterized. EF-Tu Mutant
Properties
References
Ribosome-induced GTPase affected Ribosome-induced GTPase affected Poly(Phe) synth. affected, binding of aa-tRNA affected Nucleotide binding is affected Nucleotide binding is affected, aa-tRNA binding is affected, Ribosome binding is defective aa-tRNA binding is affected Ribosome and aa-tRNA binding is affected Protein degradation; cell growth affected Slower GTPase; ribosome binding is affected Protein degradation As wild type Protein degradation aa-tRNA binding affected Ribosome induced GTPase affected
unpublished unpublished [47] unpublished
unpublished unpublished [54] [54] [54] [54] unpublished [54] unpublished unpublished
Protein degradation; normal cell growth aa-tRNA binding defective
unpublished unpublished
Thermostability decreases Thermostability decreases Higher intrinsic GTPase; thermostable No functional Thermostable; EF-Ts interaction Thermostability decreases; no ribosome interaction
[85] [85] [85] [85] [72] [85]
A) Single Mutants Glu55Leu Glu56Ala Arg59Thr Thr62Ser Thr62Ala
His67Ala His85Gln His85Gly His85Leu Asp81Ala Cys82Ala Arg300Ile EF-Tu(NHis6) EF-Tu (CHis6) B) Double Mutants His85Gly/Gly233Asp Cys82Ala/Thr394Cys C) Deletion Mutants 181–190 212–405 (Dom. I) 313–405 (Dom. I/II) 1–316 (Dom. III) 1–208 (Dom. II/III) 212–405/His85Gly
396
19.4 Summary and conclusions specific function of the protein. Thus the GTPase was affected by replacement of His85 or Thr62, the binding of aminoacyl-tRNA was impaired when Arg59 or Thr294 were mutated, and the stability of protein was decreased by mutations at Asp81 or Arg300. Remarkably, the deletion of domain III led to the stimulation of intrinsic GTPase. Some mutations were introduced in order to increase the stability of EF-Tu (Arg59) against proteolysis to improve the quality of crystals (His67) or to allow crystallisation of EF-Tu with affinity reagents and transition state analogues (Cys82). Many of these investigations are still in progress.
19.4
Summary and conclusions
The size or properties of the molecule often do not allow the use of direct physical or biochemical methods to study the structure and function of biological macromolecules. Even in the case of the tertiary structure determination by NMR or X-ray crystallography the structure dynamics, mechanism of catalytic reactions, and their regulation remain to be studied by site-specific methods. In the present work we concentrated our effort on the study of RNA structure, structure and mechanism of a regulated GTPase (represented by an elongation factor Tu) and on a protein-RNA complex (formed by aminoacyl-tRNA, EF-Tu, and GTP). Fluorescence spectroscopy (after specific labelling of RNA), electron microscopy (after attachment of a gold cluster to tRNA), NMR spectroscopy (using specifically introduced stable isotopes), ESR spectroscopy (with nitroxyl radicals bound to substrates of enzymatic reactions), affinity labelling of proteins (with reactive substrates), and site-directed mutagenesis in combination with the X-ray structure analysis of proteins were the utilized methods. The results of this investigation provided insight into the regulation and function of GTPases and allowed the discovery of some rules governing the RNA-protein interaction. Despite the large amount of new information which was gained from this research, most rewarding is the fact that we are now better prepared to ask new relevant questions for our future work.
Acknowledgement
The research laboratory in Bayreuth was supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 213, D4, and D5. We thank R. Hilgenfeld and J. Nyborg for cooperation on the study of EF-Tu 7 GppNHp structure, M. Daniel for help with preparation of the manuscript, and all co-workers who participated in the work. Their names can be found as co-authors in the referred work from our laboratory. 397
19
Site-Directed Spectroscopy and Site-Directed Chemistry of Biopolymers
References
1. H. Berchtold, L. Reshetnikova, C.O.A. Reiser, N.K. Schirmer, M. Sprinzl, R. Hilgenfeld: Nature, 365, 126 (1993) 2. M. Sprinzl: TIBS, 19, 245 (1994) 3. S. Limmer, C.O.A. Reiser, N.K. Schirmer, N.W. Grillenbeck, M. Sprinzl: Biochem., 31, 2970 (1992) 4. G. Varani, I. Tinoco Jr.: Q. Rev. Biophys., 24, 479 (1991) 5. S.A. White, M. Nilges, A. Huang, A.T. Brünger, P.B. Moore: Biochem., 31, 1610 (1992) 6. C. Cheong, G. Varani, I.J. Tinoco: Nature, 346, 680 (1990) 7. H.A. Heus, A. Pardi: Science, 253, 191 (1991) 8. M. Chastain, I.J. Tinoco: Biochem., 31, 12 733 (1992) 9. A.A. Szewczak, P.B. Moore: J. Mol. Biol., 247, 81 (1995) 10. F.H.T. Allain, G. Varani: J. Mol. Biol., 250, 333 (1995) 11. A. Heerschap, C.A.G. Haasnoot, C.W. Hilbers: Nucleic Acids Res., 10, 6981 (1982) 12. A. Heerschap, C.A.G. Haasnoot, C.W. Hilbers: Nucleic Acids Res., 11, 4483 (1983) 13. A. Heerschap, C.A.G. Haasnoot, C.W. Hilbers: Nucleic Acids Res., 11, 4501 (1983) 14. D.R. Hare, N.S. Ribeiro, D.E. Wemmer, B.R. Reid: Biochem., 24, 4300 (1985) 15. M. Sprinzl, F. Cramer: Proc. Natl. Acad. Sci. USA, 73, 3049 (1975) 16. P. Schimmel: Interpretation of experiments that delineate transfer RNA recognition in vivo and in vitro, in: F. Eckstein, D.M. Lilley (eds.): Nucleic acids and molecular biology, Springer, Berlin, p. 274 (1990) 17. L.H. Schulman: Progr. Nucl. Acids Res. Mol. Biol., 41, 23 (1991) 18. R. Giegé, J.D. Puglisi, C. Florentz: Progr. Nucl. Acids Res. Mol. Biol., 45, 129 (1993) 19. W.H, McClain: J. Mol. Biol., 234, 257 (1993) 20. Y.-M. Hou, P. Schimmel: Nature, 333, 140 (1988) 21. C. Francklyn, K. Musier-Forsyth, P. Schimmel: Eur. J. Biochem., 206, 315 (1992) 22. W.H. McClain, K. Foss: Science, 240, 793 (1988) 23. W.H. McClain, K. Foss: Science, 241, 1804 (1988) 24. H.W. McClain, Y.-M. Chen, K. Foss, J. Schneider: Science, 242, 1681 (1988) 25. C. Francklyn, P. Schimmel: Nature, 337, 478 (1989) 26. K. Musier-Forsyth, N. Usman, S. Scaringe, J. Doudna, R. Green, P. Schimmel: Science, 253, 784 (1991) 27. K. Musier-Forsyth, P. Schimmel: Nature, 357, 513 (1992) 28. M. Sprinzl, F. Cramer: Progr. Nucl. Acids Res. Mol. Biol., 22, 2 (1979) 29. K. Nagamatsu, Y. Miyazawa: J. Biochem., 94, 1967 (1983) 30. H. Paulsen, W. Wintermeyer: Eur. J. Biochem., 138, 117 (1984) 31. J.D. Puglisi, I.J. Tinoco: Methods Enzymol., 180, 304 (1989) 32. S. Limmer, H.-P. Hofmann, G. Ott, M. Sprinzl: Proc. Natl. Acad. Sci. USA, 90, 6199 (1993) 33. G. Ott, L. Arnold, S. Limmer: Nucleic Acids Res., 21, 5859–5864 (1993) 34. L. Arnold, J. Smrt, J. Zajicek, G. Ott, M. Schiesswohl, M. Sprinzl: Collect. Czech. Chem. Commun., 56, 1948 (1991) 35. G. Ott, L. Arnold, J. Smrt, M. Sobkowski, S. Limmer, H.-P. Hofmann, M. Sprinzl: Nucleosides & Nucleotides, 13, 1069 (1994) 36. C.W. Hilbers: Hydrogen-bonded proton exchange and its effect on NMR spectra of nucleic acids, in: R.G. Schulman (ed.): Biological applications of magnetic resonance, Academic Press, New York, p. 1 (1979) 37. J.L. Leroy, D. Broseta, N. Bolo, M. Guéron: NMR studies of base-pair kinetics of nucleic acids, in: P.H. van Knippenberg, C.W. Hilbers (eds.): Structure and dynamics of RNA, Plenum Press, New York, p. 31 (1986)
398
References 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60.
61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76.
D.B. Arter, P.G. Schmidt: Nucleic Acids Res., 3, 1437 (1976) S.R. Holbrook, C. Cheong, I. Tinoco, S.-H. Kim: Nature, 353, 579 (1991) O. Jardetzky, G.C.K. Roberts: in: NMR in molecular biology, Academic Press, Orlando, (1981) K. Wüthrich: in: NMR of proteins and nucleic acids, Wiley, New York, (1986) S.S. Wijmenga, M.M.W. Mooren, C.W. Hilbers: NMR of nucleic acids; from spectrum to structure, in: G.C.K. Roberts (ed.): NMR of Macromolecules, IRL Press, Oxford, p. 217, (1993) W.R. Croasmun, R.M.K. Carlson: in: Two-dimensional NMR spectroscopy, VCH, Weinheim, (1994) S. Limmer, B. Reif, G. Ott, L. Arnold, M. Sprinzl: FEBS Lett. 385, 15 (1996) R. Boelens, T.M.G. Koning, G.A. van der Marel, J.H. van Boom, R. Kaptein: J. Magn. Resonance, 82, 290 (1989) M. Taiji, S. Yokoyama, T. Miyazawa: Biochem., 22, 3220 (1983) C. Förster, S. Limmer, W. Zeidler, M. Sprinzl: Proc. Natl. Acad. Sci. USA, 91, 4254 (1994) Y. Kaziro: Biochim. Biophys. Acta, 505, 95 (1978) R.C. Thompson: TIBS, 13, 91 (1988) M.V. Rodnina, R. Fricke, L. Kuhn, W. Wintermeyer: EMBO J., 14, 2613 (1995) L. Seidler, M. Peter, F. Meissner, M. Sprinzl: Nucleic Acids Res., 15, 9263 (1987) M.V. Milburn, L. Tong, A.M. de Vos, A. Brünger, Z. Yanaizumi, S. Nishimura, S.-H. Kim: Science, 247, 939 (1990) M.S. Marshall: TIBS, 18, 250 (1993) W. Zeidler, C. Egle, S. Ribeiro, A. Wagner, V. Katunin, R. Kreutzer, M. Rodnina, W. Wintermeyer, M. Sprinzl: Eur. J. Biochem., 229, 596 (1995) C. Alexander, N. Bilgin, C. Lindschau, J.R. Mesters, B. Kraal, R. Hilgenfeld, V.A. Erdmann, C. Lippmann: J. Biol. Chem., 270, 14 541 (1995) M. Kjeldgaard, J. Nyborg: J. Mol. Biol., 223, 721 (1992) A. Wittinghofer, R. Frank, R. Leberman: Eur. J. Biochem., 108, 423 (1980) M.E. Peter, N.K. Schirmer, C.O.A. Reiser, M. Sprinzl: Biochem., 29, 2876 (1990) W. Zeidler, R. Kreutzer, M. Sprinzl: FEBS Lett., 319, 185 (1993) K.-W. Hwang, F. Jurnak, D.L. Miller: A mutation that hinders the GTP induced aminoacyl-tRNA binding of elongation factor Tu, in: L. Bosch, B. Kraal, A. Parmeggiani (eds.): Guanine-nucleotide binding proteins, Plenum Press, New York, , p. 77 (1989) A. Louie, F. Jurnak: Biochem., 24, 6433–6439 (1985) G. Ott, H.G. Faulhammer, M. Sprinzl: Eur. J. Biochem., 184, 345 (1989) F. Janiak, V.A. Dell, J.K. Abrahamson, B.S: Watson, D.L. Miller, D.L., A.E. Johnson: Biochem., 29, 4268 (1990) G. Ott, M. Schiesswohl, S. Kiesewetter, C. Förster, L. Arnold, V.A. Erdmann, M. Sprinzl: Biochim. Biophys. Acta, 1050, 222 (1990) C. Förster, G. Ott, K. Forchhammer, M. Sprinzl: Nucleic Acids Res., 18, 487 (1990) S. Kiesewetter, G. Ott, M. Sprinzl: Nucleic Acids Res., 18, 4677 (1990) B. Blechschmidt, V. Shirokov, M. Sprinzl: Eur. J. Biochem., 219, 65 (1994) J. Rudinger, B. Blechschmidt, S. Ribeiro, M. Sprinzl: Biochem., 33, 5682 (1994) J.P. Abrahams, J.J.C. Acampo, G. Ott, M. Sprinzl, J.M. de Graaf, A. Talens, B. Kraal: Biophys. Acta, 1050, 226 (1990) K. Bench, U. Pieper, G. Ott, N. Schirmer, M. Sprinzl, A. Pingoud: Biochimie, 73, 1045 (1991) R.L. Joshi, H. Faulhammer, F. Chapeville, M. Sprinzl, A.-L. Haenni: Nucleic Acids Res., 12, 7467 (1984) M.E. Peter, C.O.A. Reiser, N.K. Schirmer, T. Kiefhaber, G. Ott, N.W. Grillenbeck, M. Sprinzl: Nucleic Acids Res., 18, 6889 (1990) G. Ott, J. Jonak, I.P. Abrahams, M. Sprinzl: Nucleic Acids Res., 18, 437 (1990) A. Heerschap, J.-R. Mellema, H.G.J.M. Janssen, J.A.L.I. Walters, C.A.G. Haasnoot, C.W. Hilbers: Eur. J. Biochem., 149, 649 (1985) C. Förster, S. Limmer, S. Ribeiro, R. Hilgenfeld, M. Sprinzl: Biochimie, 75, 1159 (1993) A. Heerschap, A.L.I. Walters, J.R. Mellema, C.W. Hilbers: Biochem., 25, 2707 (1986)
399
19
Site-Directed Spectroscopy and Site-Directed Chemistry of Biopolymers
77. S. Chládek, M. Sprinzl: Angew. Chem. Int. Ed., 24, 371 (1985) 78. L.K. Duffy, L. Gerber, A.E. Johnson, D.L. Miller: Biochem., 20, 4663 (1981) 79. P. Nissen, M. Kjeldgaard, S. Thirup, G. Polekhina, L. Reshetnikova, B.F.C. Clark, J. Nyborg: The three-dimensional structure of the ternary complex. 16th International Workshop, May 27–June 1, 1995, University of Wisconsin-Madison, Abstract, 51 (1995) 80. H.J. Adkins, D.L. Miller, A.E. Johnson: Biochem., 22, 1208 (1983) 81. N.K. Schirmer, C.O.A. Reiser, M. Sprinzl: Eur. J. Biochem., 200, 295 (1991) 82. Y.-W. Hwang, M. Carter, D.L. Miller: J. Biol. Chem., 267, 22 198 (1992) 83. J.P. Noel, H.E. Hamm, P.B. Sigler: Nature, 366, 654 (1993) 84. S. Nock, A. Lechler, S. Ribeiro, R. Kreutzer: Biotechniques, 18, 608 (1995) 85. S. Nock, N. Grillenbeck, M.R. Ahmadian, S. Ribeiro, R. Kreutzer M. Sprinzl: Eur. J. Biochem., 132 (1995)
400
20
Spectroscopic Probes of Surfactant Systems and Biopolymers Alexander Wokaun
20.1
Introduction
Characterization of surfactant systems is a challenging task because these systems, which often represent stable and homogeneous phases on the macroscopic scale, are typically consisting of microscopic or mesoscopic aggregates, such as micelles. Tremendous advances in the understanding of these systems has been made through direct visualization by freeze-fracture electron microscopy and by the application of scattering techniques rendering a wealth of information on aggregate shapes and sizes. In order to investigate the microscopic properties of the individual aggregates or compartments constituting the surfactant systems toposelective methods of spectroscopy have been applied in this study. Focusing on the microscopic dynamics, three complementary methods will be discussed. First, the self-diffusion coefficients of individual components have been distinguished by FT-NMR pulsed gradient spin echo experiments. This technique has been applied to characterize the diffusion of micelles with solubilized hydrocarbons and residual mobilities in cubic phases (ringing gels) that are formed by aggregation of swollen micelles. An alternative method for tracing the motion of micellar aggregates is their selective labelling by photochromic dye probes. Diffusion coefficients are determined by monitoring the time dependence of laser induced gratings using the forced Rayleigh scattering (FRS) technique. In this way, the influence of alcohol content on the diffusion of a system forming multilamellar vesicles has been investigated. In a further study, the influence of solubilized water on the diffusion of reverse micelles in the AOT/octanol/water system and diffusion in the corresponding bicontinuous gel phase have been studied. A fascinating aspect of surfactant systems resides in the fact that within the same chemical system one-dimensional aggregates (rod-like micelles), two-dimensional bilayers, and space-filling continuous structures may be formed. This aspect of dimensionality has been probed by studying the time-dependence of fluorescence, as induced by lipophilic quencher molecules diffusing within dimensionally restricted regions of space. The second part of this study was devoted to the spectroscopic probing of biopolymers. Transitions between right and left-helical conformations of oligonucleotide duplexes were monitored by Raman and 2D NMR techniques. Macromolecular Systems: Microscopic Interactions and Macroscopic Properties Deutsche Forschungsgemeinschaft (DFG) Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
401
20
Spectroscopic Probes of Surfactant Systems and Biopolymers
The conditions for the applicability of surface enhanced Raman spectroscopy for studies of oligo and polynucleotides has been assessed. In particular, surface enhanced resonance Raman spectroscopy (SERRS) has been used to study the binding of chromophores to DNA strands at probe concentrations as low as 10–6 M.
20.2
Diffusion in surfactant systems
Three different spectroscopic techniques have been used to determine molecular and micellar diffusion coefficients in surfactant systems. Each of these methods is probing a different aspect of molecular mobilities. First, the NMR pulsed gradient spin echo (PGSE) method is used to determine composition-induced changes in the diffusion coefficients of micelles and inverse micelles in ternary surfactant systems. In particular, differences between the diffusion coefficients in micellar and cubic phases are monitored. Second, the diffusion of absorbing dye probes in light-induced gratings is monitored by the method of FRS, providing information on the structure of multilamellar vesicles. Third, the dynamics of fluorescent probes due to diffusing quenchers is investigated, rendering information on the dimensionality of diffusion pathways and hence on the structure of lyotropic mesophases.
20.2.1 Structural characteristics of micellar solutions, cubic phases, and multilamellar vesicles from NMR self-diffusion measurements Ternary systems consisting of surfactant, hydrocarbon and water exhibit a rich variety of phases. With increasing fraction of the hydrophobic components, frequently a sequence of structural changes of the type micelles ) hexagonal phase ) lamellar phase ) inverse hexagonal phase ) inverse micelles is observed [1]. Additional to the mentioned phases, the occurrence of cubic phases has recently been discovered [2, 3] which include both bicontinuous structures [4] and cubic or disordered packages of micellar aggregates. Of particular interest are the so-called ringing gel phases [5], which exhibit a metallic sound upon mechanical excitation, and have been extensively investigated within the Sonderforschungsbereich 213. In our experiments, we have focused attention on a homologous series of ternary systems consisting of alkyl-dimethylaminoxide surfactants, CnH2n+1(CH3)2N+O– (CnDMAO), a linear hydrocarbon (CmH2m+2) or a cyclic hydrocarbon (CmH2m), and water as the solvent. A systematic investigation of the concentration dependence in the system C14DMAO/C6H12 / 402
20.2
Diffusion in surfactant systems
H2O [6] showed that water constituted the continuous phase in both, the micellar solution and the ringing gel phase. In the latter, the diffusion coefficients of hydrocarbon and surfactant were lower by one and two orders of magnitude, respectively, as compared to the micellar solution. Thus, we are dealing with an aggregated network where the residual mobilities are caused by exchange of surfactant and solute molecules between neighbouring micelles in the gel. In order to investigate whether the mentioned results are generic properties of the mentioned class of ringing gels, a series of micellar solutions and of gel samples was prepared where the water weight fraction was fixed to 70 % and 57.50 %, respectively, varying the surfactant and hydrocarbon chain lengths. Self-diffusion coefficients have been determined by the stimulated echo experiment described by Tanner [7]. Details of the experiment as well as the exact composition of the samples with the corresponding phase diagrams of the ternary surfactant systems have been given recently [8]. The influence of the surfactant chain length on the self-diffusion of the solvent (here D2O) is shown in Fig. 20.1. For both, the micellar phase (rhombs) and the cubic phase (squares), the solvent diffusion coefficient is seen to rise with the alkyl chain length of the surfactant. In this series, the hydrodynamic radius of the micelles is known to increase with the surfactant length. Hence, at constant weight fraction of the micelles, the system C16DMAO/C6H12 /D2O contains fewer but larger micelles, as compared to C12DMAO/ C6H12 /D2O. As these micelles may be considered as obstacles for the diffusion of water the observed trend is then easily interpreted in terms of a standard model [9].
Figure 20.1: Diffusion coefficient of water in ternary CnDMAO/C6H12 /D2O systems, as a function of the surfactant chain length n. Rhombs: micellar phase; squares: ringing gel phase. Details of the compositions are given by Panitz et al. [8].
The self-diffusion coefficient of the solubilized hydrocarbon was determined by Fourier transformation of the second half-echo [8]. In the micellar phase an interesting trend is observed (Fig. 20.2): the diffusion coefficient first increases with the hydrocarbon chain length up to m = 10, then a pronounced decrease for the larger dodecane and tetradecane solutes follows. Light scattering experiments by Oetter and Hoffmann [5] have shown that the hydrodynamic radius of the micelles decreases (together with the total capacity for solubilization) 403
20
Spectroscopic Probes of Surfactant Systems and Biopolymers
Figure 20.2: Influence of the hydrocarbon chain length m on the micellar self-diffusion coefficient D in ternary C14DMAO/CmH2m+2 /D2O surfactant systems. For m = 6, cyclohexane C6H12 has been chosen as the solute.
as the chain length of the hydrocarbon solute increases from C6H12 to C10H22. This result agrees with the increase in the diffusion coefficient for the same series in Fig. 20.2. In order to account for the subsequent decrease in the diffusion coefficient for longer hydrocarbon chains we have to recall that the C14DMAO surfactant by itself forms rod-like micelles in the binary solution in water. Due to the addition of a critical amount of hydrocarbon these rods are transformed into spherical micelles. As mentioned above, the solubilization capacity is rapidly decreasing with the size of the solute. Thus, for very long hydrocarbons the solubility limit is reached prior to induction of the rod ) sphere transition. In fact, for the C14DMAO/C14H30 /D2O system the amount of solubilized hydrocarbon in the investigated sample was only slightly above the critical concentration required for the mentioned transition. In such a situation small dynamic deviations from an average spherical shape of the micelles are expected to occur. Thus a lower diffusion coefficient is observed for this system – and to a lesser extent also for the homologous system containing C12H26 – which is attributed to shape fluctuations of the micelles [8]. The NMR investigations have been complemented by studying the solvent diffusion in multilamellar vesicles in a related system. When the ionic surfactant C14H29N+(CH3)3Br– (CTAB) and the cosurfactant n-C6H13OH are added to the above-mentioned C14DMAO/ H2O system vesicles and lamellar phases are formed [10]. The aim of this study was to complement the rheological measurements in these systems, where a decrease of both, the storage modulus G' and the yield stress, with increasing salinity of the water had been found. Results are shown in Fig. 20.3 for three systems containing NaCl (c = 0, 10, 100 mM). The measured signal may be attributed to extravesicular water. The diffusion coefficient of the solvent D2O increases by about 50 % upon the addition of NaCl (c = 10 mM), in agreement with a lowering of the yield stress observed in the rheological measurements [10]. Using the equation DH2O = Do (1 – Fv), where Do is the diffusion coefficient of free water and Fv is the volume fraction of the micelles, an increase of DH2O appears to be accompanied by a decrease in Fv. Assuming a constant total surface area, a decrease of Fv must be due to deformation of the vesicles from the maximum volume spheres. 404
20.2
Diffusion in surfactant systems
Figure 20.3: Diffusion coefficient of the solvent D2O in a system consisting of C14DMAO (c = 90 mM), CTAB (c = 10 mM), and n-C6H13OH (c = 220 mM), which forms multilamellar vesicles. The dependence on diffusion time D is investigated. Circles: no salt added; rhombs: NaCl (c = 10 mM); squares: NaCl (c = 100 mM).
Increasing the NaCl concentration to 100 mM gives rise to a further strong increase in the diffusion coefficient. Interestingly, the apparent coefficient depends now on the diffusion time set by the experiment, i. e. the time t between the two gradient pulses. Such a dependence is standardly interpreted as being due to restricted diffusion. Both, the absolute increase and the diffusion time dependence, hint to a further deformation of the multilamellar vesicles into planar lamellar stacks or aggregates, which are partially ordered within the NMR tube.
20.2.2 Probing of mobilities in multilamellar vesicles by forced Rayleigh scattering Recently, increasing use has been made of laser-induced gratings for visualizing static and dynamic properties of gases and condensed phases [11]. In FRS two coherent laser beams are made to interfere in the volume of the sample. In our case that was a surfactant solution doped with a chromophoric dye probe. Excitation of the absorber molecules in the regions of high light intensities of the interference pattern gives rise to a spatial modulation of both, the absorption and of the refractive index of the sample. The lattice constant L of this grating is determined by the wavelength l and by the angle y between the two beams (Fig. 20.4), L l=2 sin
y=2 :
1
The hologram induced in this manner may be probed by a second laser of wavelength l', which is diffracted by the grating (Fig. 20.4). For thick holograms it is essential that the grating is probed at an angle j for which the Bragg condition (Eq. 2) for diffraction on the planes within the sample is fulfilled (Fig. 20.4), 2 L sinj m l0
2 405
20
Spectroscopic Probes of Surfactant Systems and Biopolymers
ka q
ϕ
kb Λ
Figure 20.4: FRS experimental setup. Beam (1) is the writing laser gated by a Kerr cell (5) between crossed polarizers (4, 7). The probe laser (10) is diffracted by the grating in the sample (11); the diffracted order is isolated by an iris diaphragm (13), and recorded by a pin diode (15). The Bragg condition for reading out the grating in a thick hologram is illustrated on the right-hand side.
In order to monitor the diffusion of the probe molecules the sample is illuminated for a short time interval gating the laser with a Kerr cell (Fig. 20.4). The length of the excitation pulse must be carefully matched to the intrinsic relaxation time of the photochromic probe. After terminatio of excitation the grating is washed out by diffusion of both, excited and unexcited dye molecules. Hence, the diffraction efficiency of the grating is diminished and the intensity I (t) of the diffracted probe laser beam decreases according to the equation I
t
A e
t=s
B2 C 2 :
3
The decay constant s is influenced by both, the intrinsic relaxation time sdye of the probe (i. e. the time constant of relaxation of the photoisomer produced) and the diffusion constant D, according to the equation s1 sdye1 D q2 ;
4
where q = 2 p/L. Hence, if the grating constant L is varied via the angle of intersection y (Fig. 20.4), a plot of s –1 vs. q2 will yield the desired diffusion coefficient as the slope. This procedure is illustrated in Fig. 20.5, using as an example the diffusion of the hydrophobic dye methyl red (2-carbohydroxy-4'-dimethylamino-azobenzene) in simple linear chain aliphatic alcohols. This photochromic probe molecule is transformed from the trans into the cis form by irradiation with the Ar+ laser wavelength of 514.5 nm. The lifetime sdye varies between about 20 ms and 1 s, depending on the medium and its viscosity. The dependence of the diffusion coefficient on the viscosity is clearly seen from the different slopes in Fig. 20.5. The power of the method will be illustrated by two applications. First, we investigate the influence of the alcohol content on the quaternary system for which the formation of multilamellar vesicles has been discovered by Hoffmann et al. [10]. Second, diffusion in the water-in-oil microemulsion and in the bicontinuous cubic gel phase of the AOT/octanol/ water system is investigated. The multilamellar vesicle system was investigated without the addition of salt. The surfactant mixture was held constant at 90 mM C14DMAO and 10 mM C14H29N+(CH3)3Br–, 406
20.2
Diffusion in surfactant systems
Figure 20.5: Evaluation of diffusion coefficients from angle-dependent FRS measurements. The grating decay rate s–1 is plotted against q2 (Eq. 4). The diffusion of methyl red in simple linear chain alcohols was investigated.
with the hexanol (C6OH) concentration being varied. Methyl red was added in a concentration of 7610–5 M. Polarization microscopy clearly shows [12] that the dye molecules are attached to the vesicles and hence are probing diffusion processes within the vesicles. The results presented in Fig. 20.6 comprise diffusion measurements both, in the micellar L1 phase (c (C6OH) = 0, 30, and 60 mM) and in the La phase (c (C6OH) = 150 mM). The diffusion coefficient is largest for c (C6OH) = 0 M; it drops rapidly upon the addition of 30 mM of hexanol as rod-like micelles are formed. Interestingly, the diffusion coefficient increases again if the alcohol concentration is doubled. This behaviour has been successfully interpreted [12] by noting that increasing the hexanol concentration from 30 to 60 mM causes a shortening of the rod-like micelles, which according to the theory of Doi and Edwards [13] gives rise to an increase in the diffusion coefficient.
Figure 20.6: Influence of the hexanol concentration on the diffusion coefficient of methyl red in the C14DMAO/CTAB/C6H13OH system (see text).
407
20
Spectroscopic Probes of Surfactant Systems and Biopolymers
In the La phase (c (C6OH) > 150 mM in Fig. 20.6), the diffusion coefficient of methyl red is found to be low and approximately constant. Freeze-fracture microscopy has revealed that the system consists of densely packed multilamellar vesicles (Hoffmann 1995). Their average size (1 µm) is smaller than the spacing of the light-induced grating (15–25 µm), but some large vesicles with sizes up to 30 µm are present as well. Hence, the observed decay of the FRS signal is due to the exchange of dye molecules between surfactant bilayers on the one hand and due to diffusion within giant vesicles on the other hand. Variation of the alcohol content within the La phase changes the number of the vesicles but does not alter the qualitative nature of the diffusional processes, thus understanding the constancy of the measured diffusion coefficient. The section is concluded by reporting results for the AOT/octanol/water system. The water-soluble dye congo red – a structurally related photochromic azo dye [12] – has been used as a probe. Compositions corresponding to two cross sections through the ternary phase diagram [14] are investigated. Details are given by Hahn et al. [12]. First, a series of four samples within the region of water-in-oil (w/o) microemulsion phase L2 was investigated (Fig. 20.7 a). The diffusion constant of the probe (residing within the water phase) is seen to decrease strongly with the water content, to stay constant above a water concentration of 50 wt%. This behaviour is interpreted in terms of the structural models of Scriven [15] and of Fontell [1]. The microemulsion is thought to undergo continuous
a)
b)
Figure 20.7: Diffusion coefficients in the AOT/octanol/water system determined by FRS. a) variation of the water content within the L2 phase; b) transition from the L2 into the I2 phase induced by decreasing the octanol weight fraction.
408
20.2
Diffusion in surfactant systems
structural changes with increasing water content, i. e. swelling of inverse rod-like micelles and formation of long narrow water channels within which the diffusion of the dye probe would be strongly impeded. As mentioned above, the spacing of the light-induced grating amounts to 15–25 µm. Over such distances, diffusion within the water channels may be excluded. Thus, the observed decay is ascribed to motion of the aggregates as a whole, which is slowed down as the channels are swelling due to the addition of water. The transition from the L2 into the cubic I2 phase was investigated by varying the octanol content (Fig. 20.7 b). The sample containing 20 wt% octanol corresponds to an L2 phase, whereas the three samples with 10 and 14 wt% of octanol, respectively, are cubic gels (I2 phase). The diffusion coefficient is distinctly higher in the gels, however, the increase with respect to the w/o microemulsion appears to be a gradual one, rather than a stepwise change. The highest value recorded (5610 –11 m2 s –1) is still four times smaller than the diffusion coefficient of congo red in water. The observed behaviour would not be consistent with the model of a gel consisting of closely packed, aggregated micelles. Rather, it suggests a bicontinuous structure of the gel, in which water channels are opening up with decreasing octanol content. This result is in agreement with the characterization of the gel by Gradzielski et al. [16], who found that the I2 phase is characterized by higher long-range order. The mentioned description is also consistent with the structural models developed by Fontell [1] for related systems.
20.2.3 Dimensionality of diffusion in lyotropic mesophases from fluorescence quenching Fluorescence quenching has been successfully used in surfactant research to determine aggregation numbers in micellar systems [17, 18]. Fluorescence quenching in a homogeneous phase leads to the well-known monoexponential decays but not if quencher diffusion is limited to restricted regions in space (e. g. the lipophilic regions in a surfactant system). It has early been realized [19] that the precise shape of the time dependence could be used to determine the dimensionality of the space available for diffusion. The probability for diffusive encounter between a fluorescent molecule and a given quencher decreases exponentially with time for three-dimensional diffusion. Whereas the integrated probability of encounter approaches unity in one or two dimensions the time dependence is distinctly different for each case. The aim of the present study was to demonstrate this concept within a single surfactant system forming both, one-dimensional aggregates (rod-like micelles) and a two-dimensional a-lamellar phase. The C14DMAO/C7H15OH/H2O system was chosen for this purpose because it forms in addition to the above-mentioned phases two distinct L3 mesophases of different viscosity [20]. The structure of these cubic phases has been the subject of intense investigations. Freeze fracture electron microscopy demonstrated their bicontinuous character [21]. In our study, the C14DMAO concentration was held constant at 100 mM, while the heptanol concentration was varied between 0 and 200 mM in order to realize compositions that correspond to the mentioned phases. With view to comparability of earlier studies in rod-like micelles [22], pyrene (c = 10 –5 M) was chosen as the fluorescent probe and benzo409
20
Spectroscopic Probes of Surfactant Systems and Biopolymers
phenone as the quencher diffusing exclusively within the lipophilic regions of the surfactant system. Details of the experiment and of the data reduction procedure have been described in the thesis work of Wiesner [23] and Meyer [24]. Experimental results for a micellar phase consisting of rod-like micelles and of an alamellar phase in the same system are compared in Fig. 20.8. Fluorescence decay curves are presented as logarithmic plots. The deviations from exponentiality in the micellar system (Fig. 20.8 a) are evident as all traces are curved even for long times except for the system containing no quencher (top trace). Evidently, the decay is faster in the a-lamellar phase (Fig. 20.8 b) when comparing runs at the same quencher concentration. This result meets the intuitive expectation that the approach of quenchers from all sides within a lamellar plane should give rise to a faster decay as compared to one-dimensional random walk within a cylindrical or rod-like micelle.
a)
b)
Figure 20.8: Fluorescence decay of pyrene in the C14DMAO/C7H15OH/H2O system. a) c (C7H15OH) = 0 M, rod-like micelles; benzophenone quencher (top to bottom: c = 0, 0.1, 0.3, 0.5, 0.7 mM). b) c (C7H15OH) = 62 mM, a-lamellar phase; benzophenone quencher (top to bottom: c = 0, 0.1, 0.3, 0.5, 0.7 mM).
Corresponding representative results for the L3 phase (Fig. 20.9) show again a distinctly non-exponential decay. In view of the poorer signal-to-noise ratio, obtained with this sample, an extensive data set was acquired with this sample. For the analysis the analytic result of Almgren et al. [22], of Alsins and Almgren [25] for 1D diffusion, and the model of Owen [19] for 2D diffusion have been used. Their equations were implemented in a fitting routine which performed, for a given sample, a global analysis of the entire data set recorded at different quencher concentrations. Prior to deriving values of the diffusion coefficients, it was checked whether the quality of the non-exponential decays recorded was sufficient to warrant a distinction of one and two-dimensional diffusion. In fact it was found that the use of the inappropriate model (e. g. 2D diffusion for the rod-like micelles) resulted in considerably higher mean square deviations, thus demonstrating the significance of the fits. Technically, the intrinsic fluorescence decay time of pyrene was derived from the curve without added quencher, the quenching rate constant kq was adjusted as a global parameter and an average diffusion constant was then derived using the known quencher concentrations [24]. 410
20.2
Diffusion in surfactant systems
Figure 20.9: Fluorescence decay of pyrene in the L3 phase of the C14DMAO/H2O system (c (C7H15OH) = 190 mM). Temperature was held constant at 25 8C. Concentration of benzophenone quencher (top to bottom: c = 0, 0.3, 0.6, 0.67, 0.75, 0.83, 1.0, 1.33 mM).
The selection of the appropriate model for the L3 phase deserves some comment. According to the accepted model [21,26], it consists of a bicontinuous structure featuring inner and outer water channels separated by surfactant bilayers. An idealized structure resembling a Gaussian minimal surface with positive and negative curvature at each point has been proposed [27]. The real structure probably corresponds to a multiply branched, irregular analogue of the Gaussian surface. The decisive feature for the present analysis is that the surfactant molecules are arranged in a (positively and negatively curved) surface in space, such that the application of the 2D model of diffusion is appropriate, see below. Results for the diffusion coefficients derived from the fits are compiled in Tab. 20.1. Surveying the results we see that the diffusion coefficient of the benzophenone quencher is 2–3 times higher in the two investigated lamellar phases than in the micellar phases. At first sight, this result seems to contradict the higher viscosity of the a-lamellar phase. However, we have to consider that we are dealing with the motion of the comparatively large benzophenone molecule, containing two phenyl rings, within the lipophilic region consisting of the surfactant alkyl chains. A simple model suggests that a preferred orientation of the plane of the benzophenone molecule is perpendicular to the rod axis and that diffusional motion parallel to the rod axis is severely impeded. In contrast, within a surfactant bilayer there are many more ways for the translational motion of benzophenone within the lipid region, so that the higher diffusion coefficient obtained for the latter phase may be understood. Most interesting is the result that the diffusion coefficient in the L3 phase, 6.7610–11 m2 –1 s , is significantly smaller than the one in the a-lamellar phase, although we have argued that diffusion in the L3 phase should be essentially two-dimensional. Resolution of this apparent contradiction comes from a recent theoretical analysis by Anderson and Wennerström [4] who showed that the diffusion coefficient of a random walker on a Gaussian minimum surface should be only about 2/3 of its value on a plane. In view of this result, which strictly applies only for an idealized regular structure, the results in Table 20.1 are fully supporting the presently accepted bicontinuous model of the L3 phase. 411
20
Spectroscopic Probes of Surfactant Systems and Biopolymers
Table 20.1: Quencher diffusion coefficients in the C14DMAO/C7H15OH/H2O system. C7H15OH concentration/mM
Diffusion coefficient D/10 –11 m2 s–1
0
4.9 ± 2.2
20
3.0 ± 0.9
62
12.4 ± 5.5
170
11.6 ± 2.4
190
6.7 ± 3.4
Phase and model rod-like micelles 1D diffusion rod-like micelles 1D diffusion lamellar phase 2D diffusion lamellar phase 2D diffusion L3 phase modified 2D diffusion
20.2.4 Summary of results In Section 20.2 the use of three spectroscopic methods has been exemplified for obtaining information on diffusion coefficients in surfactant systems. Regarding the micellar phases both, NMR self-diffusion measurements and FRS, have been successfully used to monitor changes in the micellar diffusion coefficients resulting from size and shape changes. This information complements the results from static and dynamic light scattering experiments in a most valuable manner. Two structurally different types of cubic phases have been clearly distinguished by the diffusion measurements. In the CnDMAO/hydrocarbon/water ringing gels which consist of aggregated micelles, residual mobilities resulting from exchange processes between the micelles have been characterized by NMR. On the other hand, the increase in long-range order and the opening of water channels in the bicontinuous AOT/octanol/water gels was evidenced by the increased diffusion coefficient of the water soluble congo red dye probe, monitored by FRS. For the quaternary C14DMAO/CTAB/hexanol/water system NMR was used to investigate salt-induced changes in the volume fraction occupied by the multilamellar vesicles, as well as deformation and shape alterations. In the same system, FRS yielded information on changes in diffusion coefficients upon the transition from rod-like micelles to the vesicular phase, induced by increasing the alcohol concentration. For the C14DMAO/heptanol/water system monitoring the time dependence of fluorescence due to diffusing quenchers resulted in a clear distinction of one and two-dimensional diffusion. For the L3 phase an experimental confirmation of a recent theoretical model for diffusion on a Gaussian minimal surface has been achieved.
412
20.3 Vibrational spectroscopy and conformational analysis of oligonucleotides
20.3
Vibrational spectroscopy and conformational analysis of oligonucleotides
Three projects have been pursued, with the aim of providing spectroscopic tools for the research of Section 19. First, conformational changes in oligodeoxyribonucleotide duplexes were monitored by Raman spectroscopy and by 2D NMR techniques. The second aim was to assess the potential of surface enhanced Raman spectroscopy (SERS) for the characterization of oligonucleotides. Third, the interaction of intercalating and groove binding dye probes with DNA strands has been investigated by resonance Raman spectroscopy.
20.3.1 Spectroscopic characterization of right and left-helical forms of a hexadecanucleotide duplex Changes between the right and left-helical forms of DNA are important for understanding control mechanisms of replication [28]. In particular, Jovin et al. [29] have characterized the structural parameters of Z DNA and have indicated Raman bands that might be used as indicators for the B , Z transition. Our first goal was to assess the sensitivity of normal Raman spectroscopy for monitoring the right-to-left conformational transition. Defined substrates for this investigation, i. e. the hexadecanucleotides d (CG)8 and d (m5CG)8, have been synthesized by Weber et al. [30]. Duplexes were obtained in the right-helical B conformation from the synthesis, transforming them into the left-helical Z conformation by exposure to high salt concentration (c (NaCl) & 3 M). Distinct differences between the Raman spectra of the two forms are evident in Fig. 20.10. In particular, the phosphodiester vibration of the backbone at a frequency of 824 cm –1 is characteristic for the C 2' -endo pucker of the ribose rings in the B conformation. In the spectrum of the Z conformation, this band is downshifted to 786 cm –1 and overlaps a cytosine ring breathing vibration. Further pronounced changes concern vibrations within the guanine ring system, which is oriented more towards the exterior surface of the double helix in the Z conformation because of the C3' -endo-syn conformation at the ribose rings. The ring breathing vibration is downshifted from 680 to 612 cm –1 upon the B ) Z transition. In addition, several ring stretching vibrations of guanine in the 1300–1360 cm –1 region are affected (Fig. 20.10). 2D NMR spectroscopy is an established tool for elucidating the three-dimensional structure of biopolymers in solution. We have used 2D NOE spectroscopy as an alternative technique for monitoring the B , Z conformational transition. The procedure is exemplary illustrated in Fig. 20.11 for the oligonucleotide d (m5CG)8. As mentioned above, the orientation of guanine relative to the ribose ring is distinctly different for the right and left-helical forms. Consequently, there are in the B conformation strong cross peaks between the guanine proton GH8 and the ribose proton GH2'. In contrast, for the Z conformation, the H8 413
20
Spectroscopic Probes of Surfactant Systems and Biopolymers
Figure 20.10: Raman spectra of the deoxyribonucleotide duplex d(CG)8. Top: B conformation, c = 39 mM; bottom: Z conformation, c = 24 mM.
Figure 20.11: NOESY spectrum of d (m5CG)8 in the Z conformation (medium: c (MgCl2) = 10 mM). Experimental conditions have been given in Ref. [30].
414
20.3 Vibrational spectroscopy and conformational analysis of oligonucleotides proton of the base and the GH1' proton of the sugar are spatially close. The spectrum demonstrates that from the presence of the respective cross peak the conformation present under the given conditions can be unambiguously inferred [30].
20.3.2 SERS spectra of deoxyribonucleotides For SERS spectroscopy of biomolecules [31], the use of small sample volumes is mandatory. For this purpose, two spectroelectrochemical cells were constructed in which the liquid sample is held in a thin cylindrical volume between a silver rod electrode and the optical access window. The design of the cells (from glass or teflon) and the oxidation-reduction cycle, used for the required roughening of the silver, have been described in detail by Zimmermann [32]. The power of the technique is illustrated by the SERS spectrum of a mononucleotide, 5'-adenosine-monophosphate (5'-rAMP). The top spectrum in Fig. 20.12 (left) was recorded with a concentration of 1.5 mM 5'-rAMP in phosphate buffer, with the electrolyte concentration adjusted to 100 mM. In particular, attention is drawn to the excellent signal-to-noise ratio achieved at a millimolar concentration. The high salinity conditions that are often used to induce a transition into left-helical conformations in oligo or polynucloetides were critically considered for the envisaged application.
Figure 20.12: SERS spectra of mononucleotides excited at 530.9 nm with the power of 29 mW and recorded in a spectroelectrochemical cell using a resolution of 7 cm –1 [33]. Left diagram: 5'-rAMP (1.5 mM), top: NaCl (84 mM), KCl (16 mM), UAg/AgCl = –0.5 V; bottom: NaCl (3.3 M), KCl (80 mM), UAg/AgCl = –0.6 V. Right diagram: top: 5'-rCMP (2 mM), phosphate buffer (2 mM); bottom: 5'-rCMP (9 mM), phosphate buffer (9 mM), NaCl (0.1 M), UAg/AgCl = –0.3 V.
415
20
Spectroscopic Probes of Surfactant Systems and Biopolymers
Therefore, the SERS spectrum of 5'-rAMP was also recorded in the presence of NaCl (3.3 M). The obtained high quality spectrum (Fig. 20.12, lower trace) demonstrates that SERS spectroscopy is feasible in the media that are typically used to prepare and study the Z conformation. Corresponding experiments with 5'-cytidine-monophosphate (5'-rCMP) showed an optimum adsorption potential of –0.3 V vs. Ag/AgCl (Fig. 20.11, right). Because the silver surface is positively charged at this potential, chloride and phosphate ions as well as impurities from the solution are adsorbed much more strongly to the electrode surface as compared to the potential of –0.5 V where the 5'-rAMP adsorption had been found to be an optimum (left). Therefore, the sensitivity in the 5'-rCMP, recorded at a concentration of 2 mM, was considerably lower as compared to the 5'-rAMP spectrum [33]. An attempt to record the SERS signals of the oligonucleotide d (CG)8 resulted in an uninformative spectrum (Fig. 20.13, top). In spite of the comparatively high total concentration (2 mg/ml & 3 mM), the signal-to-noise ratio is poor and only a few vibrations that are characteristic for d (CG)8 are detected. In the bottom trace, the normal Raman spectrum recorded at a 10 times higher concentration is reproduced from Fig. 20.10 for comparison.
Figure 20.13: SERS spectrum of d (CG)8 (2 mg/mL), excited at 530.9 nm with the power 29 mW. Conditions: phosphate buffer (0.17 M), NaCl (0.33 M), UAg/AgCl = –0.1 V. In the bottom trace, the normal Raman spectrum of d (CG)8 is reproduced from Fig. 20.10 for comparison.
Several reasons may be given for the unsatisfactory sensitivity of the SERS spectrum of d (CG)8. A comparatively more positive potential (–0.1 V vs. Ag/AgCl) had to be applied in order to promote adsorption of the negatively charged exterior of the double helix on the silver electrode and to avoid destabilization of the duplex [31]. At this potential, however, the binding of d(CG)8 is severely competed by adsorption of phosphate ions from the buffer and chloride ions from the electrolyte, as shown by corresponding test experiments with 5'-rCMP. From these experiments we conclude that only at very low buffer and electrolyte concentrations favourable conditions might be found for recording sensitive SERS spectra. 416
20.3 Vibrational spectroscopy and conformational analysis of oligonucleotides
20.3.3 Studies of chromophore-DNA interaction by vibrational spectroscopy The binding of dye molecules onto DNA strands is an important process in biochemistry [28]. Depending on its nature, this interaction may result in mutagenic, carcinogenic, and cytotoxic properties of the chromophore so that specific dyes have been suggested for the use as antitumor pharmaceuticals. Furthermore DNA binding dyes have been extensively used as stains in biochemical fluorescence microscopy work. In the present project we have investigated whether surface enhanced resonance Raman spectroscopy (SERRS) could be used to study various types of DNA-chromophore interactions. In this technique, a silver colloid is added to the solution containing the molecule to be studied [34]. First, we had to test whether the complex, between dye and DNA, would remain unaltered upon adsorption to the silver colloid surface which is a necessary condition for a valid application of the method. A second system specific question concerns the useful information that can be extracted from frequency shifts and relative intensity alterations in the vibrational spectrum of the DNA-bound chromophore [35–37]. As a first example, the complex of acridine orange (AO) with calf thymus DNA has been studied [38]. The stability of the complex upon adsorption on the silver colloid was confirmed by absorption spectroscopy (Fig. 20.14, right). In solution the absorption maximum at 504 nm (trace b) of AO in the complex is red shifted as compared to 495 nm (trace a) of the spectrum of the free AO molecule. The analogous red shift is maintained when the complex is added into a solution containing the silver colloid (trace d). Highly sensitive SERS spectra have been recorded at the remarkably low AO concentration of 2.5610–6 M (Fig. 20.14, left). The spectra of the free (trace a) and of the DNAbound dye (trace b) appear to be quite similar and only small changes in relative band intensities are detected in the difference spectra (trace c). A careful analysis has been based on a
Figure 20.14: Interaction of AO with calf thymus DNA. Left diagram: SERRS spectrum of the free AO (a) and of the AO-DNA (b) complex, both excited at 476.2 nm with a power of 25 mW. The top trace (c) represents the difference between the normalized spectra (b)-(a). Right diagram: absorbance spectra of AO (2.5610 –6 M); aqueous solution (a), complex with calf thymus DNA (b), free AO adsorbed on silver colloid (c), complex with calf thymus DNA adsorbed on silver colloid (d).
417
20
Spectroscopic Probes of Surfactant Systems and Biopolymers
complete assignment of the rich vibrational spectrum [38]. The observed alterations are compatible with a geometry in which the DNA double helix is adsorbed parallel to the silver surface [31] with the plane of the intercalated dye molecule approximately perpendicular to the helix axis. As a second example, the groove binding dye Hoechst 33 258 [39] was investigated. The SERRS spectrum has been recorded at a concentration of 10–6 M. It was analyzed in detail and assigned using normal coordinate analysis [32]. The influence of protonation on the binding to the silver surface was studied and an adsorption geometry with the two-benzimidazole rings approximately parallel to the silver surface was inferred. Besides intercalation and groove binding the ionic interaction of charged reagents with the exterior surface of the double helix is a third mode of molecular interaction with DNA strands. In this context, platinum(II) complexes have been intensely investigated with respect to their cytostatic properties. In the classic representative dichlorodiamin-platinum(II) or cisplatinum the binding to DNA is initiated by dissociation of a chloride ligand. In this context, we have investigated a series of novel dichlorobis(cycloalkylamin)Pt(II) complexes [40], in which the size of the cycloalkyl ring (CnH2n–1-) of the amine ligands was varied from n = 3 to n = 8 [41]. The aim of the study was to detect influences of the ligand size on the Pt-Cl bond strength in the complexes. But analysis showed that it is fairly constant throughout the investigated series. The pronounced differences in pharmaceutical activity seem to be mainly caused by the increasing lipophilicity of the higher membered rings.
20.3.4 Summary of results Vibrational spectroscopy and, in particular, Raman scattering have been used to elucidate selected interactions of DNAs. The B , Z conformational transition of oligo-DNA duplexes and the interaction of the intercalating dye AO with calf thymus DNA have been explicitly analyzed in the preceding sections. Surface enhancement of the Raman signals due to adsorption of the analyte onto either silver electrodes or colloids provides large gains in sensitivity in suitable cases. Limitations are implicit in the requirement of adsorption on the silver surface. In the case of negatively charge DNA double helices this leads to unfavourable competition with the binding of phosphate ions from the buffer at positive surface potentials and to desorption at negative surface potentials. Therefore, for intact double helices it seems to be difficult to take full advantage of the enhancement, demonstrated for the smaller building blocks. Very intense Raman signals may be obtained if the enhancement, due to addition of silver colloids, is acting in combination with resonant enhancement of the Raman scattering cross section by close-lying electronic transitions. This favourable situation is found for many important dye molecules that have been shown to interact with DNA in a specific manner. The present results on intercalating dyes demonstrate the high potential of SERRS for studying chromophore-DNA interactions.
418
20.4
20.4
Related projects carried out within the framework of the Collaborative Research Centre
Related projects carried out within the framework of the Collaborative Research Centre
The projects discussed in Section 20.2 have been centred around the spectroscopic probing of the structure and dynamics of surfactant systems and lyotropic mesophases, in particular hydrocarbon gels. Three related investigations not yet discussed in detail due to limitations in space shall be briefly mentioned. These are the study of the conformation of surfactant molecules by surface enhanced Raman spectroscopy, the development of a lipophilic dye probe for surfactant systems, and an outlook onto a different class of reaction gels created by hydrolysis of metal alkoxide precursors. The conformation of several surface-active molecules, polymers (alkyl-trimethylammonium surfactants, polyvinyl alcohol), and of model compounds such as choline has been studied by SERS [42, 43]. In particular, information on the geometry of their binding to the surface of silver colloids was obtained. Special methods for the synthesis of monodisperse colloids have been developed [44]. The microscopic polarity in the interior of micellar aggregates is of intrinsic interest in surfactant research. A well-known approach to this information is the use of solvatochromic dye probes. For these chromophores, the wavelengths of absorption and emission (and hence the Stokes shift) depend on the environment in which they are dissolved. A large variety of dye molecules exhibiting this property has been tested with surfactant systems. However, inspection reveals that most of these dyes contain ionic groups and, therefore, are either water-soluble or tend to bind to the hydrophilic exterior of micellar aggregates. Within the present project, we have synthesized and tested [23] the lipophilic dye probe 4-dioctylamino-anthra-thiadiazol-1,2-dion (TDA), for the structure see Fig. 20.15. As a measure for the solvatochromic properties of the compound, the Stokes shift of various
O
O N
Oc
N
N S
Oc
Figure 20.15: Stokes shift as a function of solvent orientational polarization df for TDA. The structure of the molecule is shown on the right-hand side. Solvents: hexane (1), toluene (2), CCl4 (3), dioxane (4), dibutylether (5), chlorobenzene (6), diethylether (7), ethylacetate (8), 1-chlorobutane (9), tetrahydrofuran (10), CH2Cl2 (11), dimethylsulfoxide (12), dimethylformamide (13), acetonitrile (14).
419
20
Spectroscopic Probes of Surfactant Systems and Biopolymers
simple solvents is presented in Fig. 20.14. It shows a remarkable range of variation from about 500 cm –1 to about 2000 cm –1. Solvent polarity is characterized in Fig. 20.15 by the orientational polarization df, which includes the effects of both, static and dynamic polarization of the solvent [45]. Linearity in the plot of the Stokes shift vs. df, as expected according to Lippert [46], shows that TDA is a promising candidate for polarity probing in surfactant systems. The fluorescence lifetime of TDA is highest (10–12 ns) in hydrocarbon solvents and decreases to 0.6 ns in polar solvents, such as acetonitrile. A plot of the decay rate against the dimensionless parameter EN T [47], which characterizes solvent polarity, has been given by Wiesner [23]. In contrast to the pronounced polarity dependence the lifetime is found to depend only weakly on the viscosity of the environment. Chromophore doping has been used not only as a spectroscopic probe but also in order to get non-linear optical properties for Langmuir-Blodgett films. The aim of the latter project was to monitor changes in molecular order and orientation when a spreaded film of arachidic acid (AA) is transferred from the water surface to a solid substrate in the Langmiur trough. For this aim three dyes of the phenylethenyl-pyridinium and of the benzaldehyde-hydrazone-type were dissolved in the AA film. Then surface second harmonic generation (SHG) was measured before and after transferring the films to a glass slide [48]. In the context of this study both, SHG [49] and enhanced Raman scattering [50], from multilayer samples have been studied. These substrates, which consist of a silver island film separated from a silver mirror by a dielectric spacer layer, excel by remarkable reflectivity and absorption properties that have been accounted for by appropriate modelling [51]. Polymers doped with triazene chromophores were used for sensitizing non-absorbing polymers, such as PMMA, for excimer laser ablation at 308 nm [52]. This project, which provided a strong link with the polymer-related activities described in Chapter III, resulted in the development of a new class of photosensitive polymers containing the triazene (–N=N–N–) functional group in the main chain [53]. This Section is concluded by briefly mentioning structural investigations on a different type of gels, which were generated by network formation during hydrolysis of metal alkoxide precursors. Mixed oxides are materials of high technological interest and the sol-gel technique [54] is one of the most promising routes for preparing homogeneous materials at low process temperatures. In the present context, the use of these mixed oxide gels as catalysts or catalyst supports is of particular interest. Depending on the drying method employed, mixed oxides derived from the sol-gel process are designated either as xerogels (solvent evaporation) or as aerogels (supercritical drying). The properties of TiO2 /SiO2-xerogels to be used as catalyst supports were studied by Schraml-Marth et al. [55], using vibrational spectroscopy. Walther et al. [56] have characterized the connectivity of vanadia and silica in V2O5 /SiO2-xerogel catalysts by 29Si MASNMR. Mixed oxide aerogels composed of titania, vanadia, and niobia have been shown to be highly active catalysts for the selective catalytic reduction (SCR) of nitric oxides with ammonia. These gels were characterized by FTIR, Raman, photoelectron spectroscopy, and secondary ion mass spectrometry [57–59].
420
20.5
20.5
Remarks and acknowledgements
Remarks and acknowledgements
Spectroscopic tools are one of the links for our investigations of two seemingly different classes of materials, namely the surfactant systems and the biological polymers. Specific conclusions for each of the individual projects are presented in Sections 20.2.4 and 20.3.4, respectively. We now want to focus on some features common to both activities. In the studies of biopolymers, emphasis has been given to the development of methods that are sufficiently sensitive to detect small toposelective changes in large molecules often only available in small quantities. Vibrational and NMR spectroscopy have been successfully used to characterize conformational changes in oligonucleotides and chromphore-DNA interactions. Particular emphasis was devoted to the use of surface enhanced and resonant Raman scattering techniques. Structural characterization also formed the basis for the investigations of surfactants (SERS, dye probes) and of gel systems (NMR and vibrational spectroscopies). Here the main challenge was the monitoring of the dynamics of molecules and aggregates constituting the various investigated lyotropic mesophases. NMR self-diffusion measurements of micelles and solvents, diffusion of dye probes in laser induced gratings by FRS, and time-dependent fluorescence techniques were combined to unravel the mobilities of different entities. As in these mesoscopic systems a variety of motional processes is occurring simultaneously on vastly different time scales, the use of different spectroscopic techniques, delivering complementary information, was mandatory for a consistent interpretation of the fascinating dynamics of these systems. The results reported in this chapter have been obtained by the dedicated effort of past and present members of the group. The corresponding publications, mentioning the names of all persons involved, have been quoted in the respective sections. With regard to the NMR self-diffusion measurements the author is particularly indebted to J.-C. Panitz, K.L. Walther, Th. Schaller, and A. Sebald. FRS experiments have been carried out by G. Rehder and Ch. Hahn. Fluorescence decay by diffusing quenchers was studied by J. Wiesner, G. Meyer, and M. Klenke. The investigations of biopolymers and of DNA-chromophor interactions have been advanced by F. and B. Zimmermann. We are grateful to Th. Lippert for his valuable contributions. Members of the team involved in the other projects have been mentioned in Section 20.4. This spectrum of activities has only been made possible thanks to the fruitful interaction with colleagues in joint projects within the Sonderforschungsbereich. In particular, the author would like to express his gratitude to H. Hoffmann and M. Sprinzl for numerous discussions. Many of the reported investigations have been stimulated by their suggestions. We are indebted to C.D. Eisenbach for his support in developing the dye probes and to O. Nuyken for a fruitful collaboration on the subject of photopolymers.
421
20
Spectroscopic Probes of Surfactant Systems and Biopolymers
References
1. K. Fontell: Coll. Polym. Sci., 268, 264 (1990) K. Fontell: J. Colloid Interface Sci., 43, 156 (1973) 2. K. Fontell: Adv. Coll. Interface Sci., 41, 127 (1992) 3. C. La Mesa, A. Khan, K. Fontell, B. Lindman: J. Colloid Interface Sci., 103, 373 (1985) 4. D.A. Anderson, H. Wennerström: J. Phys. Chem., 94, 8683 (1990) 5. G. Oetter, H. Hoffmann: Colloids Surf., 38, 225 (1989) 6. K.L. Walther, M. Gradzielski, H. Hoffmann, A. Wokaun: J. Colloid Interface Sci., 153, 272 (1992) 7. J.E. Tanner: J. Chem. Phys., 52, 2523 (1970) 8. J.-C. Panitz, M. Gradzielski, H. Hoffmann, A. Wokaun: J. Phys. Chem., 98, 6812 (1994) 9. P.O. Eriksson, G. Lindblom, E.E. Burnell, G.J. T. Tiddy: J. Chem. Soc. Faraday Trans. 1, 84, 3129 (1988) 10. H. Hoffmann, C. Thunig, P. Schmiedel, U. Munkert: to be published (1995) 11. G. Hall, B. Whitaker: J. Chem. Soc. Faraday Trans., 90, 1 (1994) 12. Ch. Hahn, A. Wokaun: Langmuir, 13, 391 (1997) 13. M. Doi, S. Edwards: J. Chem. Soc. Faraday Trans., 74, 560 (1978) M. Doi, S. Edwards: J. Chem. Soc. Faraday Trans., 74, 918 (1978) 14. P. Mariani, V. Luzzati, H. Delacroix: J. Mol. Biol., 204, 165 (1988) 15. L. Scriven: in: K. Mittal (ed.): Micellization, Solubilization, and Microemulsions, Vol. 2, p. 877 (1977) 16. M. Gradzielski, H. Hoffmann, J.-C. Panitz., A. Wokaun: J. Colloid Interface Sci., 169, 103 (1995) 17. P.P. Infelta, M. Grätzel, J.K. Thomas: J. Phys. Chem., 78, 190 (1974) 18. P. Lianos, J. Lang, J. Sturm, R. Zana: J. Phys. Chem., 88, 819 (1984) 19. C.S. Owen: J. Chem. Phys., 62, 3204 (1974) 20. C.A. Miller, M. Gradzielski, H. Hoffmann, U. Krämer, C. Thunig: Prog. Colloid Sci., 84, 243 (1991) 21. C. Thunig, G. Platz, H. Hoffmann: Proc. Workshop on Structure and Conformation of Amphiphilic Membranes, Jülich, (1991) 22. M. Almgren, J. Alsins, E. Mukhtar, J. van Stam: J. Phys. Chem., 92, 4479 (1988) 23. J. Wiesner: PhD thesis, Universität Bayreuth (1992) 24. M. Meyer: PhD thesis, Universität Bayreuth (1995) 25. J. Alsins, M. Almgren: J. Phys. Chem., 94, 3062 (1990) 26. G. Porte, J. Marignan, P. Bassereau, R. May: J. Phys. Paris, 49, 511 (1988). 27. J.M. Seddon, J.L. Hogan, N.A. Warrender, E. Pebay-Peyroula: Progr. Colloid Polym. Sci., 81, 189 (1990) 28. W. Saenger: Principles of Nucleic Acid Structure, Springer, Berlin, (1989). 29. T.M. Jovin, L.P. McIntosh, D.J. Arndt-Jovin, D.A. Zarling, M. Robert-Nicoud, J.H. van de Sande, K.F. Jorgenson, F. Eckstein: J. Biomol. Struct. Dyn., 1, 21 (1985) 30. J. Weber, A. Wokaun: Mol. Phys., 74, 293 (1991) 31. E. Koglin, J.-M. Sequaris: Topics in Current Chemistry, 134, 1 (1986) 32. F. Zimmermann, B. Zimmermann, J.-C. Panitz, A. Wokaun: J. Raman Spectrosc., 26, 435 (1995) 33. F. Zimmermann: PhD thesis, Universität Bayreuth (1994) 34. T.M. Cotton, S.G. Schultz, R.P. Van Duyne: J. Am. Chem. Soc., 104, 6528 (1982) 35. G. Smulevich, A. Feis: J. Phys. Chem., 90, 6388 (1986). 36. T.F. Barton, R.P. Cooney, W.A. Denny: J. Raman Spectrosc., 23, 341 (1992) 37. J. Aubard, M.A. Schwaller, J. Pantigny, J.P. Marsault, G.J. Lévy: J. Raman Spectrosc., 23, 373 (1992) 38. F. Zimmermann, B. Hossenfelder, J.-C. Panitz, A. Wokaun: J. Phys. Chem., 98, 12 796 (1994) 39. C. Zimmer, U. Wähnert: Prog. Biophys. Mol. Biol., 41, 31 (1986)
422
References 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59.
H.E. Howard-Lock, C.J.L. Lock, G. Turner, M. Zvagulis: Can. J. Chem., 59, 2737 (1981) J. Kritzenberger, F. Zimmermann, A. Wokaun: Inorg. Chim. Acta, 210, 47 (1993) J. Wiesner, A. Wokaun, H. Hoffmann: Prog. Colloid Polym. Sci., 76, 271 (1988) F. Zimmermann, A. Wokaun: Progr. Colloid Polym. Sci., 81, 242 (1990) P. Barnickel, A. Wokaun, W. Sager, F. Eicke: J. Colloid Interface Sci., 148, 80 (1992) W. Liptay: Z. Naturforsch., 18A, 1441 (1965) E. Lippert: Z. Elektrochem., 61, 962 (1957) Ch. Reichardt, E. Harbusch-Görnert: Liebigs Ann. Chem., 1983, 721 (1983) M. Klenke, H.G. Bingler, A. Wokaun: to be published (1998) H.G. Bingler, H. Brunner, M. Klenke, A. Leitner, F.R. Aussenegg, A. Wokaun: J. Chem. Phys., 99, 7499 (1994) H.G. Bingler, H. Brunner, A. Leitner, F.R. Aussenegg, A. Wokaun: Mol. Phys., 85, 587 (1995) A. Leitner, Z. Zhao, H. Brunner, F.R. Aussenegg, A. Wokaun: Appl. Opt., 32, 102 (1993) Th. Lippert, A. Wokaun, J. Stebani, O. Nuyken, J. Ihlemann: Angew. Makromol. Chem., 213, 127 (1993) Th. Lippert, J. Stebani, J. Ihlemann, O. Nuyken, A. Wokaun: J. Phys. Chem., 97, 12 296 (1993) C.J. Brinker, G.W. Scherer: Sol-Gel Science, Academic Press, San Diego, (1990) M. Schraml-Marth, B.E. Handy, A. Wokaun, A. Baiker: J. Non-Cryst. Solids, 143, 93 (1992) K.L. Walther, A. Wokaun, A. Baiker: Mol. Phys., 71, 769 (1990) U. Scharf, M. Schneider, A. Baiker, A. Wokaun: J. Catal., 149, 344 (1994) M. Schneider, M. Maciejewski, S. Tschudin, A. Wokaun, A. Baiker: J. Catal., 149, 326 (1994) M. Schneider, U. Scharf, A. Wokaun, A. Baiker: J. Catal., 150, 284 (1994)
423
21
Energy Transport by Lattice Solitons in a-Helical Proteins Franz-Georg Mertens, Dieter Hochstrasser, and Helmut Büttner
21.1
Introduction
For about 40 years the question of energy transport in muscle proteins has gained considerable attention. The biological energy quantum of 0.42 eV is given by the hydrolysis of adenosine triphosphate (ATP). It is assumed that this energy is transported practically without loss and is eventually used for the contraction of muscle fibers. The soliton concept can give an elegant answer to this question because here the dispersion of the energy can be prevented by non-linear effects. The fibers of striated muscles in vertebrates contain many myofibrils consisting of sarcomers. Each sarcomer consists of parallel-running thick and thin filaments. The basic mechanism for the muscle contraction consists in a sliding of the thin filaments relative to the thick ones [1]. This sliding can be described by phenomenological models consisting of a sequence of molecular processes [2, 3]. The thick filaments consist of myosin molecules which resemble rods with a diameter of about 40 Å and a length of about 1600 Å. The myosin consists of two polypeptide chains forming a double a-helical structure. At one end there are globular heads where the ATP hydrolysis takes place. In each helix there are three chains of peptide groups coupled by hydrogen bonds. Since 1973 Davydov [4] developed a quantum theory for solitons on these hydrogen bonded chains. The idea is that the energy from the ATP hydrolysis leads to an excitation of the amide-I vibration in the first peptide groups at one end of the chain. This vibration has an energy of about 0.205 eV and an electric dipole moment of 0.30 debye which is directed approximately along the helix axis. By the dipole-dipole interaction the next peptide groups on the chain can be excited, and so on. However, in this way the energy would not be transported but only dispersed. The essential point in Davydov’s assumption is that the vibrational excitations are coupled non-linearly to a deformation of the hydrogen bonds. If the coupling exceeds a certain threshold, solitons can be formed which are solutions of a non-linear Schrödinger equation. Further assumptions must be made in order to explain how the solitons eventually produce the relative sliding between thick and thin filaments. Davydov’s model has been refined or modified by several authors, which we do not discuss here because the stability of the solitons seems questionable. Both, thermal [5] and 424
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties Deutsche Forschungsgemeinschaft (DFG) Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
21.1 Introduction quantum fluctuations [6], reduce the lifetime such that it may not be large enough for the biological energy transport. (However, for one of the modified models the stability against thermal fluctuations seems to be better [7].) Moreover, some doubts have appeared whether the non-linear Schrödinger equation can be derived properly from Davydov’s Hamiltonian [8]. In 1984 Yomosa [9] proposed an alternative model which is much simpler than Davydov’s model because only the hydrogen bonds are involved in the energy transport. The peptide groups are rigid, i. e. they are represented by single masses which are coupled by strongly non-linear hydrogen bonds. Thus the energy is transported by lattice solitons. This has several advantages: a) The conditions for the occurrence of solitary waves on a one-dimensional lattice are very weak [5]. This means that every realistic interaction potential can be used, e. g. Lennard-Jones or Morse potentials. Solitons in the mathematical sense exist only for a completely integrable model, the Toda lattice, but for simplicity we will generally use the term soliton. b) Lattice solitons are non-topological, i. e. there is no energy gap. Thus the whole energy of a soliton can be converted into the mechanical work for the muscle contraction [9]. By contrast, Davydov solitons show an energy gap and only the kinetic part of the energy can be converted. c) Using lattice solitons, a molecular interpretation can be given [9] for the phenomenological rowing boat model [3] which describes the relative sliding between thick and thin filaments. d) Molecular dynamics simulations by Perez and Theodorakopoulos [11] have shown that even at room temperature lattice solitons are very stable against thermal fluctuations. Moreover, the lattice solitons are more stable than Davydov solitons if collisions between the two types of solitons are considered [11]. e) Quantum mechanics is necessary only for the initial condition. The idea is that the energy quantum of 0.42 eV, released by the ATP hydrolysis, produces impulsively a pulse-like compression of the hydrogen-bonded lattice. According to the inverse scattering theory [12] an arbitrary initial pulse develops into a finite number of solitons plus a radiative background (phonons). This result holds for integrable classical systems. However, at least for one integrable quantum lattice model soliton-like excitations have been found. For the quantum Toda lattice in the strong-coupling regime there is a branch of excitations with a dispersion curve (energy vs. momentum) which is practically identical to that of the classical solitons, though the ground state shows large quantum fluctuations [13]. In fact, a quantum Toda lattice with parameters appropriate for the a-helix [9] can be shown to be in the strong-coupling, i. e. semiclassical, regime [14]. Reference [13] used the Bethe ansatz. This method has been complemented recently by a semiclassical quantization of the periodic Toda chain with practically identical results in the limit of an infinite chain [15]. f) The reasoning of point (a) also holds for quantum lattice models with realistic interaction potentials. Such a model can always be replaced in a good approximation by a corresponding classical model with a renormalized potential [16], which fulfills the weak conditions for the existence of solitary waves. In the case of the Toda lattice the form of the potential remains unchanged and only the parameters are renormalized. 425
21 Energy Transport by Lattice Solitons in a-Helical Proteins For the above reasons lattice solitons are very good candidates for the energy transport. In this paper we generalize Yomosa’s model [9] in two ways: 1) The peptide groups are no longer rigid. For simplicity we consider only one internal degree of freedom (Section 21.2) which leads already to interesting new features for the solitons (Section 21.4). The generalization to more degrees of freedom is straightforward (Section 21.7). 2) Contrary to Yomosa, we do not work in the continuum limit. In fact, discreteness effects turn out to be decisive. These effects are first taken into account by applying the quasicontinuum approach (QCA) of Collins [17]. We use a rederivation of the approach in Fourier space [18] which offers several advantages (Section 21.3). However, at least for a part of the relevant energy range, the QCA is not sufficient (Section 21.5). Therefore we apply an iterative method [18] where the accuracy of taking into account the discreteness effects can be increased as much as necessary (Section 21.6). Finally, we discuss the energy loss of the solitons due to the emission of optical phonons (Section 21.6). The results of the various sections are always checked by computer simulations.
21.2
The model
Following Yomosa [9], Perez, and Theodorakopoulos [11] we consider a one-dimensional lattice of hydrogen-bonded peptide groups. The non-linear interactions between neighbouring peptide groups are described by a suitable potential V, e. g. a Toda potential or a Lennard-Jones potential, with parameters from the literature (Section 21.5). In contrast to Refs. [9, 11] we do not neglect the internal vibrations of the peptide groups. In a first step we describe each group by two masses M1 and M2 coupled by a linear interaction with eigenfrequency O0. The resulting Lagrangian is L
X 1 n
2
M1 A_ 2n
1 M2 B_ 2n 2
V
An1
Bn
1 mO20
Bn 2
An 2 ;
1
where m is the reduced mass; An and Bn are the displacements from the equilibrium positions of M1 and M2 of the nth peptide. The neglection of anharmonic terms for the internal vibrations is justified by the fact that the covalent bonds within the peptide groups are considerably stronger than the hydrogen bonds between the groups. Thus the relative displacements for the internal motion, Dn
An
Bn =a ;
are expected to be much smaller than the relative displacements, 426
2
21.2 The model jn
An1
Bn =a ;
3
of the hydrogen bonds. Dn and jn are defined in units of the equilibrium distance a of neighbouring hydrogen 2 2 bonds, times in units of O–1 0 , and energies in units of m O0 a . After this scaling we write the 2 non-linear interaction in the form aV (jn ), where the harmonic part of V has the form 12 j2n . The dimensionless parameter a measures the strength of the non-linear interaction compared to the linear one. In this notation the equations of motion are j n a
dV djn
D n Dn
1
mDn Dn1 0 1m
4
a dV dV m 0 1m djn jn 1
5
with the mass ratio m = M1/M2. As Dn appears only linearly, it can be eliminated which leads to a fourth order equation in time d2 dV dV 2 j j a a 2 c n n m 2 djn djn dt
dV djn1
dV djn 1
0
6
with c2m
m m ; 2 M
1 m
7
where M = M1 + M2 is the total mass. Linearization (dV/djn % jn) yields the dispersion curves of acoustic and optical phonons o2
q
( 1a 1 1 2
16a q c2 sin2 2 m 2
1 a
1=2 )
:
8
The sound velocity is cs
r a cm : 1a
9
427
21 Energy Transport by Lattice Solitons in a-Helical Proteins
21.3
Quasicontinuum approximation
For the diatomic chain with non-linear nearest-neighbour interactions a standard decoupling technique in the continuum limit [19] can be used and yields acoustic pulse-type solitary waves and optical envelope-type solitary excitations [20]. These calculations are rather involved, even more for our model which has not only alternating masses but also alternating interactions. We prefer not to apply this method because there are two general problems: a) Only polynomial interaction potentials can be used; usually the realistic potentials are expanded up to the fourth order, assuming that the anharmonicities are small. b) In the standard continuum approximation the difference operator is expanded up to a certain order which can lead to an ill-posed Cauchy problem [21]. This difficulty occurs because the dispersion due to the discreteness of the system is not taken into account consistently. For the monoatomic chain both problems have been overcome by the QCA of Collins [17] for solitary waves and periodic modes. Here the difference operator is inverted instead of expanded. Rosenau [21] developed a still more general approximation scheme which reduces to the result of Collins in the case of solitary waves. For the diatomic chain Collins [22] considered only pulse-like solitary waves. We are not interested here in the optical solitons because for the energy transport the pulse solitons are much better candidates: 1) They are supersonic, in contrast to the optical ones. 2) The mechanism needed for the conversion of the energy into a shortening of muscle fibers seems to be simple only for pulse solitons [9]. Since the QCA overcomes the above-mentioned problems we now apply it to our diatomic model with alternating interactions. However, we use a rederivation of the QCA in Fourier space [18]. This formulation is much simpler than the original one [17] and can easily be generalized from the monoatomic chain to our model. We are interested in solitary waves jn
t j
n
ct j
z
10
with velocity c, satisfying decaying boundary conditions. Because of these conditions the Fourier transform exists, j~
q
R1 1
dz e
iqz
j
z ;
and analogously D~ (q) for D (z). Moreover, we define the force 428
11
21.3
F
z a
Quasicontinuum approximation
dV dj
z
12
and its Fourier transform F~ (q). The equations of motion now read 1 eiq m D~ 0 ; 1m
c2 q2 j~ F~
c2 q2 1 D~
1 e 1m
m F~ 0 :
iq
13
14
The elimination of D~ yields q ~ 4c2m sin2 c2 q2 j
q 2
c2 q 2 1
~ ; c2 q2 F
q
15
which can also be obtained directly from Eq. 8. According to Ref. [18] the QCA now consists in the following procedure. We write Eq. 15 as ~ A
q j~
q F
q
16
and expand A (q) in a Taylor series for |q| < qc, where qc is the radius of convergence. A truncation after the second term and an inverse Fourier transformation yields a second order differential equation a2 j00
z F
z ;
a0 j
z
17
which allows the pulse-like solutions we are interested in. In fact, Eq. 17 has the structure which occurred already in the monoatomic case [17, 18], apart from the coefficients which here have the form a0
c2 c2m
a2 a0
c2
; c2m
12
c2m
18
c2
c
2
:
19
Since F is the functional derivative of V (j (z)) the integration of Eq. 17 yields the general result 1 2 a2 j0
z Vef f
j
z 0 2
20 429
21 Energy Transport by Lattice Solitons in a-Helical Proteins with Vef f aV
j
1 a 0 j2 ; 2
21
where we have set V (0) = 0. Because of c2m ^ 1/4, a2 in Eq. 19 is always positive and can be interpreted as an effective mass. Equation 20 has pulse-like, solitary solutions if there is a range of j for which Veff (j) ^ 0, where the equality must hold at the boundaries of this range. For the considered intermolecular potentials we have Veff (j) ^ 0 for a negative range j1 ^ j ^ 0, with a0 > 0 and a < a0 (because V % j2/2 for small j). These conditions lead to c s c cm
22
which means we get supersonic, compressional solitary waves with amplitude j1. The meaning of the upper limit cm will be discussed in the next Section. The shape j (z) of the pulses is obtained by the integration of Eq. 20: Zj z
j j1
dj0 p : 2Vef f
j0 =a2
23
For the Toda or Lennard-Jones potential this integral can only be calculated numerically. However, if we use an expansion of these potentials up to the fourth order V
j
1 2 j 2
2 bj3 2gj4 ;
24
where 9 b2 < 16 g, the integral (Eq. 23) can be evaluated analytically and yields for compressional pulses j
z
1
j2
j2
j1 j1
sinh2
Qz
;
25
with j1=2
" 1=2 # b g c2 c2s c2m ; 1 12 c2m c2 2g b c2s
Q2
j1 j2
ga : 2 a2
For the relative motions within the peptide groups we obtain from Eqs. 13 and 15 430
26
27
21.4 Velocity range for the quasicontinuum approach m e iq c2 q2 D~
q 1 m 4c2m sin2
q=2
c2 q 2
j~
q :
28
In order to be consistent with the approximations leading to Eq. 17, we expand Eq. 28 to order q2 and perform an inverse Fourier transformation, which yields D
z a0 j
z
21.4
1 1 j0
z 1m 2
1 m
a0 c2m 00
z : j 12c2
29
Velocity range for the quasicontinuum approach
There are restrictions for the velocity of the solitons, both in principle and for technical reasons. In Eq. 17 the function A
q
c2 q2
1
c2 q 2 c2 q 2
4c2m sin2
q=2
30
has been expanded in a Taylor series with the implication that the Fourier transform j~ (q) of the solitary wave is negligible for |q| 6 qc. Here qc is the radius of convergence, defined by the first non-trivial solution of q 2 cm sin cq : 2
31
In Fig. 21.1 this condition is visualized as the intersection of the straight line cq with the dispersion curve om (q) = 2 cm |sin (q/2)|. Going back to the original units, om can be identified as the dispersion of a monoatomic chain of masses M = M1 + M2 with a linear interaction with coupling constant mO20. For this reason we see now that the upper limit cm for the soliton velocity in Eq. 22 results from the linear interaction in our diatomic model with alternating linear and non-linear interactions. For the usual diatomic model there is no upper limit for the velocity [22]. In practice, i. e. for solitons with a finite width, the condition (Eq. 22) must be tightened to cs < c ccm ;
32
otherwise qc is very small and the Fourier transform j~ (q) cannot be negligible for |q| 6 qc . Moreover the lower limit cs leads to a further restriction. The sound velocity (Eq. 9) is the slope of the acoustic branch o– (q) for q ? 0 (Fig. 21.1). In order to ensure a certain 431
21 Energy Transport by Lattice Solitons in a-Helical Proteins
Figure 21.1: Phonon dispersion curves. o+(q): optical, o– (q): acoustic, om (q): monoatomic (from Eq. 31). The intersections with the straight line o = cq are discussed in Sections 21.4 and 21.6. Parameters: a = 0.6, m = 2, c/cs = 1.3.
range of soliton velocities satisfying both conditions in Eq. 32 we must demand a P 1. In fact, this is fulfilled for our model since the hydrogen bonds between the peptide groups are considerably weaker than the covalent bonds within these groups (a is the ratio of the coupling strengths, see Section 21.2). In Section 21.5 we will give estimates for a. So far we have discussed only the limitations which result from the finite radius of convergence for the Taylor series of A (q). Moreover, we must test whether the truncation (Eq. 17) behind the second term of the series is justified. For a given mass ratio m, i. e. for given cm, we choose a velocity c satisfying Eq. 32 and estimate the q range for which the truncated expansion agrees well with A (q). Then we choose a potential V and calculate the solitary wave j (z) which belongs to c. Its Fourier transform j~ (q) must be negligible outside the above-mentioned q range. The results are given in the next Section.
21.5
Solitary waves for realistic parameter values
The lattice constant a for the H-bonded peptide chains in the a-helix is about 4.5 Å [9]. As a representative value for the eigenfrequencies of a peptide group we choose O0 = 3.1161014 s–1 from the amide-I vibration. The total mass M = M1 + M2 corresponds to the mass of a peptide group plus an average residue in a muscle protein, which gives together about 100 proton masses [9, 10]. The mass ratio m = M1/M2 is kept as a free parameter which is varied between 1 and 10 (our results are invariant under the transformation m 7 ! 1/m). We model the hydrogen bonds between neighbouring peptide groups by a suitable non-linear interaction, e. g. a Toda potential with parameters fitted to an ab initio self-consistent-field molecular-orbital calculation for an H bond in a formamide dimer [9]. In our dimensionless units this corresponds to 432
21.5 Solitary waves for realistic parameter values aVT
j
a exp
bj bj b2
1
33
with b = 18 and a = 0.00123/c2m . Note that in Section 21.2 the interaction was introduced in the form aV, where V = j2/2 for j ? 0. With these parameters the sound velocity is cs % 4900 m/s for 1 ^ m ^ 10. As a second example we take a Lennard-Jones potential with parameters fitted to the equilibrium distance a and the bond energy [11, 23], which gives VLJ
j
1 1 72
1 j12
2 1
1 j6
34
and a = 0.000811/c2m . The corresponding sound velocity is about 4000 m/s for 1 ^ m ^ 10. For fixed mass ratio m, i. e. for fixed cm , we choose a velocity c in the range of Eq. 32 calculating the corresponding solitary wave j (z) and its Fourier transform j~ (q). A first test shows that j~ (qc)/j~ (0) is indeed negligible for a large range of velocities (about 30 % above cs), as expected from the discussion of the radius of convergence qc in Section 21.4. However, this large velocity range is considerably reduced by the second test described in Section 21.4. For the potentials and parameter values used here the QCA is valid for velocities c which do not exceed the sound velocity by more than about 5 to 10 %. Eventually we perform a final test by comparing them with the results of a computer simulation. If we take Eq. 23 and Eq. 29 as initial conditions and integrate the difference-differential Eqs. 4 and 5 numerically then the time evolution of a single pulse (Fig. 21.2) shows that the QCA solution is a good approximation. Only very small oscillations (phonons) appear immediately after the start. After a while these phonons are left behind and the pulse travels without changing its shape (Section 21.6). Figure 21.3 shows the scattering of two solitons.
Figure 21.2: Computer simulation of time-evolution of a single pulse for a chain of 200 unit cells with m = 1 and the Toda potential (Eq. 33). As initial condition a QCA solution with c/cs = 1.05 is used.
After these tests we turn to the essential question whether the solitons of the QCA can transport enough energy. Figure 21.4 shows the energy E of a single soliton as a function of c/cs, for Toda and Lennard-Jones potentials. E must be compared with the biological energy quantum of 0.42 eV (Section 21.1). Yomosa [9] assumed that these quanta set the initial conditions for the lattice solitons. Naturally, an arbitrary initial condition generally produces several solitons plus an oscillatory background. Knowing that nature usually works fairly ef433
21 Energy Transport by Lattice Solitons in a-Helical Proteins
Figure 21.3: Collision of two QCA solitons, same parameters as in Fig. 21.2.
Figure 21.4: Soliton energy vs. velocity. QCA results for Toda (solid line) and for Lennard-Jones interaction (dashed line). Iterative solutions for Toda (x) and for Lennard-Jones interaction (y).
fectively, and in order to be on the safe side, let us assume that the energy of one quantum essentially goes into one or two solitons. Then we see from Fig. 21.4 that we need a velocity of at least 1.2cs for which the QCA clearly is no longer valid. Naturally, this velocity is only a rough estimate that depends on the energy unit mO20 a2, i. e. on our choice of O0. But in any case we need a new method in order to handle the case of larger velocities. This is treated in the next Section.
21.6
Iterative method and stability
With increasing velocity and energy the solitary waves become narrower and their width can be in the order of the lattice constant. In this case the QCA does not hold. And it would not help to take more terms of the Taylor expansion of A (q) in Eq. 16 into account. The resulting higher order differential equations cannot be integrated like Eq. 17. Moreover, even the infinite Taylor series cannot fully represent A (q) because of the finite radius of convergence. Fortunately, an iterative method [18] has been developed for the monoatomic chain. Here the accuracy of taking into account the discreteness effects is increased systematically. In the case of the Toda lattice the iteration converges to the exact one-soliton solution. 434
21.6
Iterative method and stability
This method can also be applied to the pulse-type solitary waves of our diatomic model. Our basic Eq. 15 can be written in the form j~
q A 1
q F~
q ;
35
which already suggests an iteration because F~ (q) is the Fourier transform of F (q), (Eq. 12). However, instead of Eq. 35 we use a slightly different form which will turn out later to be more convenient. We first split the linear part of the force (Eq. 12) from the non-linear part, denoted by G, F
j
z a j
z G
j
z :
36
Then we insert Eq. 36 into Eq. 15 and isolate j~ j~
q 2 2 c q
a c2 q2 o2m
q G~
q ; o2
q c2 q2 o2
q
37
where o+ (q) and om (q) are the dispersion curves from Eq. 8 and Eq. 31, respectively. Similar to the monoatomic case [18], an iteration for Eq. 37 would converge only to the trivial solution j (z) : 0. This can be prevented by keeping j~ (0) constant during the iteration [18]. This condition leads to the elimination of c from Eq. 37 and thus to a new iteration procedure a c2i q2 o2m
q G~ i
q j~ i1
z 2 2
38 o2
q c2i q2 o2
q ci q with c2i c2s
j~
0 G~ i
0 : j~
0 c2 =c2 G~ i
0 s
39
m
This implies cs < ci < cm, i. e. the same condition as for the QCA. Each iteration consists of four steps: 1) 2) 3) 4)
Gi (z) = G (ji (z)) , G~ i (q) by Fourier transformation, calculation of j~ i+1 (q) by Eq. 40, ji+1 (z) by inverse Fourier transformation.
As starting function j1 (z) we choose the QCA result. The elimination of c from Eq. 37 means that the iteration process does not select a solution with a given velocity c but a solution with a given integrated amplitude j~ 1
0
R1 1
dzj1
z :
40
435
21 Energy Transport by Lattice Solitons in a-Helical Proteins ci changes during the iteration and converges towards a value which usually is considerably lower than the initial one. By the way, this behaviour is qualitatively similar to a computer simulation. When starting with an approximate solution as initial condition, an adaptation to the lattice by the emission of phonons is observed. The resulting solitary wave has a lower velocity (and energy) than the initial wave. For the potentials and parameters, used here, a sufficient accuracy is achieved after at most 8 iterations. As a test, these results were used as input for a computer simulation. Contrary to the QCA result for a velocity outside of the validity range (Fig. 21.5 a), the shape of the pulse remains unchanged, no adaptation to the lattice is observed (Fig. 21.5 b). The soliton character of the pulses is also seen clearly in scattering experiments, even in the highly discrete regime (Fig. 21.6). Figure 21.4 shows the soliton energies. They are considerably higher than the QCA results for the same velocities.
Figure 21.5: Computer simulations for initial conditions with c/cs = 1.28 for the QCA (a) and the iterative solution (b), using the Toda potential and m = 1.
However, a complete convergence cannot be achieved, i. e. an exact solitary wave solution probably does not exist. In fact, contrary to the monoatomic case [18], our iteration procedure (Eq. 38) shows a pole, namely at the solution q0 of ci q o
q :
41
The existence of the pole means that Eq. 38 makes sense only if G~ (q) and j~ (q) are negligible for |q| 6 q0. In practice a cut-off is necessary in each step of the iteration. Contrary to the finite convergence radius qc which limits the validity range of the QCA (Section 21.4), the occurrence of the pole is not a technical problem but is connected with a physical effect. Peyrard et al. [24] showed by computer simulations and theoretical investigations that a solitary wave with velocity c cannot be stable if the line cq has an intersection with one of the phonon branches o (q). The solitary wave permanently looses energy by the emission of phonons with frequency spectrum centred around o (q0) where q0 is the intersection point. Due to the energy loss the velocity c decreases gradually and the width of the wave increases. Thus the discreteness effects which are responsible for the phonon emission become smaller and smaller. Eventually the solitary wave is practically stable because the energy loss is negligible. This scenario holds for our model, too. The condition (Eq. 41) corresponds to the intersection of ci q with the optical phonon branch (Fig. 21.1). In a computer simulation two very different time scales appear. Starting with any initial condition including the QCA re436
21.7 Conclusion
Figure 21.6: Computer simulation of the collision of two very narrow solitons (c/cs = 1.9) from the iterative solution for the Toda potential and m = 1.
sult the above-mentioned adaptation to the lattice only takes a rather short time (Fig. 21.5 a). But this adaptation does not take place when the result from our iterative method is used as initial condition (Fig. 21.5 b). Contrary to the adaptation process, the emission of optical phonons occurs for a much longer time, which in principle goes on forever, while the solitary wave disappears asymptotically. However, this effect is extremely small when a-helix parameters are used. The situation can be illustrated by drawing Fig. 21.1, now using the a-helix parameters of Section 21.5. As a is very small there is a wide gap between the acoustic and optical phonon branches and the intersection point q0 is far outside of the first Brillouin zone (e. g. q0 % 5p for c % 2 cs). Therefore the above condition for the convergence of the iteration, i. e. j~ (q0) and G~ (q0) are negligible, is indeed very well fulfilled and the energy loss due to emission of optical phonons is negligible.
21.7
Conclusion
Lattice solitons remain good candidates for the explanation of the energy transport in the a-helix if an internal vibration mode of the peptide groups is taken into account. Our theory can easily be generalized for a lattice with a basis of more than two atoms, e. g. three masses representing a peptide group N–C=O. The neglection of anharmonicities for the internal vibrations is the essential point which allows the elimination of coordinates in the equations of motion. This neglection is justified because the covalent bonds within the peptide groups are considerably stronger than the hydrogen bonds between the peptides. Discreteness effects are very important. They are partially incorporated by the quasicontinuum approximation. However, for the parameters of the a-helix, a systematic implementation of the discreteness is necessary, which is achieved by an iterative method. The convergence of this method is limited in principle by an instability of the solitons due to the emission of optical phonons. However, this effect turns out to be negligible. In our opinion mainly two questions remain: 437
21 Energy Transport by Lattice Solitons in a-Helical Proteins a) Stability of solitons The coupling between the three hydrogen bonded chains of an a-helix possibly reduces or destroys the stability of the solitons on the chain. So far this problem has not yet been tackled analytically. Molecular dynamics simulations, which include transversal fluctuations, do not show stable solitons [25]. On the other hand, we have performed simulations for a Toda chain that branches into two Toda chains with different parameters [26]. Here we choose the parameters of the first branch so that the interactions become strong and nearly linear, representing the covalent bonds between the hydrogen-bonded chains. The parameters of the second branch are chosen similar to the original chain, namely representing the weak, very non-linear hydrogen bonds. The simulations show that a soliton, which is initiated on the original chain, prefers to go into the second branch. Very little energy is lost via the first branch, unless the energy of the soliton is very small. This is an interesting result, but naturally it does not prove the stability of solitons on the very complicated structure of three coupled chains. b) Detection of solitons In contrast to the topological solitons, it was not clear for a long time whether non-topological lattice solitons yield a clear-cut signature in the dynamic form factor S (q, o), which can be measured by inelastic neutron scattering. Recently combined Monte Carlo molecular dynamics simulations yielded only one peak in S (q, o) for the thermal equilibrium and the soliton and phonon contributions could not be distinguished [27]. However, if certain nonequilibrium configurations are used as initial condition for the molecular dynamics, additional small peaks on the high-frequency side of the main peak appear which can be identified with lattice solitons. Therefore, the question now is whether the ATP hydrolysis produces such initial conditions.
References
1. H.E. Huxley, J. Hanson: Nature, 173, 973 (1954) H.E. Huxley, J. Hanson, in: Structure and Function of Muscle, Academic, New York, p. 183 (1960) A.F. Huxley, R. Niedergerke: Nature, 173, 971 (1954) A.F. Huxley: Proc. Biophys. 7, 255 (1957) 2. A.F. Huxley, A. Simmons: Nature, 233, 533 (1971) 3. J.M. Murray, A.Weber: Sci. Amer., 230, 1974 (1974) 4. A.S. Davydov: J. Theor. Biol., 38, 559 (1973) A.S. Davydov: Stud. Biophys., 47, 221 (1974) A.S. Davydov: Phys. Scr., 20, 387 (1979) A.S. Davydov: Biology and Quantum Mechanics, Pergamon, New York, (1982) 5. P.S. Lomdahl, W.C. Kerr: Phys. Rev. Lett., 55, 1235 (1985) A.F. Lawrence, J.C. McDaniel, D.B. Chang, B.M. Pierce, R.R. Birge: Phys. Rev. A, 33, 1188 (1986) 6. H. Bolterauer, M. Opper, Z. Phys. B, 82, 95 (1991)
438
References 7. L. Cruzeiro, J. Halding, P.L. Christiansen, O. Skovgaard, A.C. Scott: Phys. Rev. A, 37, 880 (1988) 8. D.W. Brown, K. Lindenberg, B.J. West: Phys. Rev. A, 33, 4104 (1986) D.W. Brown, B.J. West, K. Lindenberg: Phys. Rev. A, 33, 4110 (1986) 9. S. Yomosa: J. Phys. Soc. Japan, 53, 3692 (1984) S. Yomosa: Phys. Rev. A, 32, 1752 (1985) 10. M.K. Ali, R.L. Somorjai: J. Phys. A, 12, 2291 (1979) J.R. Rolfe, S.A. Rice, J. Dancz: J. Chem. Phys., 70, 26 (1979) M.A. Collins: Chem. Phys. Lett., 77, 342 (1981) H. Bolterauer, in: J.T. Devreese, L.F. Lemmens, V.E. van Doren (eds): Recent Developments in Cond. Matter Physics, Vol. 4, Plenum, New York, p. 199 (1981) 11. P. Perez, N. Theodorakopoulos: Phys. Lett., 117 A, 405 (1986) P. Perez, N. Theodorakopoulos: Phys. Lett., 124 A, 267 (1987) 12. C.S. Gardner, J.M. Greene, M.D. Kruskal, R.M. Miura: Phys. Rev. Lett., 19, 1095 (1967) 13. F.G. Mertens, M. Hader, in: S. Takeno (ed.): Dynamical Problems in Soliton Systems, Springer Series in Synergetics, Vol. 30, Springer, Berlin, p. 89 (1985) 14. D. Hochstrasser, F.G. Mertens, H. Büttner: Phys. Rev. A, 40, 2602 (1989), Appendix A 15. F. Göhmann, W. Pesch, F.G. Mertens: J. Phys. A, 26, 7589 (1993) 16. F. Göhmann, F.G. Mertens: J. Phys. A, 25, 649 (1992) 17. M.A. Collins: Chem. Phys. Lett., 77, 342 (1981) M.A. Collins, S.A. Rice: J. Chem. Phys., 77, 2607 (1982) 18. D. Hochstrasser, F.G. Mertens, H. Büttner, Physica D, 35, 259 (1989) 19. H. Büttner, H. Bilz, in: A.R. Bishop, T. Schneider (eds.): Solitons in Cond. Matter Physics, Springer, Berlin, p. 162 (1978) 20. St. Pnevmatikos, M. Remoissenet, N. Flytzanis: J. Phys. C, 16, L305 (1983) St. Pnevmatikos, N. Flytzanis, M. Remoissenet: Phys. Rev. B, 33, 2308 (1986) 21. P. Rosenau: Phys. Lett. A, 118, 222 (1986) Phys. Rev. B, 36, 5868 (1987) 22. M.A. Collins: Phys. Rev. A, 31, 1754 (1985) 23. M. Levitt: J. Mol. Biol., 168, 595 (1983) 24. M. Peyrard, St. Pneuvmatikos, N. Flytzanis: Physics, 19 D, 268 (1986) 25. O.H. Olsen, P.S. Lomdahl, W.C. Kerr: Phys. Lett., A 136, 402 (1989) 26. D. Hochstrasser: PhD thesis, Universität Bayreuth (1989) 27. A. Neuper, F.G. Mertens, in: M. Peyrard (ed): Nonlinear Excitations in Biomolecules, Les Editions de Physique, Les Ulis and Springer, Berlin, (1995)
439
IV Appendix
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties Deutsche Forschungsgemeinschaft (DFG) Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
22
Documentation of the Collaborative Research Centre 213 Chairmen Prof. Dr. Markus Schwoerer Prof. Dr. Heinz Hoffmann
22.1
List of Members
Name
Institute
Membership
Blumen, A. Büttner, H. Dormann, E. Eisenbach, C.D. Faulhammer, H. Fesser, K.** Friedrich, J. Friedrich, J. Haarer, D. Höcker, H. Hoffmann, H. Kalus, J. Kiefer, W. Kramer, L. Krauss, H.L. Lattermann, G. Laubereau, A. Mertens, F.G. Nuyken, O. Pobell, F. Richter, W. Rösch, P.
Experimentalphysik Theoretische Physik Experimentalphysik Makromolekulare Chemie Biochemie Theoretische Physik Experimentalphysik Experimentalphysik Experimentalphysik Makromolekulare Chemie Physikalische Chemie Experimentalphysik Experimentalphysik Theoretische Physik Anorganische Chemie Makromolekulare Chemie Experimentalphysik Theoretische Physik Makromolekulare Chemie Experimentalphysik Experimentalphysik Struktur und Chemie der Biopolymere
1987–1991 1984–1995 1984–1990 1988–1995 1984–1989 1984–1995 1984–1989 1993–1995 1984–1995 1984–1986 1984–1995 1984–1992 1984–1986 1984–1995 1984–1992 1989–1995 1984–1995 1987–1995 1988–1992 1987–1995 1990–1995 1993–1995
* “Promotion” supported by the Sonderforschungsbereich 213 ** “Habilitation” supported by the Sonderforschungsbereich 213 Macromolecular Systems: Microscopic Interactions and Macroscopic Properties Deutsche Forschungsgemeinschaft (DFG) Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
443
22
Documentation of the Collaborative Research Centre 213
Name
Institute
Membership
Schmidt, M. Schwoerer, M. Seilmeier, A. Spiess, H. W. Sprinzl, M. Strohriegl, P. Wokaun, A.
Makromolekulare Chemie Experimentalphysik Experimentalphysik Makromolekulare Chemie Biochemie Makromolekulare Chemie Physikalische Chemie
1993–1995 1984–1995 1993–1995 1984–1986 1984–1995 1989–1995 1987–1995
Rentsch, S. University of Jena
Optik- und Quantenelektronik
1992–1995
22.2
Heads of Projects (Teilprojektleiter)
22.2.1 Projektbereich A: Gemeinsame Einrichtungen A1 A5 A6
FTIR-Gerät Rasterelektronenmikroskopie Transmissionselektronenmikroskopie
Krauss Haarer Eisenbach
1984–1992 1987–1995 1989–1995
Schwoerer
1984–1986
Schwoerer
1984–1989
Haarer
1984–1989
Dormann
1984–1990
Büttner
1984–1986
Fesser Kalus
1984–1995 1984–1992
22.2.2 Projektbereich B: Festkörper B1 B2 B3 B4 B5 B6 B7
444
ENDOR in Diacetylenkristallen : die molekulare Elektronik der topochemischen Reaktion UV-Holographie und das Wachstum von Mikrostrukturen in Diacetylen-Einkristallen Photoleitung und Redox-Photochemie Molekularer und hochpolymerer Festkörper Disubstituierte Diacetylene mit besonderen di- und pyroelektrischen Eigenschaften Kommensurable und inkommensurable strukturelle Übergänge in Charge-Transfer-Kristallen Nichtlineare Anregungen in konjugierten Polymeren Untersuchung elementarer Anregungen mit hochausgelöster inelastischer Röntgenstreuung
22.2 B8
B9 B 10 B 11 B 12 B 13 B 14 B 15
Heads of Projects (Teilprojektleiter)
Elektron-Phonon-Wechselwirkung und elektronische Korrelation in quasi-eindimensionalen ChargeTransfer Systemen Lochbrennspektroskopie an excitonischen Zuständen in Gläsern Spektroskopie und Kalorimetrie an polymeren und nicht-kristallinen Substanzen Hierarchische Modelle zum Ladungs- und Massentransport in ungeordneten Medien Lochbrenn-Spektroskopie in Polymeren: Elektrische Feldeffekte und Druckeffekte Nichtlineare Optik in Makromolekülsystemen: Materialien und Bauelemente Lichtinduzierte spektrale Diffusion von Farbstoffmolekülen in glasartigen Matrizen Lochbrennspektroskopie an excitonischen Zuständen von Aggregatketten
Büttner
1990–1995
Friedrich
1987–1989
Haarer/Pobell 1987–1995 Blumen
1987–1991
Haarer
1990–1995
Schwoerer
1990–1995
Richter
1990–1995
Friedrich
1992–1995
22.2.3 Projektbereich C: Funktionale Systeme – Mizellen, Oberflächen und Polymere C C C C
1 2 3 4
C5 C6 C7
C8 C9 C 10 C 12 C 13 C 14
Viskoelastische Tensidlösungen Lyotrope nematische und cholesterische Mesophasen Flüssigkristalline Haupt- und Seitenkettenpolymere Stabilität und Selektion in den strukturbildenden Instabilitäten von flüssigen Kristallen Struktur und Reaktivität von koordinativ ungesättigten Oberflächenverbindungen CARS-Spektroskopie an fluoreszierenden Materialien Darstellung und Untersuchung von Oligomeren und Polymeren mit definierter Struktur und besonderen Eigenschaften NMR-Untersuchungen der molekularen Bewegung und der molekularen Ordnung in festen Polymeren Zeitaufgelöste Schwingungsspektroskopie an Polymeren mit ultrakurzen Laserimpulsen Zeitaufgelöste toposelektive Spektroskopie von assoziierten Flüssigkeiten mit ultrakurzen Laserimpulsen Spektroskopische Charakterisierung flüssigkristalliner Tensidphasen und lichtempfindlicher Polymere Untersuchungen zur Konformation und Segmentbeweglichkeit von Polymeren Neue flüssigkristalline Modellsubstanzen und Polymere
Hoffmann Hoffmann Höcker Kramer
1984–1995 1984–1995 1984–1986 1984–1995
Krauss
1984–1990
Kiefer Höcker
1984–1986 1984–1986
Spiess
1984–1986
Laubereau
1984–1995
Laubereau
1984–1986
Wokaun
1987–1995
Nuyken
1988–1992
Lattermann
1988–1995 445
22
Documentation of the Collaborative Research Centre 213
C 15 Photoleitende Polymere C 16 Segmentierte Poyurethan-Elastomere mit und ohne Wasserstoffbrückenbindungen: Synthese von Modellsystemen, Struktur, Morphologie und Eigenschaften C 17 Spektrales Verhalten von Photochromen und Fluorophoren in Polymeren und deren Verwendung als Molekülsonden C 18 Brummgele C 19 Spin-Korrelation bei magnetischen Lipidschichten C 21 Toposelektive Reaktionen zur Darstellung und Charakterisierung supramolekularer Strukturen C 22 Molekulare Verstärkung von Polymeren durch Polydiacetylen-Propfcopolymere C 23 Untersuchung des Energietransfers in Copolymeren durch ultraschnelle Thermometrie nach Anregung im mittleren Infrarot
Strohriegl Eisenbach
1988–1995 1988–1991
Eisenbach
1988–1995
Hoffmann Mertens Schmidt
1990–1995 1990–1995 1993–1995
Eisenbach
1993–1995
Seilmeier
1993–1995
Haarer
1984–1990
Friedrich
1984–1986
Faulhammer
1984–1989
Sprinzl
1984–1995
Sprinzl
1984–1995
Wokaun
1987–1995
Mertens Rösch
1987–1989 1993–1995
Rentsch
1992–1995
Laubereau
1992–1995
22.2.4 Projektbereich D: Biopolymere D1 D2 D3 D4
D5
D6 D7 D8
Optische Spektroskopie an DNA-Daunomycin Intercalationskomplexen Stark-Effekt an optisch anisotropen photochemischen Löchern: Antennenpigmente in organischen Gläsern Elektronenspinresonanz-Spektroskopie mit Proteinen und Nukleoprotein-Komplexen Magnetische Kernresonanzspektroskopie spezifisch markierter Ribonukleinsäuren, Ribonukleoproteinkomplexe und Oligodesoxynukleotide Spektroskopische Reportergruppen zur Untersuchung der Struktur und Dynamik der Protein-Nukleinsäure-Wechselwirkung Strukturelle Untersuchungen an Oligonukleotiden und DNA-Assoziationsverbindungen Gittersolitonen auf Polypeptid-Ketten in Proteinen Die Struktur von Makromolekülassoziaten in Lösung am Beispiel von Protein-Nukleotid Komplexen
Ye 1 Untersuchung photoleitender Polymerthiophene auf der Pikosekunden- und Subpikosekunden Zeitskala Yw 1 Untersuchung photoleitender Polythiophene auf der Piko-Subpikosekunden-Zeitskala mit optischen und elektrischen Methoden
446
22.3
22.3
Guests
Guests (Guests with a duration of stay longer than 2 weeks)
Name
Acad. grad. Institute
Allakhverdiev, K. Prof. Aranson, I. Dr.
Year
Academy of Sciences, Baku Uni Jerusalem, Israel
1991/95 1990/92 1994 Arnold, L. Dr. Insitute of Organic Chemistry. 1989/90 Czech. Acad. of Sc., Prague 1991/93/94 Bachvarov, I. Dr. Uni Sofia, Griechenl. 1993 Bara, M. Dr. Université Paris 1985/86 Bartusch, G. Dipl.-Chem. TU Dresden 1990/91 Boesch, R. Dr. Universität Dijon 1992 Brezesinski, G. Dr. Martin-Luther-Universität Halle Wittenberg 1989 Buka, A. Prof. Inst. f. Physics, Budapest 1989/90/91 1992/93/94 Caceres, M. Dr. Centro Atomico, Bariloche 1990/92 Cameron, J.H. Dr. Heriot-Watt University Edinburgh 1995 Chen, S.H. Prof. Massachusetts inst. of Technology, USA 1987/88 Chigrinov,V. Prof. Organic Intermediats & Dyes Inst., Moscow 1992/94 Danielius, R. Dr. Vilna, Litauen 1989 Delev,V. Dr. Baschkirische Uni, Ufa 1994 Dobrov,V. Dr. State University Moscow 1994 Eber, N. Dr. Academy of Sciences Budapest, Hungarian 1994/95 Favorova, O. Dr. Soviet Academy of Sciences, Moscow 1989 Fidy, J. Dr. Inst. of Biophysics Budapest 1994 Foltynowicz Poznan 1991 Gadonas Dr. Universität Vilnius 1989/90 Gaididei, Y. Prof. Academy of Sciences Kiev, Ukraine 1994 Gammel, J.T. Dr. National Laboratory, Los Alamos 1990 Getautis, V. Dr. Uni Kaunas, Litauen 1994 Golov, A. Dr. Academy of Sciences Chernogolovka, Moscow 1994 Gouvèa, M.E. Dr. Universidade Federal de Minas Gerais, Brasilien 1993/94 Grazulevicius, J. Dr. Technicol University Kaunas 1992/95 Herenyi, L. Institut of Biophysics University of Budapest 1994 Hiltrop, K. Dr. University Paderborn 1987 Imae, T. Prof. Nagoya University Japan 1991 Imrich, J. Dr. P.J. Safarik University Kosice 1994 Ivanov, B. Dr. Institute for Metical Physics Kiev, Ukraine 1994 Jonak, J. Dr. Institute of Molecular Biology 1988 Czech. Acad. of Sc., Prague Joshi, R.L. Université Paris VII 1986
447
22
Documentation of the Collaborative Research Centre 213
Name
Acad. grad. Institute
Year
Kasperczyk, J.
Dr.
Zawiercie, Polen
Katunin,V.
Dr.
Kauffmann, H. Kharlamov, B. Kikas, J. Kim, H. S. Kohler, B.
Prof. Dr. Dr. Prof. Prof.
Inst. of Nuclear Physics, Soviet Acad. of Sc., St. Petersburg Universität Wien Academy of Sciences, Moscow Inst. of Physics, Tartu University of Seoul Südkorea University California
1989/90 1991/92 1993/95
Korrovits,V. Kotomin, E. Kozar, T. Krekhov, A.P.
Dr. Dr. Dr. Dr.
Academy of Sciences, Tartu Universität Riga Academy of Sciences Kosice Slovakia Baschkirische Uni,.Ufa
Kumar, A. Kurlat, D.H. Li, W. Lin, T.L.
Dr. Prof. Dr. Prof.
Calcutta-University, Indien University Buenos Aires Argentinien
Matyjaszewsky, K. Prof. Mikhin, N. Dr. Miller, C.A. Prof. Mistriotis, A. Dr. Möhle, L. Dr. Muthukumar, M. Prof. Nawrot, B. Dr. Nesrullajev, A.N. Prof. Nikogosyan, D.N. Prof. Noolandi, J. Prof. Ofengand, J. Prof. Ollikainen, O. Personov, R. Prof. Planer-Kühner, G. Quèmarais, P. Prof. Ragunathan, V. Dr. Rebane, A. Dr. Renge, I. Dr. Reshetnikova, L. Dr. Riecke, H. Prof. Rieckhoff, K. Dr.
448
1991 1992/94 1994/95 1991 1984/85/ 88/90 1988 1990/91 1993 1993/94 1995 1988/89/90 1987 1991/92 1991
Universität Taiwan Mainz Carnegia-Mellon-Uni. Pittsburgh, USA 1991 Academy of Sciences Kharkov, Ukrainian 1995 Rice University Dep. Houston Texas, USA 1989/95 Universität Kreta 1991 Leuna-Werk, Merseburg 1990/91 University of Massachusetts 1994 Technical University Poznan Poland 1994/95 Universität Baku 1989 Inst. of Spectroscopy 1993 Academy of Sciences, Moscow Xerox Research Centre of Canada 1991/92 Roche Inst. of Mol. Biology Nutley, New Jersey 1988 Academy of Sciences, Tartu 1991 Inst. of Molecular Spectroscopy, Dep. Moscow 1987/91 MPI f. Polymerforschung Pushchino, Russian 1990/91 LEPES – CNRS, Grenoble 1994/95 Bangalore, Indien 1989 Academy of Sciences, Tartu 1987/88/92 Academy of Sciences, Tartu 1991 Inst. of Mol. Biology Acad. of Sciences, Moscow 1990/91 Uni Northwestern, USA 1991/94 Simon Frazer University North Vancouver 1987/90
22.3
Guests
Name
Acad. grad. Institute
Year
Rötger, A. Ruckenstein, E. Rudinger, J.
Dr. Prof. Dr.
1994 1985/86 1992/93
Salaev, F. Santa, I. Sarbak, Z.M. Schott, M. Schneider, A.
Prof. Dr. Dr. Prof.
Sédlak, E. Serra i Albet, A. Dr. Shalaby, G.A. Shchipunov, Y.A. Prof. Shirokov,V. Dr. Sigler, P. B. Prof. Silber, M. Smrt, J.
Dr. Dr.
Sobkowski, M.
Dr.
Spirin, A.
Prof.
Stasko, A. Steinberg, S. V.
Prof. Dr.
Stoylov, S.P. Suzuki, H.
Prof. Dr.
Svitova, T. Szkaradkiewicz, Tamori, K. Tezak, D. Thelakkat, M.
Prof. K. Dr. Dr.
Trommsdorff, P. Prof. Tunkin,V. Dr. Ueda, T. Dr. Vainer, Y. Dr. Valiente-Martinez, M. Dr.
Universitè Fourier, Grenoble University Buffalo, USA Centre National de la Recherche Scientifique, Strasbourg Academy of Sciences of Baku, Aserbaijan Academy of Sciences Budapest, Ungarn Universität Poznan, Polen Université Paris VII Martin-Luther-University Inst. of Biochemistry, Halle-Wittgenstein University Kosice University Barcelona Faculty of Science Minoufia University Shebin El-kom, Ägypten Academy of Sciences Vladivostok, Russia Academy of Sciences Dep. of Molecular Biophysics and Biochemistry, Yale University, New Haven Passadena, Caltech, USA Inst. of Organic Chemistry Czech. Acad. of Sc., Prague Inst. of Bioorganic Chemistry, Polish Acad. of Sc., Poznan Inst. of Protein Research, Academy of Sciences, Pushchino, Russia Technical University, Bratislava, CSFR Inst. of Molecular Biology Soviet Acad. of Sc., Moscow Academy of Sciences Sofia, Bulgarien NTT Opto-Electronic Laboratories Tokai-Mure, Naka-Cun1 Academy of Sciences, Chem. Dep., Moscow University of Poznan Academy of Sciences, Tokyo University Zagreb, Croatia NSS College, Manjeri, University of Calicut, India University of Grenoble University Moscow University of Tokyo Academy of Sciences Moscow Universidad de Alcala de Henares, Spanien
1992 1990/91 1989 1985 1991 1994 1987 1995 1993 1991 1995 1991 1988/89 1990 1991 1989 1990/91 1991/92 1993/94 1993/94/95 1989/92/93 1994 1993/97 1990 1988 1995 1990 1991/92 1989 1993/94 1992
449
22
Documentation of the Collaborative Research Centre 213
Name
Acad. grad. Institute
Year
Vitukhnovski, A. Vodopyanov, K Wada, Y. Wysin , G. Xu, J.
Dr. Dr. Prof. Dr.
1990 1990/92 1988/93 1990/91 1990
22.4
Co-workers
Name Adam, D. Aechtner, Aggarwal, K. Ahmadian, R. Amberger, E. Ambrosch, A. Angel, G. Angel, M. Angel, S. Angstl, R. Bächer, R. Bachmann, S. Bara, M. Baranowski, D. Barnickel, P. Barth, K. Bauer, F. Bauer, H.-D. Beck, M. Beginn, Ch. Beginn, U. Beikmann, M. Bettenhausen, J. Bieger, P. Bittl, Th. Blank, J.
450
Academy of Sciences Moscow Inst. of General Physics Moscow University of Tokyo Kansas State University Chengdu University, China
Acad.grad
Dr. Dr. Dr.* Dr.* Dr.* Dr.* Dr.* Dr.* Dr.*
Dr.* Dr.* Dr.* Dr.*
Dr.*
Institute
Project
Period
Exp.Physik Exp.Physik Makro.Chem. Biochem. Anorg.Chem. Makro.Chem.. Exp.Physik Phys.Chem. Phys.Chem. Exp.Physik Phys.Chem. Biochem. Exp.Physik Theo.Physik Phys.Chem. Exp.Physik Exp.Physik Exp.Physik Biochemie Makro.Chem. Makro.Chem. Biochemie Makro.Chem. Biochem. Biopolymere Biochemie
B3 C9/10 C13 D5 C5 C14 C9 C1/C2 C2 B2 C2 D4/D5 B1 B8 D6 B14 B13 B2/B13 D4/D5 C15 C14 D4/D5 C15 D5 D8 D5
1991/92/93/94 1984/85/86/89 1989 1989/91/92 1989/90/91 1990/91/92 1984–89 1984/85/86 1987/88/89 1985–89 1986/87/88 1987/88/89 1985/86 1990–94 1989/91 1991/92/93/94 1994/95 1985/86/88–91 1985–89 1993 1993 1993/94/95 1993/94/95 1984–89 1993 1991/92
22.4
Co-workers
Name
Acad.grad
Institute
Project
Period
Blechschmidt, B. Blumen, A. Bodenschatz, E. Bojer, B. Bratengeier, K. Breinl, W. Bronold, F. Buchwald, E. Büttner, H. Burger, A. Burkhardt, V. Caceres, O. Carron, K. Cuc, T. T. Dahinten, T. Decker, W. Deeg, M. Denninger, G. Denzner, M. Diemer, E. Dirnberger, K. Döge, B. Dörfler, S. Dohlus, R. Domes, H. Dormann, E. Dreßel, U. Ebert, G. Eisenbach, C.D. Ernst, U. Esquinazi, P. Faulhammer, H. Fehn, T. Fehske, H. Feile, M. Feng, Q. Fesser, K. Feyerherd, B. Ficht, K. Fickenscher, M. Fiebig, A. Fischer, K.
Dr. Prof. Dr.*
Biochemie Exp.Physik Theor.Physik Anorg.Chem. Exp.Physik Exp.Physik Theo.Physik Exp.Physik Theor.Physik Phys.Chem. Makro.Chem. Theor.Physik Phys.Chem. Makro. Chem. Exp.Physik Theor.Physik Theor.Physik Biochemie/Exp.Phys. Anorg.Chem. Phys.Chem. Makro.Chem. Phys.Chem. Biochemie Exp.Phys. Exp.Phys. Exp.Phys. Exp.Phys. Phys.Chem. Makro.Chem. Anorg.Chem. Exp.Phys. Biochemie Exp.Phys. Theor.Phys. Exp.Phys. Theor.Phys. Theor.Phys. Makro.Chem. Makro.Chem. Exp.Phys. Exp.Phys. Makro.Chem.
D5 B11 C4 A1/C5 C9 D2 B6 B13 B6/B8 C1 C13 C4 D6 C3/C7 C23 C4 B8 D3/B4 C5 C18 C17 C1 D4/D5 C9 B3 B4 B4 C1 C16/C17 A1/C5 B10 D3 B13 B8 B4 C4 B6 C14 C17 C9 B4 C17
1994/95 1987–91 1985–89 1987/88/89 1984/85/86 1986 1991/95 1995 1984–1995 1987/88/89 1989/90/91/92 1990 1986/87/88 1984/85 1993/94 1991/92/93 1993/94/95 1984–89 1988/89 1988–92 1993/94/95 1986/87/88/89 1994/95 1986/87 1984–89 1984–1990 1988/89 1986/87/88/89 1988–1995 1985–89 1988/89 1984–1989 1992/93 1990–1995 1988/89 1990/91 1984–1995 1992 1990/91/92 1985/86/89/90 1986–90 1986/87/88/89
Dr.*
Prof. Dr.* Dr.* Dr. Dr. Dr.
Dr.**
Dr.* Dr.* Prof. Dr.* Prof. Dr. Dr. Dr.** Dr.* Prof.** Dr.*
Dr.*
451
22
Documentation of the Collaborative Research Centre 213
Name
Acad.grad
Institute
Project
Period
Fischer, W. Flöser, G. Förster, Ch. Franzke, D. Friedrich, J.
Dr.* Dr.* Dr.* Dr.* Prof.
Biochemie Exp.Phys. Biochemie Phys.Chem. Exp.Phys.
D4 D1 D4/D5 C12 B9/B15
Theor.Phys. Exp.Phys. Exp.Phys. Theor.Phys. Exp.Phys. Bioochemie Exp.Phys. Exp.Phys. Exp.Phys. Exp.Phys. Makro.Chem. Exp.Phys. Exp.Phys. Phys.Chem. Exp.Phys. Biochemie Makro.Chem. Anorg.Chem. Exp.Phys. Exp.Phys. Exp.Phys. Exp.Phys. Phys.Chem. Phys.Chem. Biochemie Phys.Chem. Biochemie Anorg.Chem. Phys.Chem. Exp.Phys. Makro.Chem. Phys.Chem. Exp.Phys. Biochemie Phys.Chem. Exp.Phys.
B5 B15 C9/10 B8 B4 D5 B12 B7 B12 A6 C14 B2 D2 C18 C9 D4/D5 C21 C5 B4 A3 A5/B3/B12 C6 C1/D6 D6 D3 D6 D4/D5 C5 C18 B4 C3 D6 B13 D8 C2 B9
1984/85 1984/85/86/88 1990/91/92/93 1987/88/89/91 1984–1989 1993–1995 1984/85 1994/95 1986–91 1990 1988 1985/86 1990–94 1991/92 1994/95 1986/87/88/89 1988/89 1987 1987/88/89 1988/89/90/91 1986/87/88/89 1988–92 1993/94/95 1988/89 1986–91 1984–89 1984–1995 1984–89 1984–88 1986/87/88/89 1985/86/87/88 1995 1987/88/89 1986 1993/94/95 1986/87 1986 1988/89 1992/93/94 1994/95 1986/87/88/89 1987/88/89
Frosch, H. Gafert, J. Gagel, R. Gammel, T. Gebhardt, H. Gebhardt-Singh, E. Geissinger, P. Geist, F. Giering, T. Gläser, D. Gollner, G. Gotschy, B. Gradl, G. Gradzielski, M. Graener, H. Grillenbeck, N. Grottenmüller, R. Grotz-Green, C. Gruner-Bauer, P. Güttler, W. Haarer, D. Häfner, W. Haegel, F.-H. Häfner, W. Härtl, P. Hahn, Ch. Halbritter, A. Hammon, W. Hecht, E. Heindl, D. Heitz, T. Henglein, F. Herold, M. Herrmann, F. Hertel, G. Hirschmann, R.
452
Dr.*
Dr.* Dr.*
Dr.* Dr.* Dr.**
Dr.* Dr. Prof. Dr. Dr.* Dr.*
Dr.**
Dr.* Dr.* Dr.*
22.4
Co-workers
Name
Acad.grad
Institute
Project
Period
Hochstraßer, D. Höcker, H. Hoff, H. Hoffmann, H. Hoffmann, K. Hofmann, J. Hofmann, M. Hofmann, M. Hofmann, W. Hoffmüller, P. Hopfmüller, H. Huber, G. Hübner, J. Hüser, B. Hums, E. Illner, J.- Ch. Juarez, M. de L.T. Kaiser, M. Kalus, J. Kanischka, G. Kaul, H. Kiefer, W. Klenke, M. Knöchel, F. Köhler, G. Köhler, M. Köhler, W. Kohles, N. König, R. Krämer, U. Kramer, L. Krapf, M. Krauss, H.L. Kreutzer, R. Kuang, W. Lattermann, G. Laubereau, A. Lenk, M. Lenz, U. Lieberth, M. Li, J. Limmer, St.
Dr.* Prof. Dr.* Prof. Dr.*
Theor.Phys. Makro.Chem. Makro.Chem. Phys.Chem. Anorg.Chem. Makro.Chem. Biochemie Exp.Phys. Exp.Phys. Biochemie Exp.Phys. Phys.Chem. Exp.Phys. Makro.Chem. Anorg.Chem. Phys.Chem. Theor.Phys. Theor.Phys. Exp.Phys. Makro.Chem. Exp.Phys. Exp.Phys. Phys.Chem. Makro.Chem. Exp.Phys. Exp.Phys. Exp.Phys. Exp.Phys. Exp.Phys. Phys.Chem. Theor.Phys. Exp.Phys. Anorg.Chem. Biochemie Theor.Phys. Makro.Chem. Exp.Phys. Biochemie Phys.Chem. Exp.Phys. Exp.Phys. Biochemie
D7 C3/C7 C17 C1/C2/C18 C5 C22 D4/D5 C9 B7 D4/D5 C6 C1/C2 B13 C8 C5 C1 C4 C4 B7 C3/C7 B2 C6 C12 C7 B11 B15 D2 C9 B10 C2 C4 B7 A1/C5 D4/D5 C4 C14 C9/10/Yw1 D4/D5 C1 B14 B10 D4
1986/87/88/89 1984–1986 1988/90/91 1984–1995 1986 1993/94/95 1985/86 1992/93/94/95 1984–88 1995 1984/85/86 1984–89 1993/94/95 1984/85/86 1984/85 1991–95 1988/89 1987/88/89 1984–1992 1984/85/86 1988/89 1984–1986 1992/93/94/95 1986 1987–92 1995 1984–88 1984/85/86/87 1988–93 1987/88/89 1984–1995 1990/91 1984–1992 1990/91/92 1986/87 1989–1995 1984–1995 1991–95 1988/89/90/92 1990/91 1993/94 1990–94
Dr.* Dr.*
Dr.*
Dr. Dr.* Dr.* Dr.* Prof. Dr.* Prof.
Dr.* Dr.* Dr. Dr.* Prof. Prof. Dr. Dr. Prof. Dr.*
Dr.
453
22
Name Lindenberger, F. Löbl, M. Löw, A. Lorenz, W. Luding, St. Maier, H. Martins, J. M. N. Materny, A. Matuschek, D. Mertens, F.G. Meyering-Vos, M. Merkel, K.-R. Meyer, G. Meyer, H. Mirke, M. Morys, P. Müller, K.-P. Müller, R.G. Müller-Horsche, E. Müller-Nawrath, R. Munkert, U. Nattler, G. Nawrot, B. Nefzger, H. Nittke, A. Nomayo, M. Nuyken, O. Oetter, G. Ott, G. Paap, H.-G. Pecher, U. Pesch, W. Peter, M. Platz, G. Pöhlmann, E. Pöhlmann, T. Pößnecker, G. Pobell, F. Pschierer, H. Purucker, H.-G. Raible, Ch. Rauscher, A.
454
Documentation of the Collaborative Research Centre 213
Acad.grad
Dr.*
Prof. Dr. Dr.* Dr.* Dr.* Prof. Dr. Dr.* Dr.* Dr.* Dr. Dr.* Dr. Dr.*
Prof. Dr.* Dr.
Prof. Dr. Dr.* Prof. Dr.
Dr.* Prof. Dr.* Dr.*
Institute
Project
Period
Exp.Phys. Phys.Chem Biochemie Exp.Phys. Exp.Phys. Exp.Phys. Makro.Chem. Exp.Phys. Exp.Phys. Theor.Phys. Biochemie Anorg.Chem. Phys.Chem. Exp.Phys. Makro.Chem. Anorg.Chem. Exp.Phys. Makro.Chem. Exp.Phys. Exp.Phys. Phys.Chem. Exp.Phys. Biochemie Makro.Chem. Exp.Phys. Phys.Chem. Makro.Chem. Phys.Chem. Biochemie Theor.Phys. Theor.Phys. Theor.Phys. Biochemie Phys.Chem. Exp.Phys. Makro.Chem. Phys.Chem. Exp.Phys. Exp.Phys. Exp.Phys. Exp.Phys. Phys.Chem.
C20 C1/C2 D3 B10/B13 B11 B10 C14 B2 B11 C19/D7 D4/D5 A1/C5 C12 B3 C21 C5 B10 C15 B3 B1 C2 C9 D4/D5 A6 B10 D6 C13 C1 D4 C4 B8 C4 D5 C1/C2 B4 C13 C2 B10 B15 C9 B13 C1
1991/1992 1984/85 1989 1990/91/92 1990/91/92 1992/93/94/95 1994/95 1987/88 87/88/89/90 1987–1995 1995 1985/86 1988/89/92 1989/90/91/92 1993/94/95 1986/87/88/89 1986–91 1988/89 1984/85 1984/85/86 1995 1988/89 1994 1986/87/88/89 1993/94 1993/94 1988–1992 1986/87/88 1988–1994 1985/86 1990/91/92/95 1984–88 1986/89 1984/85 1985/86 1990/91/92 1987/88/89/90 1987–1995 1993/94 1987/88/92 1991/92 1987–92
22.4
Co-workers
Name
Acad.grad
Institute
Project
Period
Rehage, H. Rehberg, I. Reichl, J. Reichstein, W. Reiser, Ch. Rentsch, S. Renner, H. Reul, St. Ribbe, A. Richter, W. Riederer, W. Robisch, P. Roder, S. Rösch, P. Röthlein, P. Röttger, J. Rothemund, P. Schellenberg, P. Schellhorn, M. Schießwohl, M. Schirmer, N. Schmelzer, U. Schmerbeck, S. Schmid, W. Schmidt, H. Schmidt, St. Schmidt, M. Schmutzler, M. Schnabel, E. Schneider, J.M. Schnitzer, H.-J. Schnörer, H. Schörner, H. Schultes, H. Schulz, S. Schumann, Ch. Schwandner, B. Schwarz-Schultz Schwoerer, M. Seidler, L. Seilmeier, A. Sesselmann, R.
Dr.** Dr. Dr.*
Phys.Chem. Theor.Phys. DipL.Phys. Exp.Phys. Biochemie Exp.Phys. Phys.Chem. Exp.Phys. Makro.Chem. Exp.Phys. Anorg.Chem. Phys.Chem. Exp.Phys. Struktur und Chem. d. Biop. Exp.Phys. Makro.Chem. Phys.Chem. Exp.Phys. Makro.Chem. Biochemie Biochemie Exp.Phys. Anorg.Chem. Exp.Phys Anorg.Chem. Makro.Chem. Makro.Chem. Exp.Phys Phys.Chem. Makro.Chem. Theor.Physik Exp.Phys. Makro.Chem. Exp.Phys. Phys.Chem. Anorg.Chem. Phys.Chem. Biochemie Exp.Phys. Biochemie Exp.Phys. Exp.Phys.
C1/C2 C4 B5 A5 D5 Ye1 C2 B10 C17 B14 C5 C2 B7 D8 B7 C3/C7 C1 B15 C14 D4 D5 B7 C5 B2/B13 A1 C13/C14 C21 C9 C1 C15 C19 B11 C15 B4 C1 A1 C1 D4 B1/B2/B13 D5 C23 D1
1984–89 1988/89 1984–86/88/89 1985–89 1987–91/93 1992–1995 1994/95 1991/92 1992 1990–1995 1986/87/88/89 1988/89 1986 1993–1995 1990 1984/85/86 1986/87 1994 1990/91/92/93 1990/91/92 1991/92 1986/87/88/89 1984/85/86 1994 1986/87/88/89 1990/92 1993–1995 1985/86 1986/87 1992 1993/94/95 1987–92 1986/87/88/89 1984/85/86/87 1990/91/92 1990/91/92/93 1984/85/86 1985/86/87 1984–1995 1984/85/86 1993–1995 1987/88
Dr.* Doz., Dr. Dr.* Dr.* Dr.* Dr. Dr.*
Prof.
Dr.* Dr.* Dr.* Dr. Dr.* Dr.*
Prof.
Dr.* Dr.* Dr. Dr.* Dr. Prof. Dr.* Prof. Dr.*
455
22
Name
Documentation of the Collaborative Research Centre 213
Acad.grad
Siebenhaar, B. Dr.* Spiess, H.W. Prof. Sprinzl, M. Prof. Sühler, G. Staufer, G. Dr.* Stöcklein, W. Strohriegl, P. Dr.** Struller, B. Dr.* Sum, U. Dr.* Terskan-Reinhold, M. Thaufelder, H. Dr.* Thom, W. Thurn, H. Dr. Thurn, R. Trapper, U. Treiber, M. Troeger, P. Dr.* Überla, H. Dr.* Ulbricht, W. Dr. Völkel, A. Vogtmann, T. Dr.* Voit, J. Dr.** Voit, S. Vollstädt, K.-U. Vornlocher, H.-P. Voss, D. Waas,V. Dr.* Wagner, R. Dr.* Walther, K.L. Dr.* Wanka, G. Weber, R. Dr.* Weber, S. Dr.* Wefing, St. Weidlich, K. Dr.* Weiss K. Dr.** Weigel, J. Weißhaar, M. Dr. Wiesner, J. Dr.* Wietasch, H. Winter, H. Wittenbeck, P. Wokaun, A. Prof.
456
Institute
Project
Period
Anorg.Chem. Makro.Chem. Biochemie Phys.Chem. Makro.Chem. Exp.Phys. Exp.Phys. Phys.Chem. Theor.Phys. Makro.Chem. Makro.Chem. Theor.Physik Phys.Chem. Exp.Phys. Theor.Phys. Theor.Phys. Exp.Phys. Theor.Phys. Phys.Chem. Theor.Phys. Exp.Phys. Theor.Phys. Exp.Phys. Exp.Phys. Biochemie Exp.Phys. Theor.Phys. Exp.Phys. Phys.Chem. Phys.Chem. Phys.Chem. Theor.Phys. Makro.Chem. Exp.Phys. Anorg.Chem. Phys.Chem. Biochemie Phys.Chem. Makro.Chem. Exp.Phys. Phys.Chem. Phys.Chem.
C5 C8 D4/D5 C1/C2 C14 B1 B2/B4 C2 B6 C17 C16/17 C4 C1 C6 B8 C4 C10 B5 C1/C2 C19 B13 B8 B13 B13 D4/D5 B3 B8 B2 C12 C2 C2 B5/B8 C8 C9 A1 C12 D4/D5 C12 C14 B4 C12 C12/D6
1987/88/89 1984–1986 1984–1995 1993/94/95 1986/87/88/89 1985 1984–89 1986/87/88/89 1984–89 1992/93/94 1986–91 1987/88 1987 1984–89 1995 1992//93/94/95 1984/85 1984/85/86 1984–89 1990/91/92/93 1987/88/89/92 1986/87/88/89 1990 1990/91/92 1992/93 1995 1986/87/88/89 1987–91 19898/89 1988/89 1984/85/86 1984–89 1984/85/86 1994/95 1986/87/88/89 1993/94 1987/88/89 1986–90 1994/95 1987/88/89 1988/89 1987–1995
22.5
International Cooperation
Name
Acad.grad
Institute
Project
Period
Wolf, M. Wolff, P. Wolfrum, K. Wunderlich, I. Yamaguchi, Y. Ye, T. Zahn, M. Zimmermann, F. Zimmermann, H. Zimmermann, W. Zollfrank, J.
Dr.*
Theor.Phys. Anorg.Chem. Exp.Phys. Phys.Chem Phys.Chem. Exp.Phys. Anorg.Chem. Phys.Chem. Exp.Phys. Exp.Phys. Exp.Phys.
B6 C5 C9 C1/C2 C1 C9 C5 D6 B7 C4 B9
1989–94 1987/88/89 1986/87/92 1984–89 1993/94/95 1987–91 1986/87/88/89 1988–93 1984/85 1984–89 1987/88
Dr.* Dr.* Dr.* Dr* Dr.*
Support of young scientists * “Promotion” supported by the Sonderforschungsbereich 213 ** “Habilitation” supported by the Sonderforschungsbereich 213 and about 200 diploma theses
22.5
International Cooperation
A6 Prof. Hashimoto, Kyoto University, Kyoto, Japan Dr. H.G. Schmeler, Miles Inc. Pitsburgh, PA, USA B3 NTT, Tokyo, Japan Universität Wien, Österreich B6 Los Alamos National Laboratory T-11, USA Prof. Y. Ono, Japan B8 Dr. Bishop, LANL, USA Los Alamos National Laboratory, Theoretical Group, Los Alamos, NM, USA B 10 MIT, Boston, USA Prof. Yu Kogan, Dr. Burin, Kurchatov Institut, Moskau, Rußland Dr. R. Nava, Universidad Central de Venezuela, Caracas 457
22
Documentation of the Collaborative Research Centre 213
B12 Prof. R. Silbey, MIT Cambridge, USA Prof. Skinner, Dept. of Chemistry, University Wiscousin, USA C1/C2/C18 Prof. Dr. J.S. Candau, Universite Louis Pasteur, Strasbourg, Frankreich Dr. M.E. Cates, Cavendish, Laboratory Cambridge, Großbritannien Prof. Dr. C.A. Miller, Rice-University, Houston USA Prof. Dr. D. Langevin, Universite Paris VI, Paris, Frankreich Dr. K. Esumi, Nagoya University Tokyo, Japan Dr. Kell Mortensen, Risö Laboratorium, Dänemark Doz. Dr. P. Stern, Akad. d. Wissenschaften Tschechien Dr. D. Roux, CNRS Pessac, Frankreich C4 Weizmann Institut, Dept. of Physics, Israel Group de Physique de Solides, Orsay, Frankreich Dept. of Physics, University California, Santa Barbara, USA Dept. of Physics, Hebrew-University, Jerusalem, Israel Dept. of Physics, CALTECH, Passadena, USA Dept. of Physics, Northwestern University, Evanston, USA Centro Atomico, Bariloche, Argentinien Prof. G. Ahlers, Santa Barbara, USA Prof. A. Chuvyrov, Ufa, Russland Prof. V. Steinberg, Rehovot, Israel Dpto. Estructura y Constiuyentos de la Materia, Facultät de Fisica, Universität Bacelona, Spanien C10 Laser Research Center,Vilnius, Litauen C12 ETH Zürich Prof. F.R. Aussenegg/Univ. Doz. Dr. A. Leitner, Institut für Experimentalphysik, Graz, Österreich Prof. A. Baiker, Chemieingenieurwesen und Industrielle Chemie, ETH Zürich, Schweiz PD. Dr. A. Leitner, Institut für Experimentalphysik, Universität Graz, Österreich C13 Slowakische Techn. Hochschule, CSFR C14 Dr. A.M. Levelut, Universite Orsay, Frankreich Dr. B. Gallot, Laboratoire des Materiaux Organiques, CNRS, BP 24,Vernaison, Frankreich C16/17/22 Prof. Blackwell, Case Western Reserve University, Cleveland, Ohio, USA Prof. Mac Knight, University of Massachusetts, Amherst, USA Dr. J. Noolandi, Xerox Research Centre Canada, Missisauga, Ontario, Canada Prof. Newkomo, Univ. of South Florida, Tampa, Florida 458
22.6 Funding C19 Center of Nonlinear Studies, Los Alamos, National Laboratory, Los Alamos USA Servece National des Champes Intenses, Grenoble, Frankreich C21 N. Pogodina, Institute of Physics, Universite St. Petersburg, Russia D4/D5 Prof. Dr. A. Redfield, Brandeis University, USA Prof. Dr. O. Uhlenbeck, University of Boulder, USA Prof. Dr. A. Spirin, Academy of Science, Russia Prof. Dr. M. Boublik, Roche Institute of Molecular Biology, Nutley, New Jersey, USA Prof. Dr. B. F. Clark, Aarhus, Denmark Prof. Dr. J. Nyborg, Aarhus, Denmark Prof. Dr. O. Lavrik, Russia Prof. Dr. L. Bosch, Leiden University, The Netherlands Prof. Dr. P. Sigler, Yale University, USA Prof. Dr. J. Heinfeld, USA Dr. M. Makinen, USA Dr. L. Arnold, Czech Republic Prof. Dr. L. Spremulli, USA Prof. Dr. R. Giegé, France Prof. Dr. A. Barciszewski, Poland D6 Prof. Dr. H. Eicke, Institut f. Phys. Chemie, Universität Basel, Schweiz D8 Prof. K. Kirschner, Schweiz Prof. A. Yaniv, Israel Prof. M. Breitenbach, Österreich Dr. M. Auer, Österreich
22.6
Funding
The Collaborative Research Centre 213 was supported by grants of the Deutsche Forschungsgemeinschaft totalling DM 28 814 200 in the period 1984–1995.
459
