編集者のお薦め: 編集者のサマリー この論文では6回対称性を持つソフトマター準結晶の研究が述べられます。 長方形と3角形からなる2種のタイリングを調べて、著者らは 2つの長さスケールを含む安定した非周期的なソフトマター系をデザインする方法を示します。 彼らの研究は、ソフトマター系でより多様な準結晶を見つける方法を示唆しています。

準結晶の発見以来、近似結晶は実験および理論の両面から有用な道具となっています。準結晶に存在する典型的なクラスターを格子定数の大きな単位胞に並べることによって得られる近似結晶は、周期結晶と準結晶の間の溝の橋渡しを行ってきました。この論文では周期結晶の領域が大きくなってゆく準結晶列を提案し、周期結晶を近似する「近似準結晶」という考え方を提案します。大小の正3角形と長方形で構成されたタイリングは3の倍数の金属比に基づき、六方対称性を持っており、また4次元超格子からの射影と見なせます。金属比ではない六方タイリングと合わせ、二元ナノ粒子や酸化物フィルム、金属間化合物などの様々な複雑構造を特徴付ける一般的な理論的枠組みを提供します。

Self-sorting, in which multiple components selectively assemble themselves by recognising self from others, is an attractive approach to produce supramolecular assemblies with controlled structures. Lock-and-key type complementary physical interactions are required for self-sorting because selective affinity is necessary to distinguish self from others. Here we show self-sorting behaviour based on a principle of geometrical complementarity by shape during our investigation of assembly of pentagonal pillar[5]arenes and hexagonal pillar[6]arenes on a surface. In the homoassembly systems, anionic pillar[5]arenes and pillar[6]arenes are adsorbed onto positively charged layers of cationic pillar[5]arenes and pillar[6]arenes, respectively, through cationic-anionic electrostatic interactions. In contrast, ionic pillar[5]arenes are adsorbed onto layers constructed from oppositely charged pillar[5]arenes, but ionic pillar[6]arenes are not. Equally, for the reverse combination, ionic pillar[6]arenes are adsorbed onto layers constructed from oppositely charged pillar[6]arenes, but ionic pillar[5]arenes are not. The geometrical complementarity by shape realises effective self-sorting even in non-directional multivalent ionic interactions.

従来の準結晶の最も顕著な特徴は、正20面体、正12角形、正10角形または正8角形で特徴付けられる非伝統的な回転対称性である。これらの物質の対称性および非周期性は、その構造を制御する2つ以上の長さスケールの無理数比に由来する。最もよく知られている例は、黄金比および白銀比に関連する2次元モデルとしてのペンローズ・タイリングおよびアンマン-ビーンカー・タイリングである。驚くべきことに、これまでに他の金属比タイリングは発見されていなかった。本論文では、自己相似的な青銅比六方準結晶パターンを提案する。このパターンは、コッホ曲線のような窓で制限された高次元の周期格子からの射影と見ることもできる。数値シミュレーションを用いることによって、コア・シェル構造を有するソフト高分子コロイド粒子系でこの準結晶のランダム・タイリングが実現できることを示す。さらに、パターンの幾何学的形状を連続的に変えることによって新たなパターンを生成できるが、これは複数の金属・シリコン合金で観察される準結晶の近似結晶に対して別の解釈を提供する。 編集者のサマリー 青銅比六方パターンに基づく２次元準結晶タイリングが提案される。

Regular structures of equal spheres on the triply periodic minimal surfaces known as primitive (P), gyroid (G) and diamond (D) surfaces are enumerated as obtained through Monte Carlo simulations of hard spheres undergoing the Alder transition. Remarkably, there exist magic numbers producing the regular structures, which are simply explained by means of hexagulation numbers defined as H = h(2) + k(2) - hk, in analogy with the Caspar and Klug's triangulation numbers, T = h(2) + k(2) + hk for icosahedral viruses, where h and k are equal to nonnegative integers. Understanding the significance of symmetry of the surfaces, the total number of spheres per cubic unit cell N is represented by N = 8H, 16H, and 32H for P-, G- and D-surfaces, respectively. Accordingly, these arrangements are analyzed in terms of space groups, equivalent positions (Wyckoff positions), and polygonal-tiling representations. The key is that there is only a limited number of efficient physical design possible even on the triply periodic minimal surfaces.

We investigate the origin of the 18-fold symmetry in quasicrystals. We perform Monte Carlo simulations of 2D disks interacting with the hard-core/square-shoulder potential, and we analyze the bond-orientational order and the local vertex configurations. We find that the so-called special clusters of particles play a crucial role in the mixing of the three sets of bond directions 20 degrees apart, producing the 18-fold diffraction pattern.

The ordered phases formed by polymeric nanocolloidal particles with a core-shell architecture are typically very different from those seen in classical hard colloids. The differences must invariably be related to softness of particles. Here we discuss the T = 0 phase diagram of a 2D model of core-softened particles represented by the hard-core/square-shoulder pair potential. The aim of the analysis is to show that a simple generalization of the hard-core potential is sufficient to induce several rather complex crystalline and quasicrystalline phases, and to illustrate how these structures can be systematically constructed and compared using a geometric scheme based on canonical local packings of the particles.

Over the past decade, quasicrystalline order(1) has been observed in many soft-matter systems: in dendritic micelles(2), in star(3) and tetrablock(4) terpolymer melts and in diblock copolymer(5) and surfactant micelles(6). The formation of quasicrystals(7-9) from such a broad range of 'soft' macromolecular micelles suggests that they assemble by a generic mechanism rather than being dependent on the specific chemistry of each system. Indeed, micellar softness has been postulated(7) and shown to lead to quasicrystalline order(10). Here we theoretically explore this link by studying two-dimensional hard disks decorated with step-like square-shoulder repulsion that mimics, for example, the soft alkyl shell around the aromatic core in dendritic micelles(2). We find a family of quasicrystals with 10-, 12-, 18- and 24-fold bond orientational order which originate from mosaics of equilateral and isosceles triangles formed by particles arranged core-to-core and shoulder-to-shoulder. The pair interaction responsible for these phases highlights the role of local packing geometry in generating quasicrystallinity in soft matter, complementing the principles that lead to quasicrystal formation in hard tetrahedra(11,12). Based on simple interparticle potentials, quasicrystalline mosaics may well find use in diverse applications ranging from improved image reproduction(13) to advanced photonic materials(14).

「Soft quasicrystal」という言葉がはじめて登場したのは，2005年アイオワ州エイムズで開催された第９回準結晶国際会議の高分子化学者V.~Percecのバンケット講演のことである．この「ソフト準結晶」＝「ソフトマターの準結晶」とは何だろうか．まず最初にソフト準結晶が発見されるまでの背景から説明してゆくことにしよう．

We review our recent numerical simulation result that 48/64 hard spheres per unit cell on the gyroid surface are self-organized by the Alder-transition mechanism. Striking evidence of entropy-driven ordering is obtained in terms of the acceptance ratio of Monte Carlo moves and order parameters. The regular tessellations of the spheres on the triply periodic minimal surface can be viewed as hyperbolic tilings on the Poincare disc.

We find that 48/64 hard spheres per unit cell on the gyroid minimal surface are entropically self-organized. Striking evidence is obtained in terms of the acceptance ratio of Monte Carlo moves and order parameters. The regular tessellations of the spheres can be viewed as hyperbolic tilings on the Poincare disc with a negative Gaussian curvature, one of which is, equivalently, the arrangement of angels and devils in Escher's Circle Limit IV.

This is a progress review of an emerging research front: soft quasicrystals including liquid crystalline dendrons, nanoparticles, mesoporous silica, colloids, ABC star and linear terpolymers, and even water and silicon. As an aid to readers, we explain the basics of quasicrystals developed in solid-state physics: orders in quasicrystals, higher dimensional crystallography, approximants, phason randomness, and the origin of quasicrystal formation. Then we review some numerical studies from early to recent ones. Our main purpose is to elucidate how to construct quasicrystalline structures: The introduction of additional components or a new length-scale is the key to discover new quasicrystals. As a case study, we describe our recent studies on ABC star terpolymer systems and present the results of simulations of dodecagonal polymeric quasicrystals. In the case of dodecagonal quasicrystals, one easily finds that the key is to search square-triangle tiling structures with changing components. Application to photonic quasicrystals is reviewed as well. Our hope is that this review will contribute furthering quasicrystal chemistry. (C) 2011 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 50: 155-167, 2012

Two decades after the first publication on quasicrystals in aluminum manganese alloys by Shechtman et al., a soft quasicrystal was found in nature. Since then a number of materials successively joined the family of soft quasicrystals, including liquid crystals, polymers, nanoparticles, colloids, mesoporous silica, and even water and silicon. In this article, important experimental advances are reviewed to introduce readers to the emerging research front of soft quasicrystalline materials. The world of quasicrystals opened by Daniel Shechtman extends its reach into the twenty-first century chemistry.

Star-shaped terpolymers of the ABC type composed of incompatible polymer components give a variety of ordered structures with mesoscopic length scales depending on their composition ratio. Their peculiar features are summarized in this report. Polymer components adopted are polyisoprene (I), polystyrene (S) and poly(2-vinylpyridine) (P), and many monodisperse samples of the IXSY P-Z type were anionically prepared. Firstly our focus is on molecules of the I1.0S1.0Px1 type, where x(1) is only a variable. The complex but systematic morphology change was displayed within the range 0.2 <= x(1) <= 10, that is, their structures change from spherical plus lamellae structure for I1.0S1.0P0.2 to periodic tilings (0.4 <= x(1) <= 1.9), then to lamellae-in-lamella (3.0 <= x(1) <= 4.9) and lamellae-in-cylinder (7.9 <= x(1) <= 10) structures with increasing x(1). Here if we pay attention to the structural variation of the P domain inclusively, it transforms from sphere to cylinder, lamella and then to matrix, which is the same as that for linear polymers. Among them, several periodic Archimedean tiling patterns can be naturally formed when the relative lengths of the three chains are close to one another. Moreover, it has been found that the tiling zone is spread out widely. For example, the series I1.0S1.8Px2 (with 0.8 <= x(2) <= 2.9) and the other series I1.0SyP2.0 (with 1.1 <= y <= 2.7) show mostly Archimedean tilings. Additionally, block copolymer/homopolymer blends with a composition of I1.0S2.7P2.5 reveal a quasicrystalline tiling with dodecagonal symmetry. Furthermore, a zinc-blende-type four-branched network structure was created just a little outside of the tiling region for a block copolymer/homopolymer blend of I1.0S2.3P0.8. When some more asymmetry in chain length is introduced, hyperbolic tiling on a gyroid membrane has successfully been constructed for the sample I1.0S1.8P3.2 and it transforms into a hierarchical cylinders-in-lamella structure with further increase in P content to I1.0S1.8P6.4. Thus, kaleidoscopic morphologies have been generated from ABC star-shaped terpolymers and their structural change has turned out to be very sensitive to relative compositions.

ソフト準結晶が近年話題になっている.この小論では,まずソフト準結晶創成には多元系,複雑分子を用いるなど5回対称クラスターや新たな長さスケールの導入が必要であることを計算機実験の立場から論じた.また,高次元結晶学を解説し,準結晶探索には近似結晶探索が鍵となっていることを記した.準結晶の起源論には2つあり,エネルギーとエントロピーの2つの立場があるが,エントロピー説に関連して準結晶特有のフェイゾン乱れを解説した.さらに,ABC星型高分子系のタイリング構造を説明し,格子高分子の計算機実験の予測を経た高分子準結晶発見への道筋を解説した.最後に,フェイゾン動力学の考慮も準結晶探索には重要な要素になっていることも論じた.

３次元および２次元結晶対称性の観点から、線状、星型２種類の３成分系高分子が作るメゾスケールの共連続立方相およびタイリング構造を紹介した。

We have found a giant zincblende structure with a 45 nm lattice constant in an ABC star polymer / homopolymer blend system. Alternating four-coordinate cellular domains of components A and B form a diamond lattice by using space-division property of ABC star polymers.Transmission electron micrographs reveal spherical domains of A and B with space-filling component C, and small-angle X-ray scattering confirms the corresponding F＼bar{4}3m space group symmetry. The present result opens a new way to construct mesoscale non-close-packed lattices, which have a relevance to photonic applications.

We present theoretical arguments and experimental evidence for a longitudinal instability in core-shell cylindrical polymer brushes with a solvophobic inner (core) block and a solvophilic outer (shell) block in selective solvents. The two-gradient self-consistent field Scheutjens-Fleer (SCF-SF) approach and Monte Carlo (MC) simulations are employed to study a conformational transition which occurs upon a decrease in the solvent strength for the inner block from Theta to poor solvent conditions. It is found that a decrease in the solvent strength for the core block leads to an instability in the cylindrically uniform structure and the appearance of longitudinal undulations in the collapsed core of the molecular brush. This result of our modeling is in excellent agreement with experimental observations on core-shell brushes with poly(acrylic acid) (PAA) core and poly(n-butyl acrylate) shell, where the core forms pearl-necklace-like structures due to either a bad solvent for PAA or complexation with multivalent ions.

2種類の3成分系高分子がバルク状態でつくるメゾスケールの構造を対象として,高分子と結晶対称性について述べた。一つはABC線状分子の3相共連続周期構造であり,もう一つはABC星型分子の二次元周期構造,アルキメデスタイリングである。また星型分子が示す構造の特異例として,結晶に当てはまらない秩序構造である準結晶タイリングも紹介した。

In a previous paper, we reported the formation of a dodecagonal quasicrystal (DDQC) in a quasi-two-dimensional lattice Monte Carlo simulation of a star-shaped polymer. In this paper, we show a series of Archimedean and quasicrystalline phases (4.8(2)) -> (3(2).4.3.4) -> DDQC -> (4.6.12) with increase of one component of ABC star polymers. The phase behaviour can be regarded as a transition between square tiling and triangle tiling via square-triangle tiling. We compare the specific heat for the phases and find that the DDQC sample possesses higher specific heat at high temperatures, which may be attributed to phason dynamics.

We present a study of photonic band structures of two-dimensional Archimedean tiling patterns. The tilings we have investigated are (4.8(2)), (6(3)), (4.6.12), and (3(2).4.3.4), which have been discovered computationally and experimentally in self-assembled microphase separation of ABC star block terpolymer systems. Using plane-wave method, we have calculated eigenvalue equations for various combinations of dielectric contrast on the complex patterns. We demonstrate the existence of complete photonic band gaps in the (4.6.12) structure. Furthermore, we find that complete photonic bands readily open in the (3(2).4.3.4) structures in the same way as in dodecagonal quasicrystals. Complex tilings open up a way to construct photonic crystals.

A mesoscopic tiling pattern with 12-fold symmetry has been observed in a three-component polymer system composed of polyisoprene, polystyrene, and poly(2-vinylpyridine) which forms a star-shaped terpolymer, and a polystyrene homopolymer blend. Transmission electron microscopy images reveal a nonperiodic tiling pattern covered with equilateral triangles and squares, their triangle/square number ratio of 2.3 (approximate to 4/root 3), and a microbeam x-ray diffraction pattern shows dodecagonal symmetry. The same kind of quasicrystalline structures have been found for metal alloys (similar to 0.5 nm), chalcogenides (similar to 2 nm), and liquid crystals (similar to 10 nm). The present result (similar to 50 nm) confirms the universal nature of dodecagonal quasicrystals over several hierarchical length scales.

A simple Landau theory of three-component alloy systems under incompressible condition is investigated, which appears to give regions of the phase diagram in which Archimedean tiling phases are stable in two dimensions. Moreover, we find regions where dodecagonal and decagonal quasicrystals appear to be stable. The Alexander-MacTague and Mermin-Troian theories of weak crystallization are revisited.

本文ファイルの1ページ目は、本記事が掲載された当該号の表紙である。

We report the formation of an approximant of a dodecagonal quasicrystal in a quasi-two-dimensional lattice Monte Carlo simulation of a star-shaped three-component polymeric alloy. It is associated with the recent striking experimental manifestation of the complex Archimedean tiling (3(2).4.3.4) consisting of triangles and squares, related to the sigma phase in the Frank - Kasper family, but whose edge length is about 80 nm. The simulation box with periodic boundary conditions (128 x 128 x 10) can be regarded as the Stampfli inflation of the (3(2).4.3.4) tiling, an approximant of the dodecagonal quasicrystal. The corresponding edge length of deflated squares and triangles is thought to be about 300 nm. Furthermore, the phason dynamics of the deflated square - triangle tiling is observed at an elevated temperature.

The Archimedean tiling (3(2).4.3.4) is a regular but complex polygonal assembly of equilateral triangles and squares. This tiling pattern with mesoscopic repeating distance has been found for an ABC star-branched three-component polymer composed of polyisoprene, polystyrene, and poly(2-vinylpyridine). In this structure the environment of a molecule splits into multiple sites and two microdomains with different sizes and shapes, are formed for one component. This complexity is the first observation in complex polymer systems and can lead to a new type of mesoscale self-organization. The tiling pattern has been observed for the other materials on much shorter length-scale; therefore, the experimental fact observed in the present study is demonstrating that the complexity is universal over different hierarchies. (c) 2005 Wiley Periodicals, Inc.

（概 要）平田森三著、キリンのまだら―自然界の統計現象をめぐるエッセイ (ハヤカワ文庫NF―数理を愉しむシリーズ) (文庫)の書評を行った。

We investigated the dynamics of long deoxyribonucleic acid (DNA) migrating through curved channels under electric fields. Long DNA exhibits large conformational changes in the curved channels because of the inhomogeneity of the electric fields around curves. Two kinds of channel shapes were used for the examination. One (type I) has the same width in the curved region as in the straight region. The other (type II) is wider in the curved region than in the straight region. The difference in migration rates between long DNA and short DNA was larger in type II than in type I chips. We discuss the separation mechanism of the type II chip. (C) 2004 American Institute of Physics.

デザインされた分子による自己組織化という概念および技術は益々深く広く進化する概念であり技術である。ここでは、混じり合わない三種類以上の高分子を連結したブロック共重合体系の作るミクロ相分離構造の多様な可能性を幾何学の観点から論ずる。対角線法という格子モンテカルロ計算法ではじめて可能になった研究である。

We investigated the curvature effect on the dynamics of long DNA using microfluidic devices. Long DNA has larger configurational entropy in a curved channel than in a straight channel. Under weak hydrodynamic flows, long DNA exhibited a curvature entropy trapping effect. The effect disappeared as the hydrodynamic flow was increased.

We propose new polymer models for Monte Carlo simulation and apply them to a polymer chain confined in a relatively thin box which has both curved and flat sides, and show that either an ideal or an excluded-volume chain spends more time in the curved region than in the flat region. The ratio of the probability of finding a chain in the curved region and in flat region increases exponentially with increasing chain length. The results for ideal chains are quantitatively consistent with a previously published theory. We find that the same effect appears with excluded-volume chains and a similar scaling relation can be applied to them up to a certain length of the polymer.

In terms of Voronoi division we study the local geometry of a grafted polymer having 52 ends in united-atom molecular dynamics simulations. The volume of a Voronoi polyhedron for a chain end is larger than that for an internal or junction atom, and that it is the most sensitive to temperature. Chain ends dominantly localize at the surface of the globule: While the ratio of surface atoms is only 24% of all atoms, the ratio of ends at the surface is 91% out of all ends. The shape of Voronoi polyhedra for internal atoms is prolate even in the bulk. We find that two specific faces play a significant role in the faces occupy 38% of the total surface area of a Voronoi Voronoi space division of covalently bonding polymers: Two bonding polyhedron and determine the prolate shape.

We propose polymer models for Monte Carlo simulation and apply them to a polymer chain confined in a thin box which has both curved and flat sides, and show that either a Gaussian or an excluded-volume chain spends more time in the curved region than in the flat region. The ratio of the probability of finding a chain in the curved region and in flat region increases exponentially with increasing chain length up to a certain length defined by the size of box.

In order to investigate the topological effects of chain molecules, united-atom molecular dynamics simulations of a 500-mer polyethylene linked by 50 hexyl groups (a grafted polymer having 52 ends) are carried out and analyzed in terms of Voronoi space division. We find that the volume of a Voronoi polyhedron for a chain end is larger than that for an internal or junction atom, and that it is the most sensitive to temperature, both of which suggest higher mobility of chain ends. Moreover, chain ends dominantly localize at the surface of the globule: The striking evidence is that while the ratio of surface atoms is only 24% of all atoms, the ratio of ends at the surface is 91% out of all ends. The shape of Voronoi polyhedra for internal atoms is prolate even in the bulk, and near the surface it becomes more prolate. We propose the concept of bonding faces, which play a significant role in the Voronoi space division of covalently bonding polymers. Two bonding faces occupy 38% of the total surface area of a Voronoi polyhedron and determine the prolate shape. (C) 2004 American Institute of Physics.

Using the Monte Carlo lattice-simulation technique, we present numerical evidence of the formation of gyroid and nongyroid tricontinuous cubic phases in high polymeric systems of ABC/A/C triblock copolymer and homopolymer blends. By increasing the volume fraction of homopolymer, a remarkable phase sequence G (gyroid) --> D (diamond) --> P (primitive) is observed, which is common to certain surfactant systems. Our results indicate that the ABC triblock copolymer system with blending homopolymers may be a zoo of cubic phases, suitable for comparative studies of these phases.

The microphase-separated morphology of ABC three-arm star-shaped copolymers with arm-length ratio 1:1:x is investigated by a recently proposed simulation method, the diagonal bond method. Five kinds of two-dimensional (cylindrical) phases, three kinds of lamellar-type phases and two kinds of continuous matrix phases are discovered. The phase diagram is presented: The progression of the morphologies as a function of x is the following: lamella + sphere; five polygonal cylinders, [8.8.4], [6.6.6], [8.6.4; 8.6.6], [10.6.4; 10.6.4; 10.6.6], [12.6.4]; perforated layer; lamella + cylinder; columnar piled disk; lamella-in-sphere. Two remarkable features of ABC star polymer systems are found: first, in all phases junction point monomers gather on lines where three interfaces meet, and second, the section of the cylindrical phase becomes the tessellation of even-numbered polygons. The free energy of the system is also calculated in the strong segregation limit for four kinds of simple phases, and the results are consistent with the simulations.

We consider quantum particle or Gaussian polymer confinement between two surfaces and in cylinders with sinusoidal undulations. In terms of the variational method, we show that the quantum-mechanical wave equations have lower ground-state energy in these geometries under long wavelength undulations, where bulges are formed and waves are localized in the bulges. It turns out correspondingly that Gaussian polymer chains in undulated boxes or tubes acquire higher entropy than in exactly flat or straight ones. These phenomena are explained by the uncertainty principle for quantum particles, and by a polymer confinement rule for Gaussian polymers. If membrane boxes or tubes are flexible, polymer-induced undulation instability is suggested. We find that the wavelength of undulations at the threshold of instability for a membrane box is almost twice the distance between two walls of the box. Surprisingly, we find that the instability for tubes begins with a shorter wavelength compared to the "Rayleigh" area-minimizing instability.

格子上の高分子シミュレーション手法である対角線法の特徴と対角線法で複雑なブロック共重合体ミクロ相分離構造の形成に成功した理由を議論した. また, 実験的には未発見の構造である対称ABCD星形ブロック共重合体の作る胞晶 (Cell Crystal) 構造と非対称ABC星形ブロック共重合体作るシリンダー構造のコンピューター画像を示し, それらの幾何学的特徴を述べた. ABCトリブロック共重合体については, 4種類 (Gyroid, Double-diamond, ラメラ, シリンダー) の構造が得られた.

空間分割の観点からブロック共重合体のミクロ相分離を考察し、新たな物質構造Cell Crystal「胞晶」構造が存在しうることを示し、計算機実験で確かめた。胞晶の極小曲面構造は、19世紀ケルビン卿がシャボン玉の充填構造に対して示唆した多面体構造に他ならない。

Using a new lattice polymer simulation method for block copolymers, we have studied a dense polymer system composed of monodisperse linear A-B-C triblock copolymers. We confirm that a tricontinuous structure for linear A-B-C block copolymers observed by Mogi et al. (Macromolecule 25, 5408 (1992) and ibid. 25, 5412 (1992)) could be, indeed, an ordered tricontinuous "double-diamond" (OTDD) structure. In addition, two (symmetric and shifted) OTDD phases and a lamellar phase have been observed in the same simulations. The two OTDD phases are conceivable by shifting one diamond network against the other diamond network.

A new lattice model for Monte Carlo simulations of dense polymer melts, developed in the spirit of Verdier-Stockmayer algorithm on square and simple cubic lattices, is presented. By introducing diagonals of squares and cubes as bonds, the lattice model acquires a large number of configurations and wiggling local moves. While it maintains the excluded Volume interactions of monomers, it allows bond crossings and phantom moves, which result in a high mobility of polymers. For an application, we carry out simulations of symmetric A-B block copolymer melts and observe a first-order transition. We also show the stretching of the chains, namely, the non-Gaussian character, as a function of temperature. A quicker evolution towards thermal equilibrium enables us to form an ordered tricontinuous double-diamond (OTDD) phase for linear A-B-C triblock copolymers and a new cylindrical phase for star A-B-C triblock copolymers. (C) 1996 American Institute of Physics.

We show that in terms of phason flips collective motions of atoms in quasicrystalline tilings are possible: atomic flows and large deformations. Approximants or large density fluctuations are associated with the atomic flows that cause plastic deformations. In this scheme no dislocations are needed. We view random tilings as metallic rubbers exhibiting plast-elasticity in analogy with the visco-elasticity of polymeric materials.

In this paper, we present results of Monte Carlo simulations of icosahedral quasicrystals, which investigate the structural change from a Penrose-like phase to a random tiling phase. We demonstrate a finite temperature transition (k(B) T(c) = 1.52 +/- 0.05 in units of mismatch energies), which implies that equilibrium quasicrystalline materials can be energetically stabilized at finite temperature and can undergo a transition to a random tiling phase at high temperature. The structural change characterized by ''sheet'' statistics and ''sheet magnetization'' is illustrated. Dynamic phasons are locked below T(c).

Using Monte Carlo computer simulations, we find strong evidence for a novel, finite-temperature, Ising-like transition in the alignment of unit cells in icosahedral quasicrystals. The Ising-like transition may be related to the transition in phason behavior from a locked (Penrose-tiling-like) phase at low temperatures to an unlocked (random-tiling-like) phase at high temperatures.

We illustrate that locking/unlocking transitions between locked (Penrose tiling-like) and unlocked (random tiling-like) phason dynamics for D-dimensional quasicrystals closely relate to the order/disorder transitions for (D-1)-dimensional Ising models. Namely, a hidden Ising-like symmetry in quasicrystals is found. The transition is observed in the ordering/disordering in worms (sheets) for 2D (3D) quasicrystals. In this paper, a finite-temperature phase transition with the 2D Ising critical exponents in 3D icosahedral quasicrystals is demonstrated using Monte Carlo computer simulations. We also propose a new type of quasicrystals called "absolutely locked quasicrystals" in which uniform phason shifts are prohibited.

Keywords: Penrose Tilings, Icosahedral Quasicrystals, Phase Transformation, Locked Phason Dynamics, Ising Models, Computer Simulation

Dynamical systems in SL(2, R) or SL(2, C) naturally appear in the transfer matrix method for quasiperiodic chains characterized by arbitrary irrational numbers. We show new sub-dynamical systems and invariants that are related to full diagonal and off-diagonal components of the transfer matrices; they are analogous to formulae of Chebyshev polynomials of the first and second kinds. Applying them to an electronic problem on the Fibonacci chain, we obtain sets of self-similar polynomials, quasiperiodic extension of the Chebyshev polynomials of the first and second kinds with self-similar properties. Two scaling factors of the self-similarities coincide with ones obtained by the perturbative decimation renormalization group method.

フィボナッチ準結晶の固有値問題における６サイクルと呼ばれる自己相似な波動関数を自己相似多項式を応用して計算した。

フィボナッチ準結晶の自己相似性から得られるエネルギースペクトルの自己相似構造を直観的に説明し、自己相似多項式の物理的意味を議論した。また、準周期性が弱い場合にブラッグ反射、力学系のアノソフ系、チェビシェフの多項式との関連を議論した。

Distribution of zeros of partition function Z without magnetic field is studied for some two-dimensional Ising models with nearest-neighbor interactions. The distributions are presented graphically for the honeycomb, triangular, diced and Kagome lattices. It is shown that an asymptotic form of high temperature expansion for lnZ is closely related with the distribution of zeros. The expansion coefficients are derived up to large orders by computer for the honeycomb and Kagome lattices. It turns out that their oscillatory behaviors are understood very well by studying the zeros off the positive real axis, in particular the period of oscillation for the Kagome lattice is proved to be about 5.25.

We present some results from an ongoing analysis of "decapod defects" in Penrose tilings. This class of topological defects, which appears to be unique to quasiperiodic systems, may play an important role in the rapid growth of large, near-perfect quasicrystals and in the phase transformation from quasicrystals to crystals.

The critical compressibility factor Zc at the gas-liquid critical point is defined by Zc=pcVc/NkBTc (pc: critical pressure, Vc: critical volume, Tc: critical temperature, kB: Boltzmann's constant, N: number of molecules). The Zc of the lattice gas on the Kagome or diced lattice is discussed. First, on the basis of exact solution for the partition function Z of the Ising model without magnetic field on these lattices, the Zc is calculated exactly. The results are Zc=0.08330842... (Kagome lattice) and Zc=0.14284554... (diced lattice). Secondly, high temperature expansion for Zc is studied and the expansion is carried out up to w13 for the Kagome lattice and up to w12 for the diced lattice. Extrapolation procedures for obtaining Zc are discussed. A similarity between the diced and Penrose lattices and the one between the Kagome and dual Penrose lattices are pointed out.

High temperature expansion for ln Z(Z : the partition function in the absence of a magnetic field) of the Ising model on the dual Penrose lattice is discussed. The terms up to w10 are derived by using projection method. The critical compressibility factor Zc and the average correlation function C are calculated in terms of duality relation. The final results are Zc = 0.0846 +-0.007 and C= 0.7403+-0.0024, which are confirmed by the expansion. Lattice constants are shown in some tables.

Keywords: critical phenomena, critical compressibility factor, dual Penrose lattice, duality, high temperature expansion, Ising model, lattice constant, projection method, theory

High temperature expansion for ln Z (Z: the partition function in the absence of magnetic field) of the Ising model on the Penrose lattice is discussed. The terms up to the order of w8 are derived. To illustrate an extrapolation procedure employed here, the critical compressibility factor Zc and the correlation function C at Tc, for the neighboring spin pair are first treated in the case of two-dimensional square lattice. It turns out that the terms up to w8 for ln Z lead to the results within the error of 0.2 ~ 0.30% as compared to exact values. Along the same line, Zc and C (average correlation function) are calculated for the Penrose lattice. The final results are Zc=0. 138+-0.002 and C=0.673+-0.003.

Keywords: high temperature expansion, Ising model, Penrose lattice, critical phenomena, theory

We present new polynomials with self-similar properties, which are obtained from the Fibonacci-chain model. The crystalline analogs are the Chebyshev polynomials of the first kind. The polynomials are akin to the fixed points of a renormalization-group equation. The structure of the zeros of the polynomials forms a tree, which we call the Fibonacci tree. Using this tree, we discuss the electronic spectrum of tight-binding models.

フィボナッチ準結晶の電子状態を繰り込み群によって解析し、電子エネルギースペクトルの自己相似構造を議論した。

熱力学の基礎から量子統計などの基礎的な統計力学を経て相転移などの応用までを扱う．理解促進のための小見出しと，金属・高分子・ソフトマター・環境関連問題ほか，多くの具体的な例題と図を用いて解説．

（全ページ）p.181 全15章（分 担）1-7,15章 （概 要）統計熱力学を講述した。 1章「熱と温度」、2章「分子の熱運動」、3章「マクスウェルの速度分布」、4章「統計力学の考え方」、5章「マクスウェル－ポルツマン分布と応用」、6章「相転移」、7章「量子統計」、15章「熱伝導」

（全ページ）p.265 全15章（分担執筆）堂寺知成 3,4,5章 （概 要）大学院講義「複雑システム科学」を概説した。3章「準結晶：複雑さの中の美しさ」4章「複雑分子：秩序構造の建築家」、5章「共連続相：ミクロ世界のラビリンス」

全ページ）p.450 （共著者）高野敦志, 堂寺知成 （分担執筆）p.155-160 （概 要）『自己組織化により形成される複合高分子系の凝集構造形成シミュレーション』

（全ページ）p.450 （共著者）基礎高分子科学編集委員会 （分担執筆）堂寺知成 p.69 （概 要）『計算機で描く溶液中の高分子ブラシの姿』

（全ページ）p.906（共著者）小川泰 他編集 （分担執筆）堂寺知成 p.380-382 （概 要）『高分子のつくる空間分割』

（全ページ）p.310 全15章（分担執筆）堂寺知成 8章 pp.138-149 （概 要）計算科学のもたらすパラダイムシフトを概説した。8章「確率とモンテカルロ計算」

（全ページ）p.181 全15章（分 担）1～6章 （概 要）熱統計力学を講述した。1章「各種のエネルギーとエネルギー変換」、2章「温度と熱」、3章「熱力学第一法則」、4章「熱機関」、5章「熱力学第二法則」、6章「熱伝導」

（全ページ）p.217 全15章（分 担）3,4,7,8,11章 （概 要）力学を講述した。 2章「運動の法則」、4章「非線形力学」、7章「計算機の応用」、8章「流体の運動」、9章「構造力学」

（全ページ）p.181 全15章（分 担）3,4,7,8,11章 （概 要）熱統計力学を講述した。 3章「温度と熱」、4章「熱力学第一法則」、7章「分子の熱運動」、8章「マクスウェルの速度分布則」、11章「熱とエントロピー」

Kazunari UEDA, Student Award

（概 要）ABC星型ブロック共重合体のミクロ相分離という自己組織化現象によってできる２次元アルキメデスタイリング構造を紹介した。その1つは複雑な合金相として知られるFrank-Kasper相（σ相）と類似な構造で、12回対称準結晶の近似結晶になっている。 同誌表紙

ABC星型ブロック共重合体の格子高分子モデルのモンテカルロ・シミュレーションによって,12回対称準結晶(近似結晶)の形成に成功した。以下の特徴が観察された。(1)フーリエ変換(構造関数)はほぼ12回対称性を示す。(2)Stampfli変換による自己相似性が観察される。(3)ランダムに形態変化や合体分離を繰り返しながらもグローバルな12回対称性は保存する。(4)(3)はタイリングの入れ替えに対応する準結晶特有のフェイブン.ダイナミクスとみなせる。(5)よく知られた12回対称性正三角形-正方形タイリングよりもランダウ理論から導かれる密度彼の構造と一致する。最近,合金のFrank-Kasper相(σ相)に関連した80nmの辺の長さを待つ正三角形,正方形からなる3^2434アルキメデスタイリング構造が実験的に発見されているが,この構造は常に準結晶相に隣接して現れる構造である。このことと合わせ,高分子準結晶の発見が期待される。

平面にビリヤード球を並べると、１つの球のまわりに６つの球が並ぶ。この構造は電子、原子から始まり、コロイド球、界面活性剤や高分子が作る構造に至るまで普遍的な規則構造として知られている。球状ウィルスのように、正曲率曲面である球面上の規則構造もよく知られている。しかし、ポテトチップのような馬の鞍型（負曲率）曲面上の物理的規則構造の研究はなされていなかった。本研究ではシュワルツのダイヤモンド及びプリミティブ３重周期極小曲面上で球のシミュレーションを行い、エントロピーを駆動力としたアルダー相転移を観察することによって、負曲率曲面上の規則構造を多数発見し、それらを結晶学、数学的観点から解析した。

結晶形成には引力は必要なく,剛体球ポテンシャル(斥力)だけで結晶形成するという Alderらの発見は 1957年の発見当時大きな驚きをもたらした。本研究によって,剛体球の外側にステップ状の斥力を付加した単純なハードコア -ソフトシェル模型で, 10回, 12回, 18回対称のさまざまな準結晶が生成されることがわかった。この結果はスケールや物質の特殊性に依存せず,一般的メカニズムで準結晶が生成されることを示した重要な成果で,準結晶研究および物質科学に大きな進展をもたらした。

研究題目「多元高分子配列ナノ空間物質の構造設計と物性理論」の研究目的は、多元高分子による全く新しい配列ナノ空間構造を設計構築すること、およびその新規物性の理論研究を行うことであった。特にコロイド結晶では作成できない充填密度が低いダイヤモンド構造創製の成果を世界に発信し、物性発現として未来のデバイス、フォトニック結晶への応用可能性を研究することであった。 平成21年度は、新しく発見した階層シャイロイド構造の構造解析を進め、実施計画に基づき研究成果の公表準備をした。また、フォトニックバンド計算の精度を高めるべくコンパクト差分法を導入したFDTD法の開発を進めた。この研究には数値計算の専門家に援助を受け、共同研究に発展しつつある。方法論としての有効性を検証しているところである。次に、高分子の作るシングルジャイロイド構造の螺旋軸に注目し、コレステリック液晶あるいは甲虫などに見られる左右螺旋偏光の反射率が異なるという現象が、多元高分子配列ナノ空間物質でも生ずるという仮説を立て、数値実験で確かめた。この研究は継続する予定である。 まとめると当該研究期間で、多元高分子による全く新しい配列ナノ空間構造を設計し、自己組織化を用いて2つの新規な構造(巨大ジンクブレンド構造、階層ジャイロイド構造)を構築することに成功した。これらの研究は物質科学、物理および高分子化学の進展に貢献し、学術的にも意義の高いもので、結晶学、数学への関連も重要だと考えている。また理論的には多元高分子のフォトニック結晶への応用可能性を広げた点、これまで議論されなかった偏光実験を提案するなど新しい物性発現の予言を行った点で当初の目標を達成したと言える。

古典結晶学で許されない回転対称性を持つ「準結晶」の発見は20世紀後半の物質科学上の大発見の1つであるが、その発見は合金系であり、高分子系では準結晶のような複雑な構造は発見されていなかった。本研究計画において研究代表者ははじめて高分子準結晶を発見し、さらに理論研究を推進することによって、金属(ハードマター)だけでなく高分子(ソフトマター)にも準結晶が存在することを明らかにし、準結晶構造の普遍性を示した。物理と化学の両業界から大きな反響を呼んだ。

1 金属学の常識として、大変形には転位の介在が必要とされる。しかし、本研究の過程において、転位に関わりなく、準結晶特有のフェイゾン欠陥のふるまいそのものが、大変形に本質的役割を果たしうることを発見した。従って、研究の焦点を、準結晶中の転位から、フェイゾン欠陥のふるまいに軌道修正した。 2 過去にわれわれが示したように、準結晶は高温で、ランダムタイリングと呼ばれる準周期構造に転移しうる。この相では、フェイゾン欠陥が、熱的に激しく励起され、フェイゾンフリップにより、原子移動が起こる。最近、いくつかの実験的研究がフェイゾンフリップによる原子運動の存在を示唆している。また、理論的にこの原子移動によって、原子が準結晶中を自己拡散することが、国外の研究者によって示された。 3 我々は本研究に於いて、この原子移動は、自己拡散だけでなく、集団運動を引き起こしうることを主張した。すなわち、原子の流れと大規模変形が起こりうる。今まで、ランダムタイリングについては、フェイゾン空間での特徴が詳しく研究されてきたが、我々は新たに実空間での運動に注目し、集団原子運動という極めてランダムタイリングに特徴的な性質を指摘した。実際には、(補助金を受けた設備品を利用した)コンピューターシミュレーションによって、簡単なタイリングモデルで、その動きを観察した。 4 この研究成果の意味するところは、高温での塑性変形の由来を、単に転位だけと考えるのではなく、準周期構造の特徴的性質が関与しているという新たな視点をもたらすことである。研究結果は、第5回準結晶国際会議(フランス、1995年5月)で報告される。