The Latest Development of the Weak Segregation Theory of Microphase Separation In Block Copolymers

Igor Erukhimovich

NanoScience and Technology NanoScience and Technology Series Editors: P. Avouris B. Bhushan D. Bimberg K. von Klitzing H. Sakaki R. Wiesendanger The series NanoScience and Technology is focused on the fascinating nano-world, mesoscopic physics, analysis with atomic resolution, nano and quantum-effect devices, nanomechanics and atomic-scale processes. All the basic aspects and technology-oriented developments in this emerging discipline are covered by comprehensive and timely books. The series constitutes a survey of the relevant special topics, which are presented by leading experts in the field. These books will appeal to researchers, engineers, and advanced students. Applied Scanning Probe Methods II Scanning Probe Microscopy Techniques Editors: B. Bhushan and H. Fuchs Applied Scanning Probe Methods VI Characterization Editors: B. Bhushan and S. Kawata Applied Scanning Probe Methods III Characterization Editors: B. Bhushan and H. Fuchs Applied Scanning Probe Methods VII Biomimetics and Industrial Applications Editors: B. Bhushan and H. Fuchs Applied Scanning Probe Methods IV Industrial Application Editors: B. Bhushan and H. Fuchs Roadmap of Scanning Probe Microscopy Editors: S. Morita Scanning Probe Microscopy Atomic Scale Engineering by Forces and Currents Editors: A. Foster and W. Hofer Nanocatalysis Editors: U. Heiz and U. Landman Single Molecule Chemistry and Physics An Introduction By C. Wang and C. Bai Atomic Force Microscopy, Scanning Nearfield Optical Microscopy and Nanoscratching Application to Rough and Natural Surfaces By G. Kaupp Applied Scanning Probe Methods V Scanning Probe Microscopy Techniques Editors: B. Bhushan, H. Fuchs, and S. Kawata Nanostructures Fabrication and Analysis Editor: H. Nejo Fundamentals of Friction and Wear on the Nanoscale Editors: E. Gnecco and E. Meyer Lateral Alignment of Epitaxial Quantum Dots Editor: O. Schmidt Nanostructured Soft Matter Experiment, Theory, Simulation and Perspectives Editor: A.V. Zvelindovsky A.V. Zvelindovsky (Ed.) Nanostructured Soft Matter Experiment, Theory, Simulation and Perspectives With 261 Figures Dr. A.V. Zvelindovsky (Ed.) Centre for Materials Science Department of Physics, Astronomy and Mathematics University of Central Lancashire Preston Lancashire PR1 2HE United Kingdom Series Editors: Professor Dr. Phaedon Avouris Professor Dr., Dres. h.c. Klaus von Klitzing IBM Research Division Nanometer Scale Science & Technology Thomas J. Watson Research Center P.O. Box 218 Yorktown Heights, NY 10598, USA Max-Planck-Institut für Festkörperforschung Heisenbergstr. 1 70569 Stuttgart, Germany Professor Dr. Bharat Bhushan University of Tokyo Institute of Industrial Science 4-6-1 Komaba, Meguro-ku Tokyo 153-8505, Japan Ohio State University Nanotribology Laboratory for Information Storage and MEMS/NEMS (NLIM) Suite 255, Ackerman Road 650 Columbus, Ohio 43210, USA Professor Dr. Dieter Bimberg TU Berlin, Fakutät Mathematik/ Naturwissenschaften Institut für Festkörperphyisk Hardenbergstr. 36 10623 Berlin, Germany Professor Hiroyuki Sakaki Professor Dr. Roland Wiesendanger Institut für Angewandte Physik Universität Hamburg Jungiusstr. 11 20355 Hamburg, Germany A C.I.P. Catalogue record for this book is available from the Library of Congress ISSN 1434-4904 ISBN 978-1-4020-6329-9 (HB) ISBN 978-1-4020-6330-5 (e-book) Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands In association with Canopus Publishing Limited, 27 Queen Square, Bristol BS1 4ND, UK www.springer.com and www.canopusbooks.com All Rights Reserved © Canopus Publishing Limited 2007 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means,electronic,mechanical,photocopying,microﬁlming,recording or otherwise,without written permission from the Publisher, with the exception of any material supplied speciﬁcally for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Preface “The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living.” Henri Poincaré (1854 - 1912) The ancient Greeks, quite ingeniously, realised that all materials and their (now known as macroscopic) properties, including life itself, are due to a limited number of tiny, constantly moving building blocks and the connections (now called interactions) between these blocks. Receiving both scientiﬁc and non-scientiﬁc opposition, the idea faded and, despite some renaissance of atomistic ideas in the 17-19th centuries, it still took more than two thousand years, until the time of Einstein, for the idea of microscopic building blocks to be fully accepted. These ideas, begun during the golden age of physics in the 20th century, have led to a comprehensive understanding of such states of matter as gases and solids, which in turn have completely revolutionised everyday life in the developed world by introducing technological wonders such as modern cars, air traffic, semiconductor chips for computers and nuclear power. Another state of matter, ﬂuids, appeared to be much more difficult to tackle, even in the case of simple liquids like liquid argon, a research favourite in the ﬁeld. Legend tells that Lev D. Landau, Physics Nobel Laureate, was said to have commented that there could be no theoretical physics of liquids, as they have no small parameters. Nonetheless, as the 20th century advanced, it also became possible to treat even this most slippery of subjects due, in part, to the introduction of computers and the development of computer simulation methods like molecular dynamics. The 20th century brought yet another revolution: the industrial production of novel classes of materials, which simply did not exist before. For instance, almost every aspect of our everyday life would change immeasurably if plastics should disappear and life would turn “blind”, “deaf” and rather miserable without liquid crystals for computer screens or mobile phones. Such new materials were given the name complex fluids, and their building blocks are not simply atoms or small molecules, but include block copolymers, surfactants, amphiphiles, colloids, liquid crystals, biomacromolecules, such as proteins and DNA, and various composites of the above. Complex ﬂuids possess features of both ﬂuids (for instance, they can ﬂow) and solids (they can have an internal structure often with various well VI Preface resolved symmetry groups). These structures have a characteristic scale for their building blocks which is in the range of nanometers to microns, but the building blocks can be made (synthesised) with various degrees of complexity, so more than one size scale can be involved. Some structures can be formed spontaneously from a homogeneous mixture of the building blocks, a process referred to as self-assembly, which can be hierarchical and occur on various time scales depending on the complexity of the building blocks. Self-assembly is related to self-organization, which makes complex ﬂuids similar to living matter, so they can serve as model systems for biological systems and bioinspired materials. In the last decades of the 20th century the term complex ﬂuids started to be substituted by a more general one that is better suited to the overall concept of condensed matter: soft matter. The transition between millennia was marked by a burst of soft matter research, due, in part, to the fact that computers had then reached a level of power allowing the simulation of experimental size systems, thus enabling the very ﬁrst “virtual experiments” of such complex systems to be performed. This development made the links between theory and experiment truly symbiotic. Nanostructured soft materials, even apart from future technological perspectives beyond our imagination, are fascinating and beautiful. This research ﬁeld is growing so fast that there has been no single book that provided an overview of the many diﬀerent perspectives on both fundamental concepts and recent advances in the ﬁeld. A group of very enthusiastic contributors has now ﬁlled this gap; and the present book is the ﬁrst comprehensive monograph on nanostructured soft matter. It covers materials ranging in size from short amphiphilic molecules to block copolymers to proteins and also discusses colloids, hybrids, microemulsions and bio-inspired materials such as vesicles. Each chapter is written by active world-class researchers in the ﬁeld who oﬀer the reader an interdisciplinary view from diﬀering perspectives. They combine the experimental approaches of Chemistry and Physics, e.g. scattering techniques, electron and Atomic Force microscopy, with various Theoretical Physics, Mathematics and advanced computer modelling methods. We hope the book will be useful for both active and starting researchers as well as for undergraduate students; or, citing one of the anonymous referees of the original proposal for this book: “There is something for everyone in this book and it would represent a very useful text for those both operating at the forefront of nano-science and those entering the field . . . ” I wish to thank the publishers at Canopus for assistance in the production of this book. I also thank Drs. R. McCabe, S. V. Kuzmin and N. Kiriushcheva. My editorial eﬀort is dedicated to Prof. A. V. Zatovsky (1942-2006), who ﬁrst introduced me to the wonders of Soft Matter. Preston, Lancashire, January 2007 AVZ Contents Preface A. V. Zvelindovsky (ed.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V Part I Experimental Advances Microemulsion Templating W. F. C. Sager . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Nanofabrication of Block Copolymer Bulk and Thin Films: Microdomain Structures as Templates Takeji Hashimoto and Kenji Fukunaga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Characterization of Surfactant Water Systems by X-Ray Scattering and 2 H NMR Michael C. Holmes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Polyelectrolyte Diblock Copolymer Micelles: Small Angle Scattering Estimates of the Charge Ordering in the Coronal Layer Johan R. C. van der Maarel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Structure and Shear-Induced Order in Blends of a Diblock Copolymer with the Corresponding Homopolymers I. W. Hamley, V. Castelletto and Z. Yang . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Electric Field Alignment of Diblock Copolymer Thin Films T. Xu, J. Wang and T. P. Russell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Control of Block Copolymer Microdomain Orientation from Solution using Electric Fields: Governing Parameters and Mechanisms Alexander Böker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 VIII Contents Structure and Dynamics of Cylinder Forming Block Copolymers in Thin Films Larisa Tsarkova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Part II Mathematical and Theoretical Approaches Mathematical Description of Nanostructures with Minkowski Functionals G.J.A. Sevink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Scaling Theory of Polyelectrolyte and Polyampholyte Micelles Nadezhda P. Shusharina and Michael Rubinstein . . . . . . . . . . . . . . . . . . . . . 301 The Latest Development of the Weak Segregation Theory of Microphase Separation In Block Copolymers I. Ya. Erukhimovich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Coarse-Grained Modeling of Mesophase Dynamics in Block Copolymers Zhi-Feng Huang and Jorge Viñals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 Effective Interactions in Soft Materials Alan R. Denton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 Part III Computer Simulations Ab-initio Coarse-Graining of Entangled Polymer Systems J.T. Padding and W.J. Briels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 Computer Simulations of Nano-Scale Phenomena Based on the Dynamic Density Functional Theories: Applications of SUSHI in the OCTA System Takashi Honda and Toshihiro Kawakatsu . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 Monte Carlo Simulations of Nano-Confined Block Copolymers Qiang Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 Understanding Vesicles and Bio-Inspired Systems with Dissipative Particle Dynamics Julian C. Shillcock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 Theoretical Study of Nanostructured Biopolymers Using Molecular Dynamics Simulations: A Practical Introduction Danilo Roccatano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 Contents IX Understanding Liquid/Colloids Composites with Mesoscopic Simulations Ignacio Pagonabarraga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 List of Contributors Alexander Böker (Germany) Physikalische Chemie II, Universität Bayreuth. Wim J Briels (The Netherlands) Computational Biophysics, Faculty of Science and Technology, University of Twente, Enschede. Valeria Castelletto (United Kingdom) Department of Chemistry, University of Reading. Alan R Denton (USA) Department of Physics, North Dakota State University, Fargo. Igor Erukhimovich (Russia) Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences, Moscow, and Physics Department, Moscow State University. Kenji Fukunaga (Japan) Polymer Laboratory, Ube Industries, Ltd., Chiba. Ian W Hamley (United Kingdom) Department of Chemistry, University of Reading. Takeji Hashimoto (Japan) Advanced Science Research Centre, Japan Atomic Energy Agency, Ibaraki. Michael C Holmes (United Kingdom) Centre for Materials Science, Department of Physics, Astronomy and Mathematics, University of Central Lancashire, Preston. Takashi Honda (Japan) Japan Chemical Innovation Institute, and Department of Organic and Polymeric Materials, Tokyo Institute of Technology. Zhi-Feng Huang (USA, Canada) Department of Physics and Astronomy, Wayne State University, Detroit and McGill Institute for Advanced Materials and Department of Physics, McGill University, Montreal. Toshihiro Kawakatsu (Japan) Department of Physics, Tohoku University, Sendai. XII List of Contributors Johan R C van der Maarel (Singapore) Department of Physics, National University of Singapore. Nadezhda P Shusharina (USA) Department of Chemistry, University of North Carolina at Chapel Hill. Johan T Padding (The Netherlands) Computational Biophysics, Faculty of Science and Technology, University of Twente, Enschede. Larisa A Tsarkova (Germany) Physikalische Chemie II, Universität Bayreuth. Ignacio Pagonabarraga (Spain) Departament de Física Fondamental, Universitat de Barcelona. Jorge Viñals (Canada) McGill Institute for Advanced Materials and Department of Physics, McGill University, Montreal. Danilo Roccatano (Germany) School of Engineering and Science, Jacobs University Bremen. Michael Rubinshtein (USA) Department of Chemistry, University of North Carolina at Chapel Hill. Thomas P Russell (USA) Polymer Science and Engineering Department, University of Massachusetts, Amherst. Wiebke F C Sager (Germany) IFF-Soft Matter, Forschungszentrum Jülich. G J Agur Sevink (The Netherlands) Leiden Institute of Chemistry, Leiden University. Julian C Shillcock (Germany) Theory Department, Max Plank Institute of Colloids and Interfaces, Potsdam. Jiayu Wang (USA) Polymer Science and Engineering Department, University of Massachusetts, Amherst. Qiang Wang (USA) Department of Chemical and Biological Engineering, Colorado State University, Fort Collins. Ting Xu (USA) Department of Materials Science and Engineering, University of California, Berkeley. Zhou Yang (United Kingdom) School of Materials, The University of Manchester. Andrei V Zvelindovsky (United Kingdom) Centre for Materials Science, Department of Physics, Astronomy and Mathematics, University of Central Lancashire, Preston. The latest development of the weak segregation theory of microphase separation in block copolymers Igor Erukhimovich 1. Introduction. One of the most interesting phenomena occurring in copolymer systems is so-called order-disorder transition (ODT) or microphase separation, i.e. formation of ordered morphologies possessing symmetry of a crystal lattice,1-5 which occurs with changing (typically decreasing) temperature T. With further decrease of T the ODT is often followed by various order-order transitions between the different ordered morphologies1. Obviously, the physical reason for this ordering is competition between the short-range segregation and long-range stabilization tendencies. More precisely, with decreasing T the energy gain upon local segregation grows as compared to the loss of the translational entropy accompanying such segregation whereas the immiscible blocks can not separate fully because of their covalent bonding. As a result, an ordered pattern of alternating domains, which are filled preferably by the monomers of the same sort, arises. Block copolymer melts with different structural and interaction parameters are known to form different morphologies at the orderdisorder and order-order transitions so that the ultimate goal of the theory is to determine the symmetry and geometry of the most stable ordered phases for a copolymer melt or blend given its composition, architecture and temperature as well as predict the thermodynamic, scattering and mechanical properties of the phases. The milestones in achieving this goal would be i) a general understanding of the relationships between the block copolymer architecture and the symmetry of the occurring ordered morphologies as well as scenarios of the order-disorder and order-order transitions; ii) a quantitative description and calculation of the stable ordered phases structure; and iii) calculation of various parameters of the phases. Therewith, having in mind the huge variety of the block copolymer architecture the chemists could synthesize, it is very important that the theory would be capable to provide an express information (even though a semiquantitative one) about new copolymers rather than explain the already observed systems only. Putting it by words of Reimund Stadler, “the large number of a priori imaginable combinations in ABC triblock copolymers makes it necessary that theory and experiment have to be closely linked. By such a close feedback it may be avoided that the development of new materials is merely based on accident.” 1 In what follows we use the terms “morphology”, “ordered phase” and “symmetry class” as synonyms. 1 The first ever attempt to describe theoretically the microphase separation transition was done by Meier6 within so-called strong segregation approximation (SSA). According to the SSA, the alternating domains forming the ordered morphologies are just microphases, which consist of the corresponding chemically homogeneous blocks, the width d of the inter-domain transient layer (domain interface) being rather narrow as compared to the domain size L. Therefore, the SSA is based on the idea that the ODT is controlled by a balance between an entropic loss due to confinement of the polymer blocks within (or outside of) the domains (micelles) and energetic gain (as compared to the uniform state) under formation of these micelles. Since within the SSA the unlike monomers contact and interact within the domain interfaces only, the energetic gain is basically determined by the values of the total domain interface and interface (surface) free energy. The SSA, which was further elaborated by Helfand et al.7-10 and, finally, by Semenov et al.,11-14 results naturally (in the limit of zero interface width) in the idea that the morphology of the ordered phases is fully determined by the requirement that the total inter-domain surface is minimal. Very transparent and appealing from the geometrical point of view, the minimal surface approach is rather popular among mathematicians and experimentalists.15-18 However, the condition of the narrow interface d<<L is practically never fulfilled in the real block copolymer melts above the glass transition temperature Tg . E.g., in the binary AB diblock copolymers the condition d<<L fulfills for χ ≥ 100 ,19 where χ = χN is the reduced Flory interaction parameter20, N is the total degree of polymerization of the diblock copolymer chain. In case the Flory parameter χ is related to the temperature T via the simple relationship 2χ = θ T , θ being the Flory temperature, the condition holds for rather low temperatures T ≤ θN 100 ~ TODT 10 , where TODT is the ODT temperature calculated by Leibler21 in the opposite so-called weak segregation approximation (WSA). Besides, the minimal surface approach disregards a long-range contribution11 due to micelles’ ordering. The WSA, on which this review is focused, is related to the situation at the very onset of ordering. The physical idea the WSA is based on was first stated by Landau22 as early as in 1937 as a toy model of the 2nd order phase transition accompanied by an explicit ( F {Φ ( r )} )=1 ( ΔF r, {Φ ( r )} ) dr . symmetry change. In the contemporary designations the original Landau Hamiltonian reads: Φ ( r1 ) c ( r1 − r2 ) Φ ( r2 ) d r1 d r2 + ∫ 2∫ (1.1) Here the Fourier transform c ( q ) = ∫ d r c ( r ) exp ( iqr ) has a minimum at a finite q = q* , T T the order parameter Φ is assumed to be scalar for a while, the specific excess free energy of the system in a point r 2 ( ΔF r, {Φ ( r )} ) = α Φ3 ( r ) + β Φ 4 ( r ) 3! T (1.2) 4! is determined by the value Φ(r) in the same point only and T is the temperature measured in the energetic units (the Boltzmann constant kB = 1). the Hamiltonian (1.1) is in a state described by a profile {Φ ( r )} with a probability As consistent with general principles of statistical physics,23 the system described by ( ( ( ) ) ) w {Φ ( r )} ~ Z −1 exp − F {Φ ( r )} T , (1.3) corresponding functional integral over all profiles {Φ ( r )} : the normalization constant Z being the partition function, which can be written as the ( ( ) ) Z = ∫ exp − F {Φ ( r )} T δΦ ( r ) (1.4) Thus, the total free energy of the system and any observable quantity a, which, evidently, is just a thermodynamic average, i.e. the average taken with the probabilistic measure (1.3), ( ( ) ) F = −T ln Z = −T ln ∫ exp − F {Φ ( r )} T δΦ ( r ) read a ({Φ ( r )} ) exp ( − F ({Φ ( r )} ) T ) δΦ ( r ) ) ∫ exp − F Φ r T δΦ r ∫ ( ({ ( )}) ) ( ) ( a = a {Φ ( r )} = (1.5) (1.6) Φ (r ) = Φ (r ) The most important observables are, of course, the average order parameter (1.7) S ( r1 − r2 ) = ( Φ ( r1 ) − Φ ( r1 ) ) ( Φ ( r2 ) − Φ ( r2 ) ) and the correlation function ( ) ( ( as well as the scattering factor G ( q ) = ∫ S r1 − r 2 exp i q r1 − r 2 ) ) dr1dr 2 (1.8) V, (1.9) where V is the total volume of the system. For τ>0, a minimum (at least, a metastable one) of the virtual free energy (1.1) is Φ (r ) ≡ 0 . provided by the order parameter profile (1.10) Then Φ ( r ) = 0 , whereas to calculate the correlation function and scattering factor one can keep only the first non-vanishing (quadratic) terms in the expansion of the Hamiltonian (1.1) appearing in the integrals (1.5)-(1.8). In this approximation, which is referred to as the random phase approximation (RPA), the desired expressions read: 3 ( ⎡ G ( q ) = 1 c ( q ) ≈ ⎢ τ + C q 2 − q*2 ⎣ ) ⎥⎦ 2 ⎤ −1 , (1.11a) value of the minimum τ = c ( q* ) , which plays role of a reduced temperature, measuring where the second approximate equality holds near the minimum of the function c(q). If the how close is the spatially uniform (disordered) system to the loss of its stability, is small enough, then the correlation function reads (in an approximation valid for τ ( ( ) ( r τC ) ) ( q* τC ) ⎧sin ( q*r ) exp − r τ C ( 2q* ) ⎪ S (r ) ∼ ⎨ ⎪cos ( q*r ) exp − r τ C ( 2q* ) ⎩ d =3 d =1 . q*4 ) (1.11b) The behavior of the weakly ordering systems described by the correlation function (1.11b), which is shown in Fig. 1,.is rather different from that of the simple liquids2. Since the correlation function is proportional to the response function,23,24 the equation (1.11b) implies that any perturbation arising in the weakly ordering systems is propagating as a harmonic wave (spherical, cylindrical or plane one depending on the perturbation’s dimensionality), which oscillates with the period L = 2π q* and decays on the distance rc ~ q* C τ . The waves’ amplitude is infinitely increasing (within the RPA) as τ → 0 . Figure 1. Typical behavior of the plane perturbations in the weakly ordering systems. The curves of different colors correspond to different values of the reduced (positive) temperature τ, the values of τ are decreasing and the system is approaching ODT in the downward direction. 2 S ( r ) ∼ exp ( − r rc ) r , where rc is the correlation radius. Remember that the correlation function of the simple liquids is described by the Ornstein-Zernike correlation function 4 At τ ≤ 0 the perturbations become infinite. Were there no other restrictions, it would mean that the spatially homogeneous state becomes absolutely unstable with respect to the waves as the critical ones) and the RPA is not valid anymore. In this case Φ ( r ) = 0 growth of the order parameter waves with the wave lengths L (henceforth we refer to such provided by a finite thermodynamically equilibrium order parameter profile Φ ( r ) ≠ 0 . corresponds to a maximum of the virtual free energy, whereas the minimum of the latter is In fact, however, the local order parameter cannot exceed some finite value (e. g., for diblock copolymer melts Φ is a linear function of the volume fraction of the monomers A), which implies that with increase of the fluctuations they become strongly correlated even in the disordered state, the correlation being resulted in a decrease of the correlation function S(r) and, thus, a stabilization of the disordered (spatially homogeneous) state as compared to the RPA results. Quantitatively this effect is addressed and explained by so-called Brazovskii (Hartree) fluctuation corrections25 to the mean field approximation, which are shown to transform the order-disorder transition into the 1st order one even at the critical point. new equilibrium order parameter profile Φ ( r ) ≠ 0 could be neglected, which implies In the mean field approximation, i.e. under assumption that the fluctuations around the calculation of the integral (1.4) via the deepest descent method within the so-called pre- ( F = −T ln Z = F {Φ ( r )} ) exponential accuracy, we arrive at the well-known expression for the free energy (1.12) profile Φ ( r ) is expected to be build by all the critical waves basically. However, since the It follows from the presented discussion that at the onset of ordering the equilibrium critical waves are coupled via the non-linear excess free energy (1.2), the critical waves both interfere and generate some new waves (so-called higher harmonics). As a result, only some discrete sets of these waves survive, each stable set of the standing order parameter If the symmetry of a morphology is that of a spatial lattice ℜ, Φ ( r ) , generally, is an waves being corresponded to a crystal lattice. infinite series in the Fourier harmonics corresponding to the set of the points of the lattice ℜ-1 reciprocal to ℜ:22 Φ (r ) = ∑ qi ∈ℜ −1 A ( qi ) exp i ( q i r + ϕi ) (1.13) But close to the critical point τ = 0, α = 0 , where α is the coefficient appearing in the cubic term of the Landau expansion (1.2), the coupling generating the higher harmonics is 5 small, so that one can keeps in the expansion (1.13) the main (or primary) harmonics only, Φ ( r ) = AΨ ( r ) , Ψ ( r ) = ∑ exp i ( q i r + ϕ ( qi ) ) i.e. those whose wave numbers are equal to the critical value q* : qi = q* (1.14) Then arising of the spatially periodic order parameter could proceed (in the mean field approximation) as the 2nd order phase transition or the 1st order phase transition close to the 2nd order one, which situation is referred to as the weak crystallization.22,26,27 It was shown21,28,29 basing on the universal (Gaussian) conformational behavior of the long polymer blocks that the Landau instability does occur in block copolymer systems. E.g., for the binary incompressible AB copolymers the function c(q), which appears in (1.1) g AA ( q ) + g BB ( q ) + 2 g AB ( q ) and according to (1.11a) is just the inverse scattering intensity, was shown21,28-30 to read c (q) = g AA ( q ) g BB ( q ) − ( g AB ( q ) ) 2 − 2χ , (1.15) where g ( q ) = gij ( q ) is so-called structure matrix (see section 3), which is determined by the macromolecules’ architecture only. The plots c(q) for molten diblock copolymer which were first calculated by Leibler,21 are presented n Fig. 2 for various χ (temperatures). It is easy to see that the main prediction of the theory in the disordered state is that the only change of the shape of the curve c ( q ) ~ I −1 ( q ) is its downward shift as the whole. correspond to the values of χ = 2 ( n − 1) . Figure 2. The curves c(q) labeled by numbers n Figure 3. Inverse intensity for different temperatures. The curves correspond to T of 240,220,200,180,160 and 100 oC (from top to bottom). (Fig. 5a of ref 31). As is seen from Fig. 3, where the experimental data31 for I −1 ( q ) are presented, this prediction holds very well for T >180 oC, i.e. until systems stays in the disordered state. It is worth to notice that the way the experimental data are presented in Fig. 3 is the most natural from theoretical point of view (it provides a direct opportunity to identify the region of the disordered state as the whole) but extremely rare in the experimental literature (I hardly can present any other example). 6 Moreover, it was shown by Leibler21 via a strict microscopic consideration that the Landau toy model perfectly describes the phase behavior of not too asymmetric molten diblock copolymers AnBm. His seminal theory of microphase separation in diblock copolymers20 became a real paradigm for both building the phase diagrams of the ordered phases’ stability within the WSA given the phenomenological coefficients in the Landau expansion of the free energy of the weak segregating systems in powers of an order parameter and microscopic calculation of the coefficients for block copolymers with a given architecture. In particular, Leibler21 found that the thermodynamically stable ordered morphologies for diblock copolymers are the body-centered cubic lattice (BCC) and the structures possessing hexagonal (HEX) and lamellar (L) symmetries, the sequence of the transitions being the disordered phase (DIS) – BCC- HEX – L (see Fig. 2). Further we refer to the phases BCC, HEX and L as the conventional or classic ones. The phase diagrams with the same topology were found for a variety of molten AB block copolymers with different architectures32-36 (see Fig. 4). Figure 4. The conventional phase diagrams of the molten diblock copolymers An Bm 21 (left) and star block copolymers ( An ) ( Bm ) 35,36 (right). The phase transition line DIS-BCC, BCC-HEX and HEX-L 4 4 are plotted by the blue, green and red colors, respectively. For some special sets of parameters, all the phase transitions lines (DIS-BCC, BCCHEX and HEX-L) merge at the critical point where the 2nd order phase transition from the disordered to lamellar phase occurs. For AnBm diblock copolymers the critical point corresponds to the symmetric diblock copolymer in case the repeated units of both blocks have the same excluded volumes v and Kuhn lengths a. Later Fredrickson and Helfand37 incorporated the Brazovskii fluctuation corrections into the Leibler theory and showed that 7 they considerably shift the ODT towards higher temperatures as compared to the mean field WST. It is worth to notice that according to the WSA the ODT in block copolymers shares a common physical background (the Landau weak crystallization) with various physical phenomena like the blue phases appearance in liquid crystals,38,39 charge-density waves generation upon addition of an ionic solute to a solvent in its critical region40,41 and microphase separation in weakly charged polyelectrolyte solutions,42,43 the polymer specific features of this approach being appeared at the stage of microscopic calculation of the Landau expansion coefficients only. The WSA provides also description of a rather special type of ordering predicted within the WSA also for random copolymers.44-51 Thus, the weak segregation theory of microphase separation in block copolymer systems provides a unique opportunity to test the general phenomenological concepts of the statistical theory of solidliquid transition via a rigorous microscopic consideration. Unfortunately, the region of the WSA applicability corresponds to a rather narrow vicinity of the critical point. Besides, the WST employs the so-called first harmonics approximation we discuss in more detail below and it is often believed52 that within this approximation “the predictions about ordered structures are limited to classical phases of lamellar, hexagonal, and body-centered cubic structures, and consequently the possibility of other structures such as bicontinuous structures, e.g., double gyroid, is excluded.” The double gyroid (G) phase mentioned here is an important phase characterized by Ia 3d space group symmetry, which was first discovered in lipid-water and surfactant systems53,54 and has been attracted much interest during the last decade due to bi-continuous morphology characteristic of this phase. So, during the last decade the so-called self consistent field theory (SCFT) by Matsen,55-57 which is free of these shortcomings, became dominant in understanding the behavior of the ordering block copolymer systems. The SCFT, which is considerably more polymer-specific than the WST, is based on the Edwards58-Helfand7-10 idea that inhomogeneities in the density profiles of the chemically different polymer repeating units are caused by some effective (self-consistent) fields, which, in turn, are themselves determined by the arising density profiles. The new powerful trick elaborated by Matsen and Schick55,56 was to seek for the desired density profiles and the self-consistent fields as some series in the eigenfunctions of the corresponding diffusion equation with due regard for their space symmetry. In contrast to the WST, which involves only few (1 to 12, depending on the lattice symmetry) primary harmonics, the SCFT series involve many hundreds of the eigen functions, and, thus, provide a much broader region of the SCFT applicability. It enabled Matsen and Schick55,56 to success in building the SCFT phase diagrams of molten AB diblock and star copolymers (see Fig. 3) revealing the stable 8 G phase in a reasonable agreement with experiment.59,60 A specific feature of these phase diagrams the plane ( f , χ ) is existence of two triple points f A = f1t < f crit , T = T1t and f A = f 2t > f crit , T = T2t , where three phases HEX, G and L coexist. (As usual, fA =n/N is the composition of the A monomers, n,m and N=n+m are the total numbers of the A and B units per block copolymer chain and the total degree of polymerization of the chain.) Therewith, the conventional sequence DIS-BCC-HEX-L and non-conventional one DISBCC-HEX-G-L hold for compositions within and out of the interval ( f1t , f 2t ) , respectively. The SCFT has been successfully applied to describe the stable ordered phases in various block copolymer systems (see, e.g., refs 61-63 and the references in other Chapters of the book) and it became dominant in understanding the behavior of the ordering block copolymer systems. Accordingly, the WST has been considered for some time as a sort of old-fashioned and outdated technique. In fact, however, the areas of expertise of the SCFT and WST are rather complementary than overlapping. In particular, it is worth to notice that the experimental59 and SCFT55 ( f , χ) phase diagrams are in a qualitative agreement only and there is a notable upward shift of the experimental phase transition lines in the plane as compared to the SCFT ones. The shift is due to the Brazovskii-Fredrickson-Helfand (BFH) fluctuation corrections25-27,37 neglected within the SCFT, which are far from being minor. However, these corrections are easily incorporated into the WST37,64-71 and it is within the WST that this upward shift of the phase transition lines was explained and quantitatively described3. The application of the WST requires calculation of cumbersome expressions for the socalled higher structural correlators; still, the SCFT is no less technically involved, whereas the corresponding numerical calculations are much more time consuming than those needed for the WST (the same is valid for other numerical methods discussed in the book). Besides, unlike the SCFT, a considerable part of the calculations necessary to build the phase diagrams within the WST can be done analytically. All these advantages enabled the WST analysis73-76 of the ternary ABC block copolymers, which resulted in understanding of many peculiar properties of these systems. In particular, it is easy to incorporate into the framework of the WST the effects of non-Flory interactions, which resulted in development of the WST theories of the ODT in reversibly associating52,77-80 and compressible81-83 block 3 Besides, the WST with due regard for so-called fluctuation caused q* -renormalization67 provided explanation of a shift of the scattering peak location towards lower values of q with temperature decrease, which is noticeable in Fig. 3 and was first reported in ref 72. 9 copolymers as well as the WST analysis of the ODT phase diagrams with the low critical ordering temperature.84,85 At last, it was demonstrated via a general WS analysis26,76,86 that under certain conditions the most stable phases around the critical point are not necessarily the classic ones. Instead, some others cubic phases such as gyroid G, simple cubic (SC), face-centered cubic (FCC), so-called BCC2 also called single or alternated gyroid, (we refer to all the phases but BCC, HEX, and L as the nonconventional ones) are stable, whereas the classic phases are metastable only. Moreover, the phase diagrams of the ternary ABC block copolymers (both linear and miktoarm) build via a generalization76 of the Leibler WST were found to reveal all the aforementioned non-conventional stable phases. The list of the non-conventional phases shown to be stable within the WST (we describe the phases in more detail in the Appendix) is to be appended by a new phase belonging to the symmetry class Fddd (O70) and called orthorhombic. The phase, which was found by Bates et al.87-90 in the ternary linear ABC block copolymers, is strongly degenerate to allow weak segregation as discussed by Morse et al.91,92 Remarkably, the phase was shown to be stable within the WST even in a vicinity of the critical point92 (see Fig. 5). Figure 5. The phase diagram of the molten diblock copolymers92 including the Fddd phase. Full lines are the phase boundaries calculated from the WST. Points denoted by + are the lamellar-Fddd boundary points and ◊ are the Fddd-hexagonal boundary points calculated using the numerical SCFT method. (Fig. 3 of ref 92). One more recent achievement of the WST is discovery of the so-called “structure-instructure” morphologies or two-scale microphase separation. It was found experimentally by ten Brinke, Ikkala et al.93 who studied self-assembling supramolecular structures in poly(4-vinylpyridine)-block-polystyrene (P4VPb-PS) diblock copolymer with side chains (e.g., pentadecylphenol, PDP) attached by hydrogen bonds to the P4VP block. The idea of the resulting morphologies is given by the cartoon and TEM micrograph presented in Fig. 6 10 and Fig. 7. Another (and quite different) example of 2-scale morphologies was found by Goldacker,94 Abetz and Stadler in blends of polystyrene-b–polybutadiene-b–poly(tert-butyl methacrylate S33B34T33 (total M = 1.6 ×105) and polystyrene-b–poly(tert-butyl methacrylate S47T53 (total M = 1.03 ×105), which contains 10 wt % triblock copolymer (see Fig. 8). Figure 7. Transmission electron micrograph of PS-blockP4VP(NPD)1.0, where nominally one nonadecylphenol (NPD) has been hydrogen bonded with each pyridine group. The long period of the alternating PS (light grey) and P4VP(NPD)1.0 (dark grey) lamellae equals LD~ 550 Å. The number-averaged molecular masses of the P4VP and PS blocks were 49,500 and 238,000 daltons, respectively. The P4VP-(NPD)1.0 lamellae are further ordered into alternating lamellae of nonpolar nonadecyl tails of NPD molecules and polar P4VP backbones. The long period of this structure is LC ~ 40 Å. The two sets of lamellar structures are, as expected, mutually perpendicular. (Fig. 4 of ref 93). Figure 6. Schematic illustration of the self-organized structures of PS-block-P4VP(MSA)1.0(PDP)1.0. The local structures are indicated; macroscopically, the samples are isotropic. (A) Alternating PS layers and layers consisting of alternating onedimensional slabs of P4VP(MSA)1.0 and PDP for T< TODT (14). (B) Alternating two-dimensional PS and disordered P4VP(MSA)1.0(PDP)1.0 lamellae for TODT < T < TOOT. (C) One-dimensional disordered P4VP(MSA)1.0(PDP)x (with x<<1) cylinders within the three-dimensional PS-PDP medium for T > TOOT. (Fig. 3 of ref 93). Figure 8. Transmission electron micrograph of the mixture of copolymers S33B34T33 and S47T53; dark-grey and light-grey bands correspond to ST lamellas, and black bands refer to B layers. (Fig. 1 of ref 97). 11 Despite an apparent visual difference of the structures shown in Fig. 6 and Fig. 7, it was shown95-97 that within the WST both systems possess a common feature. Namely, for these systems the function c(q) appearing in (1.1), (1.11a) and (1.15) is rather sensitive to details * * and q = qmax , of the block copolymer architecture and could have two minima4 at q = qmin * * and Lmin = 2π qmax being corresponded to the bigger and the lengths Lmax = 2π qmin smaller characteristic scales of the morphologies (see Fig. 9). Figure 9. The q-dependence of the inverse scattering intensity for the poly (A)m-block-poly(A-graft-B)n polymer shown in Figs. 6,7 and analyzed within the RPA in ref 95. Here the curves labeled I, II, III, IV and V correspond to (n,m) = (21, 3), (20, 5.5), (20, 5.69), (20, 6) and (1, 44), respectively (n is the number of the side chains per the comb-like block, m = N A d , d and N A being the degrees of polymerization of the side chains and homopolymer A block, respectively. (Fig. 3 of ref 95). h h A similar behavior was found also for some block copolymer solutions,98,99 which stimulates to seek some properly designed block copolymer solutions capable to serve as photonic crystals. The WST phase diagrams of the two-length-scale morphologies were build100-103 and supported by the SCFT calculations104-107 of the phase diagrams in a broader temperature interval. Such a two-scale behavior is closely related to formation of the nonconventional morphologies,76,100-103 and expected to be, along with using the multicomponent block copolymers, a new rather efficient route towards tuning the phase behavior of the self-assembling block copolymer nanostructures. It is to provide a better understanding of the latest advancements in the WST, which is the purpose of the rest of the Chapter. Since the basic features of the WST in the binary AB block copolymers are well described in the original papers as well as reviews,1-3 in what follows we skip the derivations as well as discussion of the BFH fluctuation corrections and 4 Note that Landau himself has strongly emphasized21 that “it is absolutely improbable” that the function c(q) vanishes simultaneously in more than one minimum and, thus, the disordered state becomes unstable simultaneously for the order parameter waves with different wave lengths. More precisely, the two-scale instability corresponds to a strongly degenerate situation. But the wealth of the parameters controlling the block copolymer fluctuation behavior makes it possible to realize such a degenerate and “improbable” situation in some properly designed systems. 12 focus only on the latest results, which are obtained in the mean field approximation. The subsequent presentation is organized as follows. I consider a simple weak segregation model enabling us to understand the physical and mathematical bases, which cause the nonconventional phases’ stability in section 2. Here I introduce also the 2nd shell harmonics approximation, which gives an example of important distinction between the strongly and weakly fluctuating fields. A generalization of the Leibler WST to the multi-component block copolymer systems, which turns out thanks to the broader application of the distinction between the strongly and weakly fluctuating fields, is outlined in section 3. Some applications of the generalized WST to the ternary ABC block copolymers are described in section 4. In Conclusion I summarize my feeling of the current state and most urgent problems of the WST. At last, in the Appendix the most typical conventional and non-conventional weakly segregated ordered morphologies are described in detail. 2. The WST and the non-conventional phases’ stability. Non-locality (forth vertex angle dependence) effects. The stability of the nonconventional morphologies in weakly segregated systems is determined by the degree and character of non-locality of their free energy as a functional of a specified profile of the corresponding order parameter (we refer to such functional as the virtual free energy). Indeed, let us start with a generalization of the original Landau Hamiltonian (1.1), which is to take into account that for polymer systems the specific excess free energy (1.2) is not ( ) ∑ n1! ∫ Γn ( r1 − r,..rn − r ) ∏ Φ ( ri ) d ri n =3,4 i =1 local. Namely, it takes the form ΔF r, {Φ ( r )} = n (2.1) where some continuous functions Γ3 ( R1 , R 2 , R 3 ) , Γ 4 ( R1 , R 2 , R 3 , R 4 ) describe how {Φ ( r )} in a vicinity of the point r rather than on the local value of Φ(r) at the very point r much the specific excess free energy at the point r depends on the whole profile of the Γ3 ( R1, R 2 , R 3 ) = α δ ( R1 ) δ ( R 2 ) δ ( R 3 ) , only. Obviously, the expressions (1.2) and (2.1) coincide in the limit Γ 4 ( R1, R 2 , R 3 , R 4 ) = β δ ( R1 ) δ ( R 2 ) δ ( R 3 ) δ ( R 4 ) . The meaning of the non-locality becomes clearer when rewriting the free energy (1.1), (2.1) in the Fourier-representation: 13 F =∫ Δ Fn = c ( q ) Φq dq 2 ( 2π ) 3 2 + ΔF3 + ΔF4 , (2.2) n Φ q dq ⎛ n ⎞ ( i) i . 1 δ Γ q q , , q ( .. ) ⎜ ⎟ ∑ ∏ i n n 1 ∫ ⎜ ⎟ n ! ⎝ i =1 ⎠ ( 2 π )3 i =1 (2.3) Hereafter, we refer to functions and their Fourier transforms as the same functions in rand q-representations, respectively, and distinguish them only by the choice of the letters used to denote their arguments; this convention is not expected to cause any misunderstandings due to the context. It is worth to notice that the functions Γn appearing in the cubic and quadric free energy contributions (2.1a) depend on the structure of the system. Substituting characteristic of the WST expression (1.14) for the equilibrium order parameter into the mean field free energy (1.12) with due regard for expressions (1.1), ( ) (α ) (2.1)-(2.3) and minimizing the final expression with respect to the amplitude A we get20 ΔF fℜ ( τ ) = ℜ = VT 2 3 α ℜ + 9α ℜ − 32τβ ℜ 3 212 β3ℜ ℜ ( 2 − 9αℜ − 32τβℜ ) Here we introduced the cubic and quadric vertices αℜ = k −3 / 2 3! ∑ q1 + q 2 + q 3 = 0, q1 = q 2 = q 3 = q* ℜ ( 3) Cℜ = ∑ 3 cos Ω j k 3 / 2 βℜ = = k2 4! q1 +q 2 +q3 +q 4 =0, q1 = q 2 = q3 = q 4 = q* λ0 ( 0 ) 4k ∑ + (2.4) Γ 3 ( q1 , q 2 , q 3 ) exp i ∑ i =1 ϕi = γ (1) Cℜ , 3 ( Γ 4 ( q1 , q 2 , q3 , q 4 ) exp i ∑ i =1 ϕi k ∑ λ 0 (hi ) + ∑ ℜ λ 4 2 ( h1, h2 ) cos Ω j 4 (2.5) ) ( 4) k (2.6) . In (2.5), (2.6) the phases Ω (j3) , Ω (j4 ) are the algebraic sums of the phase shifts ϕ appearing in (1.14) for triplets and non-coplanar quartets of the vectors {q i } involved in the definitions of α and β, respectively. Besides, we used the designations and parameters of Leibler:20 γ (h) = Γ3 (q1 , q 2 , q 3 ), q12 = q 22 = q*2 , q32 = (q1 + q 2 )2 = hq*2 , λ(h1 , h2 ) = Γ4 (q1 , q 2 , q 3 , q 4 ), (2.7) q i = q* , i = 1,..4 (2.8) 14 h1 = ( q1 + q2 ) q∗2 , h2 = ( q1 + q3 ) 2 2 q∗2 , h3 = 4 − h1 − h2 = ( q1 + q 4 ) λ 0 (h) = λ ( 0, h ) = Γ 4 (q, −q, p, −p), h = ( q + p ) The symbol ∑ ℜ n 2 2 q∗2 . (2.9) q*2 . (2.10) implies summation over all sets of n vectors for given morphology ℜ appearing in (2.5), (2.6). The first summation in (2.6) is over all pairs of non-collinear vectors q i and q j , 2k is the number of the vectors in the reciprocal space belonging to the The phase transition lines τℜ1 / ℜ2 ( γ (1) ) between the morphologies ℜ1 and ℜ2 are coordination sphere with the radius q*. ( ) ( ) fℜ1 τ, ( γ (1) ) = fℜ2 τ, ( γ (1) ) . determined by the equation (2.11) In particular, if the cubic vertex vanishes due to the symmetry for both morphologies ℜ1 and ℜ2 the phase transition line between the morphologies is determined by the equation βℜ1 = βℜ2 (2.11a) function λ (h1 , h2 ) , which appears in the expression (2.6) for the fourth vertex βℜ. Following The topology of thus obtained phase diagrams is influenced by the explicit form of the Leibler20 we refer to the h-dependence of the vertex λ as the angle one since the values of the parameters hi depend on the angles between the vectors qi. For diblock copolymers the angle dependence is rather weak,20 which enabled Fredrickson and Helfand34 to propose the following commonly accepted approximation: λ ( h1 , h2 ) ≈ λ ( 0, 0 ) = λ 0 ( 0 ) . (2.12) However, the approximation (2.12) was shown71 not to stay true for any polymer systems. crystallizing systems we approximate the function λ (h1 , h2 ) as follows:71,81 To estimate effects of this angle dependence on the phase diagram of the weakly 3 3 ⎛ 3δ ⎛ ⎞⎞ λ(h1 , h2 , h3 ) = ∑ f (hi ) = λ 0 ⎜⎜1 − ⎜ 4 2 − ∑ hi2 ⎟ ⎟⎟. i =1 i =1 ⎠⎠ ⎝ 32 ⎝ (2.13) λ (h1 , h2 ) in powers of hi, the positive (negative) sign of δ corresponds to a disadvantage The approximation (2.13) keeps only the first non-constant term in the expansion of The resulting phase diagrams in the plane (τ, γ (1)) are sets of parabolas (advantage) of the lamellar structure as compared to all other ones. τℜ1 / ℜ2 = 9τℜ1 / ℜ2 γ 2 (1) ( 32λ 0 ) , (2.14) 15 where the reduced temperatures τℜ1 / ℜ2 depend only on the angle dependence strength δ. ( ) The reduced phase diagram in the plane τℜ1 / ℜ2 , δ demonstrated71,81 that increase of the strength of the model angle dependence (2.13) results in increase of stability of the G phase (as compared to HEX and L) and that of the various non-conventional cubic phases as compared to L. Therewith, all competing phases are to be taken into account. As shown in Fig 10, the stability of the newly discovered orthorhombic phase Fddd (see refs 81-86 and Appendix) turns out to reveal a non-monotonous behavior with increase of δ, the region of the Fddd phase stability being turned out as big as that of the G phase. a) b) τ eff Figure 10. The reduced phase diagram for the model angle dependence (2.13) in the plane (the reduced temperature - the angle dependence strength δ). a) the phase diagram76 calculated without taking into account that the orthorhombic phase Fddd could exist; b) the phase diagram calculated with due regard for the phase Fddd. It was claimed,71 basing on the reduced phase diagram shown in Fig. 10a, that the conventional phase transition sequence DIS-BCC-HEX-L occurs only for δ<δ0=0.362. Now, after the Fddd phase is discovered, one can assert that this sequence never holds. Instead, the following non-conventional sequences occur: i) the sequence5 DIS-BCC-HEX-Fddd-L for δ<δ12 , δ12 = 4/9; ii) the sequence DIS-BCC-HEX-Fddd-SG for δ1 > δ > δ12, δ1 = 0.61005;; iii) the sequence DIS-BCC-HEX-G-Fddd-SG for δ2 > δ > δ1, δ2 = 0.61684; ii) the sequence DIS-BCC-HEX-G-SG for δ23 > δ > δ2, δ23 = 2/3; iii) the sequence DIS-BCC-HEX-G-FCC for δ3 > δ > δ23, δ3 = 0.822; 5 For δ=0 and for the angle dependence characterizing the Leibler molten diblock copolymers this sequence was found first by Morse et al.86,87 16 iv) the sequence DIS-BCC-G-FCC for δ34 > δ > δ3, δ34 = 5/6; v) the sequence DIS-BCC-G-SC for δ45 > δ > δ34, δ45 = 0.891; vi) the sequence DIS-BCC-SC for δ2 > δ > δ45, δ2 = 4/3; vii) the sequence DIS-BCC for δlim > δ > δ2, δlim = 1.538; At last, βBCC vanishes at δ = δlim , which is characteristic of so-called tricritical point where the 2nd order phase transition line terminates and the 1st order phase transition line starts. For δ > δlim the original weak segregation approximation does not hold anymore and the next (5th, 6th etc.) terms of the Landau expansion are to be taken into account. The approximation (2.13) could seem to be too academic but it provides a good idea of the phase behavior of some real systems. In particular, the reduced phase diagram of the molten ternary linear ABC block copolymers with the non-selective middle block (see Fig. 11a) calculated by the author71 before the Fddd phase was discovered is rather similar to that presented in Fig.9a, the composition of the middle block playing the role of the angle dependence strength parameter δ. Thus, the region of the Fddd phase stability for the ternary linear ABC block copolymers with the non-selective middle block is expected to neighbor with that for the G and SG phases in qualitative agreement with experiment.84,85 Remarkably, the phase behavior of the molten ternary star (miktoarm) ABC block copolymers with one of the arms non-selective with respect to two other ones differs strongly both from that of molten diblock copolymers and the linear ABC block copolymers middle block (see Fig. 11b). a) b) Figure 11. The reduced phase diagrams76 along the critical line f A = f C = f B 2 calculated with due regard for the actual angle dependence of the ternary ABC block copolymers in the plane (the nonselective block composition fB, the reduced temperature τ eff ) for linear (a) and miktoarm (b) ABC block copolymers. The regions of the stability of the disordered state and body-centered cubic, hexagonal, lamellar, double gyroid, single gyroid, face-centered cubic, simple cubic and BCC3 lattices are labeled by the numbers 0, 1, 2, 3, 4, 5, 6, 7 and 8 respectively. 17 The physical origin of the non-conventional cubic phases’ stability for the ternary linear ABC block copolymers with a long non-selective middle block is obvious: short strongly incompatible side blocks prefer to aggregate into small micelles rather than into thin layers, it is this tendency, which is “coded” into the angle dependence of the effective non-local vertices Γ 4 (q1, q 2 , q3 , q 4 ) and becomes apparent already at the very onset of ordering when segregation is still weak. The 2nd shell harmonics approximation. Thus, the first harmonics approximation (1.14) along with due regard for the angle dependence of the 4th vertex provides a reasonable explanation of the stability of the non-conventional phases. However, during the last decade some authors71,108-110 queried reliability of the approximation (1.14). In particular, Hamley and Podnek71 suggested that the gyroid morphology existence is due to anomalously large (by modulus) but negative contribution of the second harmonics with h = q 2 q*2 = 4 / 3 characteristic for the G morphology to the total free energy. Of course, this suggestion by itself is not sufficient to explain the G phase stability in diblock copolymers since the second harmonics with h = 4 3 are characteristic also for the BCC3 and G2 morphologies, which all belong to the G family. Moreover, it has no relation to the problem of the G stability at the very critical point, which depends only on the strength of the angle dependence of the forth vertex as shown above. However, if the angle dependence is not strong enough to provide the G stability at the critical point, we are to deal with two closely related problems: i) which phases are stable at the triple points (if any) existing near the critical point; and ii) which factors determine location of the triple points. In general, contribution of many harmonics (rather than that of the only second ones) determines location of the triple point in question, which is shown via direct calculation by Matsen and Schick.51,52 But it is natural to expect that only certain finite number of the higher harmonics is really relevant if the triple points are close enough to the critical one. To single out such selected higher harmonics, instead of the first harmonics approximation (1.14) Φ (r ) = Ψ (r ) + ∑ ψh (r ) , we chose the trial function as follows:47,76 (2.15) h≠1 where Ψ (r ) = A ∑ exp i (q i r + ϕ i ) qi ∈ℜ −1 , qi2 = q*2 ψ h (r ) = and ∑ qi ∈ℜ , −1 qi2 = hq*2 aqi exp i ((q i r ) + ϕ(q i ) ) (2.16) (2.17) 18 are the sums of the main and higher harmonics, respectively. After substituting the trial function (2.15)-(2.17) into (2.2), (2.3) the virtual free energy takes the form F = Fmain + Fhigh + Fcoupling , where Fmain = F ({Ψ ( r )}) is the contributions of the dominant harmonics (2.16), ΔFcoupling is (2.18) that due to coupling between the higher dominant and harmonics, generated by the cubic term of the original Hamiltonian: ( 3) ΔFcoupling VT = A02 ∑ γ ( q1, q2 , −q1 − q2 ) aq1+q2 exp ( i ( φ1 + φ2 ) ) 2 q ∈ℜ−1 , q2 =q 2 i (2.19) * i ( q ( h −1) + τ) a and the contribution of the higher harmonics is determined as follows: Fhigh = VT 2 qi∈ℜ ∑ −1 , qi2 = hq*2 2 4 * qi 2 . (2.20) All other terms we skipped in (2.18) are irrelevant. Indeed, minimization of the free energy (3.31) with respect to the complex amplitudes aq , q 2 q*2 = h ≠ 1 , gives aq = − where (( A02 γ ( h ) ∑ q exp i φi + φ j 2 2 q*4 ( h − 1) + τ )) (2.21) Σ q means summation over all pairs of the main harmonics given the condition qi + q j + q = 0 (2.22) According to (2.21), (2.22) the second harmonics induced by coupling (2.19) belong to all the coordination spheres of the corresponding reciprocal lattice, radius of which does not exceed the doubled radius of the dominant coordination sphere, their amplitudes being of the 2nd order of magnitude with respect to the main harmonics amplitudes A0. The number of the coordination spheres of different radii satisfying eq (2.22) depends on the lattice symmetry and varies from 1 (for the LAM) to 8 (for the G) and 10 (for the BCC3). We refer ∑ q exp ( i ( φi + φ j ) ) to all of the corresponding harmonics as the 2nd shell harmonics. For the lattices with nonzero phase shifts like SG, G and G2, the factor appearing in (2.21) and, thus, the corresponding higher harmonics vanish identically, which gives a natural derivation of the extinction rules111 within the WS theory. ( ) Substituting (2.21) into eqs (2.18)-(2.20) results in the final expression for the free ( ) energy including the higher harmonics contribution up to order of ΔF = VT τ A02 + αℜ A03 + βℜ A04 , O A04 : (2.23) 19 where βℜ is related via eqs (2.6), (2.8)-(2.10) to the forth vertex Γ 4 renormalized with due regard for the 2nd shell higher harmonics: Γ 4 (q1 , ..,q 4 ) = Γ 4 (q1 , ..,q 4 ) − ( B ( q1 , .q 2 ; q3 ,q 4 ) + B ( q1 , .q3 ; q 2 ,q 4 ) + B ( q1 , .q 4 ; q 2 ,q3 ) ) B ( q1 , .q 2 ; q3 ,q 4 ) = γ2 ( h) q*4 ( h − 1) , p = −q1 − q 2 = q3 + q 4 , h = 2 τ is omitted in the definition (2.25) since τ (2.24) p2 (2.25) q*2 q*4 is a condition of the WSA validity (see Appendix 2).. It is worth to remember now that the reduced phase diagrams build in Fig.9 and Fig.10 describe the stability of the weakly segregated phases in the very critical point only, i.e. in the limit γ→0. It follows from eqs. (2.24), (2.25) that the phase transition lines are affected by the actual dependence γ(h) in a finite vicinity of the critical point where the cubic vertex is small but finite. The advantages and limits of the 2nd shell harmonics approximation. The phase diagram of the molten diblock copolymer calculated in ref 71 with taking into account the 2nd shell harmonics contribution into the 4th vertex of the virtual free energy as described above is presented in Fig. 12. One can estimate both the advantages and deficiencies of our approximation comparing it with those of Leibler21 and Matsen and Schick.55 a) b) Figure 12. The phase diagram of the diblock copolymer in the 2nd shell harmonics approximation (without taking into account the Fddd phase existence). The designations of the phases are the same as in Figure 11. a) comparison with the Leibler21 phase diagram (shown by the dashed lines); b) extrapolation to the region of comparatively high values of χ , which demonstrates possibility of the G2 lattice (labeled by the number 9) stability. 20 As is seen in Fig. 12a, as far as the conventional phases is concerned our phase diagram would almost coincide with that of Leibler21 precisely approaching the latter in the vicinity of the critical point. The only difference would be some broadening of the BCC phase stability region (basically at cost of the HEX phase) with increase of the diblock copolymer asymmetry. The situation changes drastically as soon as we include into the list of competing phases those of the G family, which were not taken into account in the original paper.21 All three phases of the family described in section 2 become stable when the asymmetry f − f c increases and the 2nd shell harmonics effect is taken into account. It is worth to notice that when we calculate the free energy of the ordered phases within the conventional 1st harmonics approximation the phases of the G family turn out to be metastable only. Therewith, in our approximation the triple point LAM-HEX-G is located at χ = 10.88 , which is rather close to the result f=0.452, ~ χ = 11.14 obtained by f=0.462, ~ Matsen and Schick55 within the SCFT using much more harmonics. Comparing the presented numerical results for the triple point we conclude that our 2nd shell harmonics approximation somewhat overestimates the effect of the higher harmonics but, nevertheless, is in a reasonable agreement with the SCFT results obtained using the whole series of the higher harmonics.55 ( χDIS-BCC ( f ) → −∞ ) when f → f Another clear evidence of such an overestimation is that the phase transition line DISBCC3 falls down sharply 3 di = 0.4183. At the first sight this result seems meaningless, but, in fact, its physical meaning is rather clear. It could be understood by analogy with that of the spinodal of block copolymers with respect to microphase separation. The latter was defined21,28,29 as the line (surface) where the inverse scattering factor appearing in the quadratic term of the free energy (2.2) vanishes. Accordingly, the correlation function (1.11b) calculated within the RPA diverges here and, account the fluctuation corrections25-27,37,64-71 shows that the function S ( r ) stays finite and thus, the uniform state of the systems becomes absolutely unstable. However, taking into the uniform state stays stable (at least metastable) even beyond the RPA spinodal. Thus, the latter should be now understood as a crossover line between the regions with different temperature scaling of the correlation radius and the exact border of the region where the RPA does not hold even qualitatively. Quite similarly, the sharp falling down of the phase transition line DIS-BCC3 when f → fdi is shown76 to be determined by the fact that the minimal quadric vertex β ℜ = min β ℜ changes the sign in the point f = f di due to the 2nd shell harmonics renormalization of the 21 vertex, therewith ℜ = BCC 3 for molten diblock copolymer. So, β BCC 3 ( f ) < 0 for f < fdi and, therefore, the expansion of the Landau Hamiltonian in powers of the order parameter Φ up to the 4th term only becomes inapplicable. As in the spinodal case, the unphysical divergence of the leading term is to be removed by including into the expansion the terms of the higher order than that causing the divergence. In our case it means to take into account at least the terms of the 5th and 6th powers in Φ as well as the 3rd shell harmonics contributions. The corresponding generalization of the WST is expected to smooth (not eliminate!) the sharp phase transition line DIS-BCC3 (in general, DIS - ℜ ) shown in Figure 12. It is natural to refer to the line β ℜ = 0 as the WS border line since beyond it the higher harmonics effect becomes so important that the system could not be described properly even within the 2nd shell harmonics approximation of the WS theory. Two more interesting features of the modified WS phase diagram shown in Figure 12 are the phase transition lines G – BCC3 at f=0.4343 and G – G2 (see Fig. 12b) situated at relatively high values of ~ χ . It is important to stress that the WS theory can not claim responsibility for prediction of precise location of both these phase transition lines. Indeed, they lay too far from the critical point so that the stability of the phases BCC3 and G2 could be only an artifact of the WST extrapolation beyond its validity region. Nevertheless, these phase transition lines are interesting as indications of the fact that the stability of the double gyroid phase G is caused by a moderate development of the 2nd shell harmonics only, whereas a further increase of the degree of segregation results in increase of stability of other cubic phases (in our case, BCC3 and G2) at cost of the G phase. The described features of the phase diagrams of molten diblock copolymers are characteristic of binary AB block copolymers with various architectures as is exemplified by the phase diagrams of molten symmetric triblock and trigraft AmBnAm copolymers we calculated within the 2nd shell harmonics approximation (see Fig. 13). a) b) 22 Figure 13. The phase diagrams of the molten symmetric triblock (a) and trigraft (b) AmBnAm copolymers in the 2nd shell harmonics approximation (without taking into account the Fddd phase existence). The designations of the phases are the same as in Figure 11. Summarizing, the WST in the 2nd shell harmonics approximation provides a rather reasonable accuracy in locating the triple point HEX-G-LAM and interesting (much less reliable, though) hints as to stability of some other non-conventional cubic phases. 3. The WST applications to the multi-component block copolymer systems. Φ i (r ) = φ i (r ) − φi of the partial local volume fractions φi ( r ) = vρi ( r ) of the repeated units The order parameter for these systems is the n-component vector of the local deviations (monomers) of the i-th sort (here v is the excluded volume supposed to be the same for all sorts of monomers) from their values φi averaged over the whole volume of the system. As common for all polymer systems consisting of flexible macromolecules, their virtual free ({φi ( r )}) + F * ({φi ( r )}) . energy takes the form112,113 F = Fstr (3.1) where the first term is so-called structural free energy Fstr corresponding to the entropy of the inhomogeneous ideal system of the copolymer macromolecules under consideration with certain specified spatial profiles of the volume fractions of the repeated units of the i-th WS theory the free energy (3.1) is evaluated by its expansion in powers of Φ i ( r ) : sort and the second is the interaction contribution we discuss in more detail later. Within the ( ) F {φi ( r )} = F ({φi }) + ΔF2 + ΔF3 + ΔF4 + ... . (3.2) with the contributions ΔFn defined as follows: ΔF2 1 Γij ( q ) Φ i ( q ) Φ j ( −q ) dq , = ∫ 3 2 T ( 2π ) (3.3) n Φ (q ) dq ΔFn 1 ⎛ n ⎞ ) αi i i = ∫ δ⎜ ∑ q i ⎟ Γα(n1 ,.., , n = 3,4 α n (q 1 ,.., q n ) ∏ 3 T n! ⎝ i =1 ⎠ (2π) i =1 (3.4) In the disordered state the thermodynamically average values of the fluctuations Φ i ( r ) n Φ i ( 0 ) Φ j ( r ) exp ( − ΔF ({Φ l ( r )} ) T ) ∏ l =1 δΦ l ( r ) ∫ Sij ( r ) = Φi ( 0 ) Φ j ( r ) = n exp ( − ΔF ({Φ l ( r )} ) T ) ∏ l =1 δΦ l ( r ) are zero. Then the only observable quantities are the correlation functions (3.5) and their Fourier transforms (scattering factors) Gij ( q ) = ∫ Sij ( r ) exp ( iqr ) dr. 23 Random Phase Approximation. Let us assume that the fluctuations are small and keep in the free energy expansion (3.2) the quadratic contribution (3.3) only. Then, as first shown by the author114 (see also refs 115-117), the matrix G = Gij ( q ) can be expressed in terms of two independent matrices characterizing the connectivity and interactions effects: G −1 = Γ = g −1 − c , (3.7) where the matrices g-1 and -c are contributions to the matrix Γ appearing in (3.3) from the structural and energetic addendums, respectively (see eq (3.1)), matrices G-1 and g-1 are inverse to the matrices G and g, respectively, and the so-called structural matrix g is defined114-116 as follows: (S ) g = gij ( q ) , gij ( q ) = ∑ nS γ ij ( q ) . (3.8) S In the second of the definitions (3.8) nS is the number density of the (macro)molecules with structure S, summing up is carried out over all the structures S present in the system (S ) including the monomers, and the molecular form-factors γ ij ( q ) read ( ( )) S , (S ) γ ij ( q ) = ∑ exp iq r ( li ) − r n j (3.9) where r(li) is the vector-radius of the l-th repeated unit of the sort j and the symbol <...>s implies averaging over all Gaussian conformations of the macromolecule S. For monomers, (1) obviously, γ ij ( q ) = δij . The matrix g is just the matrix of the correlation functions for the ideal polymer system i.e. the system with the same structure as that under study but with no interactions between ( cij ( r ) = −δ2 F * {φl ( x )} ) ( δρi ( 0 ) δρ j ( r ) ) their repeated units, whereas (3.10) is the matrix of the direct correlation functions, which is well known in the theory of simple liquids.118 The uniform (disordered) phase stays thermodynamically stable (at least, metastable) with respect to micro- or macrophase separation when the quadratic term (3.3) is positive definite:29 min Λ ( q ) = Λ ( q* ) > 0 (3.11) where Λ(q) is the minimal of the eigenvalues λi ( q ) , i = 1,.., n of the matrix Γ of the rank n and the wave number q* of the critical order parameter waves is the location of the absolute minimum of the function Λ(q). Therewith 24 q*2 > 0 (3.12) is the condition that it is micro- rather than macrophase separation, which occurs after the uniform state becomes unstable. Accordingly, the spinodal line (surface), which delineates the region in the space of the structural and interaction parameters of the system under study where the spatially uniform (disordered) state is absolutely unstable within the RPA, reads29 min Λ ( q ) = Λ ( q* ) = 0, (3.13) The interaction term F* is naturally determined assuming that it does not depend on the polymer structure of the system. (This natural requirement is, in fact, rather subtle. Strictly speaking, it is correct only when the Lifshitz number Li = v a3 , which plays the role of the Ginzburg parameter, is small (here a is the Kuhn length). If Li is not small then the effective monomer-monomer interaction is strongly influenced by the correlation between the neighboring (along the chain) monomers and renormalized accordingly. It is this phenomenon for which Khokhlov119,120 coined the term “quasimonomers” and which is quantitatively addressed by the PRISM theory.121,122 Henceforth we believe that the interaction term F* describes the properly renormalized interaction.) In the simplest case of the compressible Flory-Huggins lattice model (when some cells of the lattice are not ( F * {φi ( r )} )= occupied by any monomer repeating units) the interaction term F* reads T ⎞ dr ⎛ 1 ⎜ ⎟, − φ − φ + χ φ φ r r r r 1 ln 1 ( ) ( ) ( ) ( ) ( ) ( ) ∑ ij i j ∫v⎜ ⎟ 2 i≠ j ⎝ ⎠ (3.14) where φ ( r ) = ∑ i =1 φi ( r ) is the total volume fraction of all sorts of monomers. Then the l respectively. In the incompressibility limit 1 − φ ( r ) → 0 the first addendum in (3.14) addendums in (3.14) describe the compressibility and van der Waals interaction effects, ( ) F * {φi ( r )} = (T 2 ) ∫ ∑ χij φi ( r ) φ j ( r ) dr v . vanishes and we get the well known quadratic expression62,117 (3.15) i≠ j Φ i ( r ) are determined by the structural entropic term only. But in general (beyond the Thus, in this limit the higher vertices in the virtual free energy (3.1) expansion in powers of incompressible Flory-Huggins model) the enthalpic (interaction) contribution into the free energy could affects the cubic and quadric terms either as found for molten diblock copolymers taking into account the “quasimonomer” renormalization123,124 and using the equation of state model.81-83 It is worth to notice that the characteristic scales of the structural and interaction contributions into the higher vertices are rather different (of the order of the macromolecule and monomer size, respectively). Thus, the main source of the 25 strong angle dependence (if any) of the 4th vertex, which may lead to stability of the nonconventional phases as discussed in section 2, is expected to be the structural contribution determined by so-called higher structural correlators as shown in detail in refs 21,47,76,117. The critical points in the multi-component block copolymer systems and the strongly and weakly fluctuating fields. Violation of the (meta)stability condition (3.11) is sufficient but not necessary to guarantee crystal ordering. Typically, a finite order parameter profile (1.13) arises via discrete 1st order phase transition when the condition (3.11) still holds and the disordered phase is at least metastable. However, the span of this profile decreases when the relative magnitude of the cubic terms decreases and, finally, the ordering transforms into a continuous 2nd order phase transition (within the mean field approximation only!) at the critical point where the cubic terms vanish. Thus, the WST certainly holds (at least in the same sense as the SCFT does) in a vicinity of the critical point(s). It is easy to locate the critical point(s) (if any) for the scalar WST describing the ODT in the incompressible binary block copolymers, where only one cubic term exists (see eq (1.2)). But the virtual free energy of the n-component block copolymer systems contains ~n3 cubic terms (that of incompressible does (n-1)3 ones), so that it could appear62 that it is hardly possible at all to apply the WST to many-component systems. Nevertheless,47,76 it is possible to reduce consideration of the n-component systems to that presented in section 2 for those with a scalar order parameter via distinguishing the minimal Λ(q) and all other (positive) eigenvalues λ i (q ) (i=2,…n) of the matrix Γ appearing in (3.3) as well as the corresponding eigenvectors Ei (q ) and ei( s ) (q ) . The projections of the vector order parameter Ψ (q ) = E i (q ) Φ i (q ), ϕ s (q ) = ei(s ) (q ) Φ i (q ), s = 2,.., n. (3.16) into the corresponding eigenvectors play the role of the strongly and weakly fluctuating fields, respectively, the account of the latter being carried out similarly to treating of the 2nd shell harmonics in section 2. As the result, one gets the effective scalar free energy of the weakly segregated multi-component block copolymer systems in the form (2.23), where the effective cubic vertex αℜ is defined by expression (2.5) with ( 3) Γ3 ( q1 , ..,q3 ) = Γ α ,α 1 2 ,α 3 (q1 , ..,q3 )Eα1 ( q1 ) Eα 2 ( q 2 ) Eα3 ( q3 ) . (3.17) All other cubic terms either renormalize the effective the quadric vertex βℜ described by a cumbersome expression71 we skip here for brevity or contribute to the terms of the order of magnitude Ο(Ψ5) and thus exceed the accuracy of the WST. 26 4. The WST predicted peculiarities in the multi-component block copolymer systems. The RPA structure factors in the disordered state. To calculate the structure factor matrix G for the Flory-Huggins model we find the matrix c for finite compressibility from ( eqs (3.10), (3.14): cij ( q ) = v (1 − φ ) −1 ) + χij 2 , (4.1) and, finally, take the incompressibility limit 1 − φ ( r ) → 0 . In this (or equivalent) way the substitute (4.1) into the r.h.s. of the general RPA equation (3.7), invert the resulting matrix structure factor matrices G(q) were found and the spinodal conditions analyzed73-76,97-99 for some ternary ABC block copolymer systems. Werner and Fredrickson74 studied the spinodal conditions (3.11), (3.12) for molten linear and comb-like ABC (monodisperse and statistical) block copolymers and found the spinodal lines as well as the q* dependence on the values of three independent Flory parameters χ AB , χ AC , χ BC . They found that increase of one of these χ-parameters (given two others are fixed) could result in a non-monotonous ordering tendency called the reentrant ODT. A similar result was found by the author et al.73 Namely, the spinodal ODT temperature Ts in the linear ABC block copolymers, in which one of blocks is much shorter and more incompatible that two others, changes nonmonotonously with increase of the short block incompatibility, the minimal value of Ts and the period of the arising ordered structure being less than those for the corresponding diblock copolymer without any third strongly incompatible block. Thus, the peculiar situation that “more incompatibility results in less segregation” is explained by an additional entropic loss related to confinement of the ABC macromolecules in a lattice with the smaller periodicity. Two-length-scale behavior in molten ABC. Cochran, Morse and Bates75 considered in more detail the scattering behavior of the linear ABC triblock copolymer melts and found that tuning the values of the architecture, scattering contrast and interaction parameters one achieves a reasonable agreement between the two-peak profile of the SAXS indicatrix observed in the disordered poly(isoprene-b-styrene-b-dimethylsiloxane) (ISD)125 and the RPA structure factor, the height of one of the peaks being increased when approaching the ODT (see Fig. 13). The authors concluded that “the RPA structure factor is representative of the true structure in disordered ABCs” and attributed such a two-peak profile “to the natural existence of multiple length scales in ABCs”. 27 Figure 14. Computed RPA scattering of a model compositionally symmetric, N=100 ISD block copolymer, at six temperatures approaching the spinodal limit for three different choices of the scattering contrast vector c. The temperatures shown in each instance are 1.0324 Ts , 1.0171 Ts , 1.0121 Ts , 1.0072 Ts , 1.0020 Ts , and 1.0002 Ts , where Ts = 302.4 K. Temperature decreases with increasing peak intensity. (Fig. 1 of Ref 75). An additional insight into the nature of such an unusual behavior is provided by the RPA analysis76 for symmetric An BmCn block copolymers with non-selective middle block ( χ AB = χ BC = χ ). For such a symmetric ABC the RPA free energy (3.3) reads ΔF2 = 1 4N ∫ ( 2π )3 ( λ + ( Φ + ( r ) ) d 3q 2 ) + λ− (Φ− (r )) , 2 λ + = b ( q ) + χ AC − 4χ, Φ + ( r ) = Φ A ( r ) + Φ C ( r ) , . λ − = a ( q ) − χ AC , Φ − ( r ) = Φ A ( r ) − Φ C ( r ) where χ = χN , χ AC = χ AC N , (4.2) N = m + 2n and the functions a(q) and b(q) defined in ref 76 both for linear and miktoarm ABC depend on the reduced squared wave number Q = q 2 a 2 N 6 only. 28 It turns out that both the character of the weakly segregated morphology occurring in such a symmetric system and the very possibility to describe it within the WS theory χ AC , ~ χ ) is depends crucially on the values of the interaction parameters. Namely, the plane (~ ~ ,χ ~ ) = 0 and λ (~ ~ divided by the lines λ − (χ AC + χ AC , χ ) = 0 into i) the stability region ( λ − > 0, λ + > 0 ), where the fluctuations of both order parameters Φ + (r ), Φ − (r ) are finite and the uniform state is stable (or at least metastable) with respect to these fluctuations; ii) the AC-modulation region ( λ − < 0, λ + > 0 ), where the uniform (disordered) state is unstable with respect to formation of certain profile Φ − (r ) ≠ 0 , the order parameter Φ + (r ) iii) the B-modulation region ( λ + < 0, λ − > 0 ), where the uniform state is unstable with respect to formation of certain profile Φ + (r ) = −Φ B (r ) ≠ 0 , being weakly fluctuating; the order parameter Φ − (r ) being weakly fluctuating; and iv) the region ( λ − < 0, λ + < 0 ), Φ + (r ) and Φ − (r ) . where the uniform state is unstable with respect to fluctuations of both order parameters Figure 15. The spinodal behavior of the symmetric ternary ABC copolymers: a) the fA-dependences of the reduced squared critical wave numbers q- (curves 1) and q+ (curves 2) for both the linear (solid) and miktoarm (dashed) symmetric ABC copolymers; b) the classification of the spinodal instability regions in the (χ, χ AC ) -plane. The solid lines satisfy eqs (4.2a), (4.2b) for fA = 0.245, the numbers 0, 1, 2 and 3 label the stability, AC-, B-modulation and twolength-scale regions, respectively; the dashed lines describe the temperature evolution of the systems with I) I ) ~ (I ) χ (I ) and ~ χ AC χ > k 0 (a), χ AC χ = k 0 (b) and χ AC χ < k 0 (c), where k 0 = χ (AC χ (AC ,χ c) the fA-dependences of the coordinates χ AC (curves 1) and χ (I ) are the coordinates of the point of intersection of the solid lines; d) the fA-dependences of the ratio k = χ AC χ (I ) (I ) (I ) (curves 2) for both the linear (solid) and miktoarm (dashed) symmetric ABC copolymers; for both the linear (solid) and miktoarm (dashed) symmetric ABC copolymers. ~ ,χ ~ ) are the straight lines The lines λ + = 0 and λ − = 0 in the plane (χ AC (( ) (q ) + (~g ) (q )) , = min a(q ) = a(q ), a(q ) = (~ g ) (q ) − (~ g ) (q ) , χ AC − 4χ = − min b(q ) = −b(q + ), b(q ) = min ~ g −1 χ AC − −1 −1 11 −1 11 12 12 (4.3a) (4.3b) 29 the critical wave numbers q+ and q- characterizing the periods of the profiles Φ + (r ) and Φ − (r ) , respectively, being the locations of the absolute minima of the function a(q) and b(q), sought within the semiaxis 0 < q 2 < ∞ . The lines (4.3a) and (4.3b) intersect in the I χ = χ( ) = ( a ( q− ) + b ( q+ ) ) 4 point with the co-ordinates ( I) χ AC = χ AC = a ( q− ) , (4.4) As is seen from Fig. 14a, the values of the reduced squared critical wave numbers q+ and q- for both the linear and miktoarm ABC copolymers are rather different. A typical separation of the plane (~ χ AC , ~ χ ) into the regions with different types of the I) spinodal instability is shown in Fig. 14b. The dependences of the coordinates χ (AC , χ (I ) and I) the ratio k = χ (AC χ (I ) of f are plotted in Figs. 14c,d, respectively. As seen from Fig. 14c, the solid curves 1 and 2 do intersect and for the compositions corresponding to the intersection point the ABC should reveal the 2-scale behavior we discussed in the Introduction, which is in a good agreement with the results of ref 75. Two-length-scale behavior in blends ABC and AC. Another way to realize the 2-scale behavior is to blend the triblock ABC and diblock AC copolymers. The author et al.97 analyzed the spinodal stability of such blends within the RPA, the famous Hildebrand approximation for the χ-parameters being used for simplicity: ( χij = v δi − δ j ) 2 ( 2T ) , (4.5) where δi is the solubility parameter of the i-th component supposed to be temperatureindependent. It is convenient73 to take as the two independent interaction parameters characterizing the ternary systems in the approximation (4.5) the following ones: χ = χ AC = v ( δ A − δC ) 2 ( 2T ) , x = ( 2δ B − δ A − δC ) ( δ A − δC ) . (4.6) χAC characterizes incompatibility of the side blocks in the ABC triblock copolymer whereas the selectivity parameter x describes how much is the middle block B selective with respect to the side blocks. Remarkably, in this approximation the spinodal condition (3.11) takes simple form73 τ = ( 2χ AC N ) max W ( Q, x ) −1 (4.7) where N is the total degree of polymerization of the triblock copolymer and the function W depends on the reduced squared wave number Q = q 2 a 2 N 6 , selectivity x and structural parameters involved in the definition of the structural matrix g(Q) as described in refs 73, 97. For the blends of ABC and AC the function W is shown97 to have one or two maxima at 30 Q = Qmin and Q = Qmax depending on the blend structural parameters and selectivity and, thus, to be capable of revealing two-length-scale behavior. A new interesting feature of these blends is that Qmin can reduce to zero6. parameters of the ABC/AC blends is provided by so-called phase portrait in the ( φ ABC , x ) Information concerning localization of various modes of instability in the space of plane as shown in Fig. 15. a) b) Figure 16. The phase portraits of the AN C N / AhN BmN ChN for h=2; a) m=4 and b) m=10. The region corresponding to the macro- and microphase separation instabilities are labeled by the numbers 1 and 2, respectively. a) the only Lifshitz line is plotted; b) the bold solid and dashed lines are stable and metastable Lifshitz lines, respectively, within the region confined by the dotted thin lines the function W(Q) has two maxima; on the thin short dash line both maxima are equal and the system reveals two-length-scale behavior. For the blends where the middle block of ABCs is not long enough the function W(Q) has the only maximum Q = Qmax , which could be located either on the boundary of the interval 0 ≤ Q < ∞ of the permissible values of Q, i.e. at Qmax = 0 (in this case the blend would undergo macrophase separation at low temperatures) ) or within the interval, i.e. at Qmax > 0 (in this case the blend would undergo microphase separation at low temperatures). The line separating these two regions is referred to as the Lifshitz line (see Fig. 15a). The Lifshitz curve has the two vertical asymptotes at φ ABC = 0 and φ ABC = φL ( φL = 0.2 in Fig. 15a). Thus, the macrophase separation of the ABC/AC blend may occur 6 The situation is reminiscent of that found by Holyst and Schick126 who carried out the RPA analysis of symmetric ternary mixture of A and B homopolymers and AB diblock copolymer. They discovered that in some situations one of the components of the matrix of the structure factors Gij ( q ) for this mixture could exhibit two equal maxima. However, this situation could be observed in the disordered phase only and both maxima never diverge simultaneously in the A/B/AB mixture how it occurs in the ABC/AC blend. 31 only in the interval 0 < φABC < φL , For the mixtures with φABC > φL the ODT (microphase separation) is only possible. For the longer middle blocks B the function W(Q) could have two maxima and, accordingly, the Lifshitz line splits into two lines corresponding to reducing to zero of location of the stable or metastable maximum. A typical phase portrait for this situation is presented in Fig. 15b. An important prediction made in ref 97 is that it could be possible to provide occurring of regular superstructures reminiscent of the pattern shown in Fig. 8 via a subtle tuning of the selectivity, composition φABC and structural parameters of the ABC/AC blends. A similar RPA analysis was carried out for solutions of the di- tri- and regular polyblock copolymers in non-selective solvents,98 where also the conditions for two-length-scale The critical lines for the ABCs. For simple temperature dependence χ AC (T ) = Θ AC (2T ) , behavior were found. χ(T ) = Θ (2T ) the states of a ternary ABC system with different temperatures are located on ~ ,χ ~ ) . As shown in Fig. 14b, the χ AC = k ~ χ , k = Θ AC Θ in the plane (χ the straight line ~ AC system leaves the stability region crossing either the line (4.3a) or (4.3b) depending on the value of k. In the first case (B-modulation), which occurs, e.g., for ABA copolymer ( χ AC = 0 ), the effective cubic vertex reads ( ) (3 ) (3 ) (3 ) (3 ) γ (1) = 2 −3 / 2 Γ111 + Γ222 + 3Γ112 + 3Γ221 , Γijk(3 ) (1) = Γijk(3 ) (q1 , q 2 , q 3 ), (4.8a) q i = q* , q1 + q 2 + q 3 = 0 . The straightforward calculation as consistent with refs 32,33,36,73 shows that there is the only critical point where the cubic vertex (4.8a) vanishes for symmetric triblock (miktoarm) copolymer An Bm An . The point is located at f B = 0.49 ( f B = 0.557 ). On the contrary, in the second case (AC-modulation) the cubic vertex vanishes identically for symmetric copolymer with any composition of the ( non-selective block since it reads ) ( 3) ( 3) ( 3) ( 3) γ (1) = 2−3/ 2 ⎜⎛ Γ111 − Γ 222 + 3 Γ 221 − Γ112 ⎞⎟ ⎝ ⎠ (4.8b) So, the ternary ABCs belonging to the AC-modulation class are expected to undergo much smoother ODT than those belonging to the B-modulation class. In the Hildebrand approximation the symmetry assumption χ AB = χ BC = χ holds if the middle block is non-selective with respect to both side blocks (x=0), which occurs, e.g., for poly(isopren-b-styrene-b-2-vinylpyridine) triblock copolymers.127 But the continuous ODT transition in the ternary block copolymers occurs not only for f A = f C and x=0. The 32 critical lines were build by the author76 via numerical solving the equation γ (1, f A , f C ) = 0 for different values of 0<x<1 (see Fig. 16). Figure 17. The critical lines for the linear ABCs in the Hildebrand approximation for different values of the selectivity x. The symmetric bold lines correspond to non-selective middle block (x = 0), the critical lines labeled by the numbers 1, 2, 3, 4 and 5 correspond to the values of the selectivity parameter x = 0.01, 0.1, 0.3, 0.5 and 0.8, respectively. The dashed lines cb and ab are the critical lines for x = ±1 . Remarkably, the line f A = f C is not the only critical line even for x=0. Another critical line is the curve ac, which is rather close to the straight line ac. For x ≠ 0 the critical lines consist of two branches. In the limit x → 1 one of the branches, which corresponds to the case fC → 0.5, χ AC − χ BC χ AC + χ BC , approaches the diblock copolymer critical line ab whereas another branch, which corresponds to AB copolymer with a short strongly interacting C block,73 approaches Bc. The WST border lines and the phase diagrams of the ABC triblock copolymers. As discussed in sections 2 and 3, taking into account the higher harmonics and other weakly fluctuating fields results in a renormalization of the effective 4th vertex β of the effective free energy, and, eventually, in vanishing β with moving off the critical lines. Therewith, the line β ( f A , fC ) = 0 has meaning of the WST border line, i.e. a crossover line confining the region beyond which the WST does not hold even qualitatively. In Fig. 17 the WST border lines calculated within the Hildebrand approximation for the linear and miktoarm ABC triblock copolymers are presented.76 It is worth to make two remarks here. First, as is seen from Fig. 17b, the WST border line can intersect the critical line as shown in Fig. 17b. The intersection point is expected76 to be the tricritical point where the line of the 2nd order phase transitions transforms into that of the 1st order ones. Thus, in general, a judgment on the validity of the RPA (and even critical point) analysis in each case requires the full WST analysis including finding of the 4th vertex. Second, the WST validity region in the composition triangle is far from being negligibly small, especially for linear ABCs. In 33 particular, in the interval 0.42 ≤ f B ≤ 0.58 the WST phase diagram can be build for any asymmetry of the side blocks (see Fig.18). a) b) Figure 18. The maps describing the WST application to the melts of the ABC a) linear and b) miktoarm triblock copolymers with the nonselective middle block (x = 0) within the Hildebrand approximation. The critical and WS border lines are shown by solid and short dashed lines, respectively. Note that the WST border line intersects the critical line f A = fC for miktoarm ABC triblock copolymers. Figure 19. TheWST phase diagram of the molten linear ABC block copolymers calculated f B = 0.55 within the Hildebrand approximation in the plane ( σ, χ ) , where σ = f A ( f A + fC ) is the asymmetry parameter. The designations of the phases are the same as in Figure 11. It is worth to stress a rather broad (as compared to that for diblock copolymers) region (4) of the double gyroid (G) phase stability, which in this case can be predicted basing on the WST only. We skip here many other phase diagrams build and discussed in ref 76 the basic information of which is already given above in Fig. 11. 5. Conclusion. Even this brief survey of the current development of the weak segregation theory brings forward, hopefully, the idea that the theory is still far from being exhausted its potential. As I tried to show in the review, the WST is capable to describe and predict non-conventional phases in multi-component block copolymer systems like gyroids (both single and double), face-centered and simple cubic phases as well as rather non-conventional new types of two34 scale-length ordering. The WST easily incorporates various ideas of the short-range thermodynamic interactions between the repeated units of polymer components. The main advantage of the WST is its capability to state analytically and solve both analytically and numerically the problems of optimization of various properties of the ordered block copolymers and identifying the most promising structures of multi-component block copolymers worth to be synthesized by chemists, which could be too special to find them by chance. Still, the WST has some rather natural and important problems to be solved, in which case its predictive capacity could much increase. First of all I mean extension of the WST to the level of the 3rd shell approximation. Such an extension seems to be a natural limit for the Landau-like theory, which would correspond to the Landau expansion up to the 6th order in powers of the order parameter and thus provide an opportunity to describe the ODT near the tricritical point and the WST border line discussed above. Thereupon it is worth to mention here the work,51 where the authors have shown that inclusion of the 6th order terms in the Landau expansion of the free energy of polydisperse copolymers results in considerable corrections to the previous results. The next problem is still to improve our understanding of the fluctuation effects. In particular, the closely related problems of the interplay between the short- and long range fluctuation corrections128,129 and theoretical explanation of the observed deviations (see ref 31 and Fig. 3) of the inverse structure factor from the RPA predictions are still open issues. One more important problem is to study coupling in the ordered phases between the density fluctuations and transversal sound waves (shear fluctuations), which until now has been considered130,131 only in the disordered state and is expected132 to be especially important in the weak segregation regime where the shear modulus is small.133,134 But the most important problem is, of course, to establish a prompt and reliable feedback between the theoreticians, experimentalists and industry since there is no good physics, which could not generate a good technology, and there is no good technology which would be not based (at least, implicitly) on a good physics. To conclude, I thank my colleagues and friends Volker Abetz, Henk Angerman, Kurt Binder, Gerrit ten Brinke, Monica Olvera de la Cruz, Andrey Dobrynin, Alexander Grosberg, Jean-Francois Joanny, Albert Johner, Alexei Khokhlov, Ludwik Leibler, Marcus Müller, Alexander Semenov, Friederika Schmidt and Reimund Stadler†, whose feedback so much helped me in my work. 35 The BCC family is determined by the set {q i } consisting of 12 vectors whose relative Appendix 1. The basic weakly segregated morphologies. directions are given, e.g., by the six vectors listed below and the same vectors taken with ( = (q ) 2 )(0, − 1, − 1), the opposite sign: q 1 = q* qI * 2 (0,1, − 1), ( = (q q 2 = q* q II * ) 2 )(−1, 0,− 1, ), 2 (−1, 0,1, ), ( = (q q 3 = q* q III * ) 2 )(−1, − 1, 0) 2 (1, − 1, 0), 20 The vectors could be visualized as the edges of octahedron or tetrahedron24 or via their planar mapping shown in Figure A1. The arrow circuits in Fig. Aa correspond to the equalities q I = q II + q3 , q II = q III + q1 , q III = q I + q 2 , q1 + q 2 + q3 = 0 . a) b) Figure A1. The planar mapping of the vectors characterizing the symmetry of the main harmonics a) for the BCC family and b) for the orthorhombic lattice Fddd (see explanations in the text). 1. For the conventional BCC all phases ϕ appearing in the definition (1.14) of the basic function Ψ are zero and the vertices read α BCC = 8 γ 6 3 / 2 , β BCC = [λ 0 (0 ) + 8λ 0 (1) + 2λ 0 (2 ) + 4λ (1,2,1)] 24 . (A1.1) 2. If the phases corresponding to three vectors q1, q2 and q3, which form a base of the tetrahedron, and those of three non-coplanar vectors qI, qII and qIII (the thin and solid lines, respectively, in Figure Xa) are equal to π/2 and 0, respectively, we get a lattice first 3) discussed in ref 12 and called there the BCC2. For this lattice Ω (ABC = 3 π 2, 3) 3) 3) 4) 4) (4) Ω (ABS = Ω (ACS = Ω (BCS = π 2, Ω (ABCSA = Ω (BCASB = Ω CABSC = π and the basic function and α BCC 2 = 0 , β BCC 2 = [λ 0 (0 ) + 8λ 0 (1) + 2λ 0 (2 ) − 4λ (1,2 )] 24 . the vertices read (A1.2) The BCC2 phase could be shown76 to possess symmetry of the I 4132 space group (No.214), which is non-centrosymmetric and closely related to so-called single gyroid surface. Thus, it seems to be the simplest (and the only up to now) cubic non36 centrosymmetric morphology that could be described (and for some case predicted) within the WS theory. The BCC2 belongs to the class of morphologies like L, FCC and SC we refer to as the degenerate ones because for them the cubic vertex (2.5) identically equals zero due to the ΔFℜ = − τ 2 (4β ℜ ). symmetry reasons. For these degenerate morphologies the free energy (2.4) reads21 (A1.3) Thus, the most stable degenerate morphology is that having the least value of the quadric vertex. For references, we present the expressions for the vertices also for FCC L and SC βFCC = ⎡⎣ λ 0 ( 0 ) + 6λ 0 ( 4 3) − 2λ ( 4 3, 4 3) ⎤⎦ 16 , βL = λ 0 ( 0 ) 4 , βSC = ⎡⎣ λ 0 ( 0 ) + 4λ 0 ( 2 ) ⎤⎦ 12. morphologies: ( = (q = (q = (q ) 6 )(− 2,−1,−1), 6 )(+ 2,+1,−1), 6 )(+ 2,−1,+1), ( = (q = (q = (q 6 (− 2,+1,+1), q 02 = q* ) 6 )(+ 1,+2,−1), 6 )(− 1,−2,−1), 6 )(− 1,+2,+1), (A1.4) ( = (q = (q = (q 6 (+ 1,−2,+1), q 03 = q* ) 6 )(+ 1,−1,+2 ), 6 )(− 1,+1,+2 ), 6 )(− 1,−1,−2 ). 6 (+ 1,+1,−2 ), The G family is determined by 2⋅12 main harmonics given by the 12 vectors q 01 = q* q11 q 21 q 31 * * * q12 q 22 q 32 * * * q13 q 23 q 33 * * * (A1.5) Figure A2. The planar mappings of the vectors characterizing the symmetry of the main harmonics for the G family (left); and of the edges of the regular icosahedron (right). a) the vectors depicted by bold lines have zero phases for all three morphologies G, G2 and BCC3, those depicted by dashed and thin solid lines have phases equal to π only for the double gyroid (G) and both for G and G2, respectively (see the definitions (2.19a) and (2.19b); b) the edges depicted by thin and bold lines correspond to the vectors to be removed and properly rotated to transform the icosahedron into the G cell. and 12 opposite ones. The planar mapping of the vectors is shown in Figure A2a. For comparison, in Figure A2b the planar mapping of the regular icosahedron is shown. 37 Obviously, the set of vectors (A.5) (and the opposite ones) is obtained via a deformation of the regular icosahedron, which involves removing 6 of 30 edges of the icosahedron and the proper rotations of the remaining edges, the resulting polyhedron being the Wigner-Zeitz cell of the corresponding crystal lattice. It is this relationship between the G family and the icosahedron symmetry which causes the famous 10 spot SAXS pattern observed in the gyroid phase. 1. The morphology arising if all the phases ϕi are set to equal zero we call the BCC3. It is just the ordinary BCC but the fact that the dominant harmonics correspond here to the 3rd (rather than the 1st!) co-ordination sphere. α BCC3 = γ 33/ 2 , βBCC3 = ⎣⎡ Λ1 + 4 ⎣⎡ λ (1 3, 2 3 ) + λ ( 2 3,5 3 ) ⎦⎤ + 2λ ( 2 3, 2 3 ) ⎦⎤ 48, ( Λ1 = λ 0 ( 0 ) + 2 λ 0 ( 4 3 ) + 2 ( λ 0 (1 3 ) + λ 0 ( 2 3 ) + λ 0 (1) + 2λ 0 ( 5 3 ) ) ) (A1.6) 2. The trial function (1.14) with the main harmonics (A.5) and the phase choice ϕ12 = ϕ23 = ϕ31 = ϕ01 = ϕ02 = ϕ03 = 0, ϕ21 = ϕ32 = ϕ13 = ϕ11 = ϕ22 = ϕ33 = π. corresponds to the bi-continuous gyroid (G) or double gyroid morphology having the α G = γ 33 / 2 , β G = [Λ 1 − 2Λ 2 − 4λ (1 3 , 2 3)] 48 , symmetry Ia 3 d . Λ 2 = 2λ (2 3 , 5 3) − λ (2 3 , 2 3). (A1.7) It is seen from (A.6), (A.7) that due to the symmetry of the BCC3 and G lattices their cubic vertices are identical and, therefore, the BCC3 - G phase transition line (surface) is determined by equation β G − β BCC3 = 0 (A1.8) 3. The trial function (1.14) with the main harmonics (A.5) and the phase choice ϕ21 = ϕ32 = ϕ13 = π, ϕ12 = ϕ23 = ϕ31 = ϕ01 = ϕ02 = ϕ03 = ϕ11 = ϕ22 = ϕ33 = 0 (A1.9) corresponds to the morphology of the symmetry I 4 3d we refer to as the G2. For this ( ) α G 2 = γ 2 ⋅ 33 / 2 , β G 2 = [Λ 1 + 2Λ 2 − 4λ (1 3 , 2 3)] 48 . morphology α HEX = 2 γ 3 3 / 2 , β HEX = [λ 0 (0 ) + 4λ 0 (1)] 12 . (A1.10) For reference, we give here also the vertices for the HEX morphology20 (A.11) The orthorhombic lattice (space symmetry group O70 or Fddd) is generated by the 2⋅4 e1 = q* ( a, b, c ) , e 2 = q* ( a, −b, −c ) , e3 = q* ( − a, b, −c ) , e 4 = q* ( − a, −b, c ) main harmonics given by the 4 vectors (A1.12) 38 where q* = q* a 2 + b 2 + c 2 , the values of the periods a, b, c of the reciprocal lattice are, in general, all different and the choice of the phases is ϕ01 = π, ϕ02 = ϕ03 = ϕ04 = 0. Thus, in general, the lattice belongs to the class of the degenerate morphologies and the corresponding 3rd and 4th vertices read ( 3 ⎡ α 0ortho = 0 , β0ortho = ⎢ λ 0 ( 0 ) + 2∑ λ 0 ( hi ) − 2λ h1 , h 2 ⎢⎣ i =1 where the designations hi = 4ai2 (a 2 1 ) )⎥⎥ ⎤ ⎦ (A1.13) 16 , + a22 + a32 are introduced. In particular, for the case a = b = c the orthorhombic lattice becomes just the FCC one. If a 2 = 2b 2 = 2c 2 then two second harmonics ( 0, ±2b, 0 ) and ( 0, 0, ±2c ) also belong to the first coordination sphere of the reciprocal lattice, which now contains (along with the main harmonics) 2⋅6 vectors and At last, if a 2 = 4b 2 = 12c 2 then two second harmonics b1 = e1 − e 2 = q* ( 0, 2b, 2c ) and could be easily checked to correspond to the BCC lattice in the co-ordinate space. b 2 = e 4 − e3 = q* ( 0, −2b, 2c ) as well as the forth harmonic c = b1 + b 2 = q* ( 0, 0, 4c ) also belong to the first coordination sphere with the radius q*2 = a 2 + b 2 + c 2 = 16c 2 . The planar mapping of the vectors is shown in Figure Xd. In this special case only those harmonics, which have the same symmetry rather than belong to the same coordination sphere of the reciprocal lattice, have equal amplitudes, so ( ( ) ( )) = 2 ( cos ( e r ) + cos ( e r ) + cos ( e r ) − cos ( e r ) ) , Ψ = 2 cos ( cr ) . that the trial function (1.14) reads Φ ( r ) = AΨ A ( r ) + BΨ B ( r ) + C Ψ C ( r ) , Ψ B = 2 cos b 2 r − cos b 1 r , ΨA 2 3 4 1 (A1.15) C The choice of the signs in (A.15) is determined by equivalent requirements to satisfy the F = VT min f ( A, B, C ) , Fddd symmetry and provide the minimal free energy, which takes the form f ( A, B, C ) = τ ( 4 A2 + 2 B 2 + C 2 ) − 4γ A2 B − 2γ B 2C where ( ) + λ ( 0 ) + 2 ( λ (1 4 ) + λ ( 3 4 ) + λ (1) − λ (1 4,3 4 ) ) A4 + ( λ (1) + (1 2 ) λ ( 0 ) ) B 4 + ( λ ( 0 ) 4 ) C 4 + 4λ (1,3 2 ) A2 BC +4 ( λ (1) + λ ( 3 2 ) ) A2 B 2 + 4λ ( 3 2 ) A2C 2 + 2λ (1) B 2C 2 . 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