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Toward the discovery of new soft quasicrystals:From a numerical study viewpoint

TOMONARI DOTERA

Department of Physics, Kinki University, Higashi-Osaka 577-8502, Japan

Dated: November 2, 2011

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Table of contents: The 2011 Nobel Prize for Chemistry was awarded for

“the discovery of quasicrystals” - crystals that are ordered without repeating

unit cells, and possess ‘forbidden’ symmetries. Such crystals have potential

photonic applications since they can be engineered with interesting optical

properties and periodic bandgaps. In this Review, the rational design and

realization of‘ soft’- including polymer, colloid, and liquid crystalline - qua-

sicrystals is examined. Approaches to finding new quasicrystals is a particular

focus.

ABSTRACT: This is a progress review of an emerging research front: soft

quasicrystals including chalcogenide, liquid crystalline dendrons, nanoparti-

cles, mesoporous silica, colloids, ABC star and linear terpolymers, and even

water and silicon. As aids to readers, we explain the basics of quasicrystals de-

veloped in solid-state physics: orders in quasicrystals, higher dimensional crys-

tallography, 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 lengthscale is the key

to discover new quasicrystals. As a case study, we describe our recent stud-

ies on ABC star terpolymer systems and present the results of simulations of

dodecagonal polymeric quasicrystals. In the case of dodecagonal quasicrys-

tals, 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.

Keywords: quasicrystal, soft matter, block copolymer, self-assembly, simu-

lation, photonic crystal.

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INTRODUCTION

In the 19th century, 17 plane groups or 230 space groups were determined based on

periodicity, then the “classical” crystallography, in which the allowed rotationally symmetry

is only twofold, threefold, fourfold or sixfold symmetry, was perfectly completed. In 1980s,

however, a forbidden five-fold electron diffraction pattern was found by Shechtman et al. in

rapidly quenched aluminum manganese alloys1, which created a enormous sensation in solid-

state physics. The Shechtman’s paper, which was regarded totally absurd in the textbook

common sense of crystallography, had not been published for two years. According to him,

he was strongly attacked by Nobel Laureate Linus Pauling even after the publication, and

confessed that he was isolated in his colleagues.

In spite of the strong objection in chemistry, a new research field arose in physics soon

after the discovery, and the concept of “quasicrystals” proposed by Levine and Steinhardt

was established as a new class of ordered state with both non-crystallographic rotational

symmetry and quasiperiodic translational symmetry2,3. Non-crystallographic symmetry re-

ported so far is icosahedral, decagonal4, octagonal5, or dodecagonal symmetry6,7. In metallic

physics in particular, the thermally stable icosahedral phases has been the central theme to be

investigated8. Tsai et al. had the notable success of systematic discoveries of stable phases

(Al63Cu25Fe12, Al70Pd20Mn10, Zn60Mg30Y10, Cd85Yb15) in terms of the Hume-Rothery rule9,

and the first structural determination of a quasicrystalline system in terms of the higher-

dimensional crystallography was achieved by Takakura et al. 10, hence the understanding

of quasicrystalline alloys is now thought to be matured. Quasicrystals are now regarded as

thermally stable structures with sharp Bragg peaks distinct from non-equilibrium states such

as glassy or amorphous materials. In fact, in 1991 the International Union of Crystallogra-

phy (IUCr) decided to redefine the term “crystal” to mean “Any solid having an essentially

discrete diffraction diagram.” Accordingly, both periodic and “aperiodic” crystals are said

to be crystals.

Meanwhile, the revolution of crystallography had not been recognized, as it seemed, in

chemistry for two decades. However, as an unprecedented streamline a new frontier “soft

quasicrystal” appeared in the twenty-first century chemistry11. More precisely, a sudden ex-

plosion of the quasicrystal world occurred in “dodecagonal” quasicrystals and related phases

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called “approximants,” which covered diverse kinds of materials: chalcogenide12,13, liquid

crystalline organic dendrons14, nanoparticles15, mesoporous silica16,17, colloids18, ABC star

polymers (Fig. 1)19,20, and ABC linear terpolymers21. We now realize that the study of soft

quasicrystals is a fundamental issue to be explored in chemistry domain including colloidal

and polymer chemistry. In addition, as for the application of soft quasicrystals, photonic

quasicrystals attract growing interests as bottom-up technology. Furthermore, there are

studies of artificial quasiperiodic structures such as quasiperiodic potentials generated by

five laser beams22,23, and optical trapped quasiperiodic colloidal particles.24

This review is organized as follows. For the first half, we provide the fundamentals of

quasicrystals, some concepts and tools for novice readers. To understand quasicrystal, we

first introduce orders in quasicrystals: quasiperiodic translational order, non-crystallographic

rotational order and self-similarity. Second, the most useful tool, higher-dimensional method

is presented. Third, randomness in quasicrystals concerning the origin of quasicrystals is

described. Then we focus our attention on the past numerical studies in statistical physics,

which have elucidated the formation mechanism of quasicrystals, and find points at issues

in order to discover designing principles of new soft quasicrystals in synthetic chemistry.

Surprisingly, one finds that there is a common principle beyond one specific research area.

Then, we discuss Archimedean tiling phases and our simulations of polymer quasicrystals

in details as a case study. In the end, we review photonic quasicrystals as the application

of soft quasicrystals. Finally, summary and future outlook are given. Since dodecagonal

quasicrystals were found mostly in soft quasicrystals, we adopt dodecagonal examples in

several explanations.

FUNDAMENTALS OF QUASICRYSTALS

Orders in Quasicrystals

Two types of long-ranged order needed to form quasicrystals are “quasiperiodic transla-

tional order” and “non-crystallographic rotational order.”2,3 The former causes Bragg peaks

and the latter represents non-crystallographic rotational symmetry. It should be noticed

that Bragg peaks do not mean periodicity, but only mean translational order. The term

“quasiperiodic” means translational order with two or more incommensurate length-scales

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in one direction, for instance, a quasiperiodic function:

f(x) = cos(x) + cos

(√6 +

√2

2x

), (1)

which gives two incommensurate δ-functions (Bragg peaks) in the reciprocal space, although

it is nonperiodic (aperiodic). Hence the term quasiperiodic does not always mean non-

crystallographic rotational symmetry. Such quasiperiodic structures are observed in modu-

lated phases and incommensurate composites.25

We also note that the center of N -fold symmetry for the entire space is not necessary,

although there are usually local clusters with N -fold symmetry. Exactly speaking, N -fold

non-crystallographic rotational symmetry exists in the reciprocal space, which is observed

in scattering experiments. Accordingly, the quasicrystalline structure can be generally de-

scribed by

ρ(r) =D∑

i=1

cos(ki · r + φi), (2)

where D is larger than space dimensions, which property is specific to quasicrystals, φi

are phases, and ki are reciprocal vectors that points to the vertices of an icosahedron, an

octagon, a decagon, or a dodecagon representing non-crystallographic rotational symmetry.

Non-crystallographic rotational angles at the same time induce quasiperiodic translational

order in each direction.

Finally let us mention the relation between algebraic numbers (Pisot numbers) and qua-

sicrystals. There are special quadratic equations appearing in quasicrystal-related geometric

objects:

x2 − x − 1 = 0, x = (1 +√

5)/2 = 1.618...,

x2 − 2x − 1 = 0, x = 1 +√

2 = 2.414...,

x2 + 4x − 1 = 0, x = 2 +√

3 = 3.732....

These numbers x are solutions of the above equations, which are called golden, silver, and

platinum mean respectively, related to the ratios of self-similarity (scaling) transformations

of pentagonal (or decagonal / icosahedral), octagonal and dodecagonal quasicrystals. Some

ideal quasicrystals have self-similarity in addition to the above two orders.

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In the case of dodecagonal symmetry, the algebraic number is related to the self-similar

transformation called Stampfli inflation, which is sn+1

tn+1

=

7 3

16 7

sn

tn

, (3)

where sn and tn are the numbers of squares and triangles for the nth generation of tilings26.

Figure 2 displays an example of the self-similar transformation. The eigenvalue of the matrix

in eq.(3) is the square of 2 +√

3, whose value is the scaling ratio of Figure 2. Another ratio

of tn/sn tends to 4/√

3 = 2.309401... as n goes to infinity. When one obtains square-triangle

tilings in experiments or simulations, this ratio should be examined as a quality index of the

tilings.

Higher Dimensional Crystallography

The state of art quasi-crystallography is called “higher-dimensional crystallography.”3,25

Why are higher dimensions needed? Why is higher-dimensional crystallography useful? Be-

cause there are several reasons: (i) Construction of quasicrystals; (ii) Calculation of scattering

intensities; (iii) Space group classification; (iv) Approximants.

(i) Higher-dimensional method is convenient to construct quasicrystals. To see this in-

tuitively, we look at a one-dimensional example, the Fibonacci lattice, which is known as

the one-dimensional analogue of decagonal and icosahedral quasicrystals. As shown in Fig-

ure 3, the Fibonacci lattice is obtained as the projection of the square-lattice points inside

the window between broken and solid lines whose tangent is the inverse of the golden ratio

[(√

5 − 1)/2 = 0.618...]. The width of the window is that of a square indicated by thick

solid lines. The number ratio of two segments A and B indicated on the solid line is the

golden ratio implying that there is no periodicity, as the ratio should be rational if a unit

cell exists. In the terminology of higher-dimensional crystallography, the space along the

Fibonacci lattice is called the physical space, and the space normal to the physical space is

called the perpendicular (internal) space.

Similarly, for the most famous quasicrystal, the Penrose lattice, lattice points r can

be represented by Z-module, meaning linear combinations of four basis vectors ei =

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(cos[2πi/5], sin[2πi/5]) with integer coefficients:

r =3∑

i=0

niei, (4)

where ni are integers. There is no need to employ five basis vectors because the fifth one

is dependant:∑4

i=0 ei = 0. As a set of integers (n0, n1, n2, n3) is regarded as a lattice

point in four dimensions, the Penrose lattice can be lifted up to the four-dimensional cubic

lattice. This observation in turn implies that two-dimensional quasicrystals are obtained

as the projection from four-dimensional lattices. The projected points are restricted by a

window function in the perpendicular space as shown in Figure 3. In the same way, the

icosahedral quasicrystal is considered as the projection of the six-dimensional simple cubic

lattice points inside a window. Depending on the size, shape, and symmetry of a window,

different quasicrystals are generated. Once a window is fixed, the structure obtained from

the window is determined and considered as a perfectly ordered structure.

Without getting into details, we list additional reasons. (ii) The Fourier transform of

a quasicrystal and its diffraction pattern are projections of the corresponding quantities in

higher dimensions. Since the relation between the edge of tiles and the Bragg peaks is not

straightforward, the higher-dimensional crystallography is indispensable. For instance, for

dodecagonal quasicrystals, the relation between looks complex:

l =2π

q

√2 +

√3

3(5)

where q is the magnitude of the prominent scattering vectors, and l is the edge length of trian-

gles and squares27. (iii) The space groups of quasicrystals are classified as higher-dimensional

groups25. (iv) Finally, it is important to notice that higher-dimensional crystallography can

be applied to traditional crystals. Namely, when the tangent is rational, crystals are ob-

tained. In particular, the most important application of this idea is that related structures

with similar alloy compositions are considered as rational approximations of quasicrystals

called “approximants.” For approximants, analyses and theories for crystalline structures can

be applied. The study of approximants has been hence very sound approach to comprehend

the structure and the physical properties of quasicrystalline materials. Moreover, the search

of approximants is the key to discover new quasicrystals, as we shall later see.

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Disorder in Quasicrystals and the Origin of Quasicrystals

At the moment of discoveries, the detailed analyses of randomness in soft quasicrystals

have not been pursued. However, randomness in quasicrystals is an astonishing feature of

quasicrystals. For future study, we briefly describe the randomness in quasicrystals.

In quasiperiodic structures the rearrangement of tiles is possible as shown in Figure 4(a)

in the case of Penrose rhombus tiles28. The corresponding tiling rearrangement process in

dodecagonal tiling is illustrated in Figure 4(b)29,30. Since the energetic difference is thought

to be small, it is thermally activated. This mode is impossible for crystals where the unit

cell is single, while it is possible for quasicrystals where unit cells are multiple. This mode

is called phason mode (phason flip), while elastic distortion of the lattice is called phonon

mode31,32. As shown in Figure 5, the phonon mode is considered as move along the physical

space, while the phason mode is considered as move along the perpendicular space. In fact,

this kind of disorder has been observed by various experimental methods33–37.

Quasicrystals occupy the position between crystalline and amorphous materials (glasses).

Therefore, it is natural to raise a question: Are quasicrystals in random states? The answer is

“No” for perfect quasicrystals, and “YES” for random quasicrystals. This has been debated

as a fundamental question. Remarkably, non-crystallographic symmetry can be generated

from randomized states. The bond directions in tilings that are obtained after successive

rearrangement of tiles is the same as those of perfectly ordered quasiperiodic tilings that

are composed of the same tiles. Furthermore, randomized tiling does not lose its non-

crystallographic symmetry, because the maximized entropic state is the highest rotationally

symmetric state, where all N -fold Bragg peaks should have the same intensity. This idea is

called “random tiling hypothesis.”29,38–40

With respect to the origin of quasicrystal formation, there are two arguments based on

energy E and entropy S. Let us consider the Helmholtz free energy

F = E − TS, (6)

where T is temperature. To lower F , there are two ways: Making perfect quasicrystals by

lowering energy E at low temperatures; Making random quasicrystals by maximizing entropy

S at high temperatures.

To describe disorder in a quasicrystalline structure, let us discuss the density picture

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again:

ρ(r) =D∑

i=1

cos(k‖i · r + φi(r)). (7)

Disorder is represented through phases φi(r). The phase is described by

φi(r) = u(r) · k‖i + v(r) · k⊥

i , (8)

where u(r) is the usual phonon variable, and v(r) is the phason variable.

As shown in Figure 5a, u(r) and v(r) describe disorder in the physical and perpendicular

spaces, respectively. Figure 5b displays wave vectors for dodecagonal quasicrystals. Notice

that the definition of wave vectors for the perpendicular space is different from that for the

physical space: this corresponds to the fact that the rotations that make the quasicrystal in-

variant have two different two-dimensional representations, under which phonon and phason

variables transform.

For perfect tilings, v(r) =const. The deviation form the correct tangent value is one type

of disorder in quasicrystals, which is represented by a phason strain tensor ∇hi(r‖). Here,

h(r‖) is a coarse-grained field of r⊥ around r‖

hi(r‖) =

∫dr‖k(r‖ − r′‖)r⊥i (r′‖), (9)

where a smearing kernel satisfying∫

dr‖k(r) = 1. When ∇hi(r‖) is finite for the entire

system, it is called linear phason strain: Approximants have finite values. The analysis of

the linear phason has been done in polymeric quasicrystal19.

There is another type of phason disorder called random phason. For random tilings, it is

assumed that an elastic free energy F from tiling configurational entropy is described by

F =K

2

∫dr‖

∑i

(∇hi(r‖))2, (10)

where K is the elastic constant and h(r‖) is a coarse grained field of r⊥ around r‖. We

assume that lowering the free energy enforces a quasicrystal to be organized. Notice that in

F no energetic term exists as in the case of rubber elasticity. For simplicity, we assume no

dependence of directions for the elastic constant.

Dimensionality of systems is one of important aspects. For two-dimensional quasicrystals,

the thermal average of phason fluctuations is estimated by

〈∆h2〉 ∼ 2π

βKln L + const., (11)

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where ∆h = h − 〈h〉, and L is the system size. Consequently, the phason fluctuation is

logarithmically divergent40, stemming from long wave-length fluctuations. Therefore, phason

variables cannot be restricted in a finite window. In other words, two-dimensional tiling

systems cannot have δ-function Bragg peaks and perfect quasiperiodicity in a strict sense. In

fact, for the Penrose tiling with a ground state by matching rules, a Monte Carlo simulation

showed that at any finite temperatures, the tiling is always in the random tiling state in

which phason fluctuations dominate41. It means that even locally perfect Penrose tilings are

globally random tiling when scaled up.

There is an experimental observation of a random tiling for a TPTC（p-terphenyl-3,5,3’,5’-

tetracarboxylic acid）molecular network system on graphite, which is mapped to a random

rhombus tiling with 60◦ and 120◦ angles (Fig. 6). Then the system can be viewed as a

two-dimensional fluctuating membrane in a simple cubic lattice. In this case, the phason

variable was just one dimension and turned out to diverge logarithmically42.

On the other hand, in three dimensions, the phason variables are confined to a finite

value, there exist three-dimensional quasicrystals having true Bragg peaks with diffuse wings

representing randomness. Bulk quasicrystals are of course three-dimensional and thus the

origin of quasicrystalline structures can be random tilings. It should be noticed that the

term “random tiling” in the field of quasicrystal physics does not simply mean a tiling with

randomness, but it usually means a random tiling subjected to the random tiling hypothesis.

Perfectly ordered tiling generated by a certain local energetic rules in three dimensions

can be a random tiling at high temperatures43–46. At low temperatures, long-wave phason

fluctuations are locked, therefore they cannot behave as the elastic modes. This situation is

similar to that of frozen polymers that cannot exhibit rubber elasticity. So far the transition

between perfect and random tilings have not been observed even in hard quasicrystals.

Recent discussion includes the relation between phason disorder and chemical disorder.

Furthermore, it is known that there is no long-range translational and thus no quasiperi-

odic translational order in two dimensions because of thermal fluctuations. Even if it exists,

the translational order becomes quasi-long range in two-dimensional crystals; however, it is

remarkable that bond-orientational order (BOO) can maintain long-range order. Moreover,

even though the translational order is lost, the BOO can still be quasi-long range. Usually

the latter bond-orientational ordered phase that exists between two-dimensional crystal and

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liquid phases is called “hexatic phase” observed in colloidal and polymeric systems47–52. In

the same spirit, the phase invariant under dodecagonal symmetry operations would be called

“dodecatic phase.”

FROM SIMPLE TO COMPLEX: TWO APPROACHES

In the past, icosahedral clusters have been discussed in simulation studies of metallic glass

using “monoatomic + simple” potentials. However, no quasicrystal formation extending to

the whole system has been reported53. To form quasicrystals, at least one of two approaches

was necessary: (I) multi-atomic + simple potentials; (II) monoatomic + complex potentials

(that has two length-scales).

The method (I) is to simulate multi-atomic simulations based on the fact that the qua-

sicrystal discoveries have been in alloys, in which the choice of atomic sizes are determined to

form local icosahedral, decagonal or dodecagonal clusters. The first example was appeared

just to investigate local stability54. Early works includes: Lancon and Billard conducting

molecular dynamic simulations55, Widom et al. performing Monte Carlo simulations to

form decagonal random tiling quasicrystals56, and Henley et al. conducting simulations for

dodecagonal quasicrystals57. The lesson we can study from this approach is that the key

to form quasicrystals is to construct local quasicrystal clusters by using multi-component

systems. The drawback in these simulations is that the positional change of different atomic

species is difficult and therefore to attain the equilibrium state becomes difficult. The same

situation may happen in soft materials, where time scale to achieve thermal equilibrium

often becomes long or impossible.

The method (II) is to add complexity for potentials. The first success is dodecago-

nal quasicrystal formation by the Dzugutov potential58, which has been used by Roth and

Gahler,59 and Keys and Glotzer et al.60 Several potentials were devised61,62, in particular,

recent progress was done by Engel and Trebin, in which Lenard-Jones-Gauss potentials:

V (r) =1

r12− 1

r6− ε exp

[−(r − r0)

2

2σ2

](12)

was devised. This adds a Gaussian valley in addition to that of usual Lenard-Jones poten-

tial. The key to form quasicrystals is to set the second valley at advantageous positions

for non-crystallographic configurations. 　 In addition, this potential avoids the triangular

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lattice. Engel and Trebin performed two-dimensional simulations, and obtained decagonal

and dodecagonal phases at high temperatures depending on the parameters ε and σ. The

merit of this approach is that the simulation cost is reduced because of monoatomic systems.

The idea is more radicalized by Rechtsman, Stillinger, and Torquato64. They thought the

optimization problem of a spherical potential that gives a target structure. According to the

method, even the diamond lattice could be constructed in terms of a monoatomic but consid-

erably complex spherical potential, which approach may hint in designing supra-molecules to

form complex lattices. In fact, designing complex molecules is promising approach to form

complex crystalline structures: Wiesner et al. obtained an A15 structure by using hybrid

dendritic polymers (dendrons) with eight arms.65,66

To materialize two-lengthscale, however, a feasible way is to synthesize coreshell particles

as shown in Figure 7(a). The simplest model for the core-shell particles is represented by

hard-core and soft-shell potential investigated by Jagla67 and Glaser et al. 68, which turned

out to form various phases. It is an open question whether of not the simple potential can

produce quasicrystals.

The length ratio r associated with dodecagonal (N = 12) or decagonal (N = 10) qua-

sicrystals is given by

r =1

2 sin(π/N)

=

√

6+√

22

= 1.932... (N = 12)√

5+12

= 1.618... (N = 10)

In Figure 7, two-length scales for dodecagonal quasicrystal are given in Figure 7(b), which

were employed by Engel and Trebin63, and Lifshitz et al. 69,73.

In view of the reciprocal space, mean field theories were devised in the same sprits that cor-

respond to (I) and (II). Alexander and MacTague theory48,70 of crystallization of monoatomic

systems has been extended to two- or multi-order parameter theory by Mermin and Troian

to understand the stability of quasicrystalline phases71, and the theory was further extended

to polymeric quasicrystals72. These approach corresponds to (I). On the contrary, corre-

sponding to (II), Lifshitz and Petrich directly used two wavelengths for one order parameter

to study Faraday waves73. The approach (I) is natural, while the approach (II) catches the

essence of phenomena.

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Other types of simulations such as a lattice polymer system74,75, a tetrahedral packing

system76, and a bilayer of water or silicon77,78 turned out to exhibit dodecagonal order

without any input of dodecagonal symmetry.

TILING BY POLYMERS

Archimedean tiling phases

In the following paragraphs, we describe how we found a dodecagonal polymeric qua-

sicrystal. Microphase separations of‘ block copolymers ’composed of chemically distinct

polymers linked together have provided magnificent crystalline morphologies such as lamel-

lar, co-continuous, cylindrical, and spherical phases79–82. We investigated the phase behavior

of ABC star block terpolymers consisting of chemically distinct three polymers linked at one

junction (Fig. 8(a)). When the molecular weights of three components are not much dif-

ferent, their melts can form two-dimensional tiling patterns, precisely, polygonal cylindrical

phases whose sections are the Archimedean tilings (Fig. 8(b)). Archimedean tilings depicted

by Kepler in Harmonices Mundi II (1619) are regular patterns of polygonal tessellation in

plane by using regular polygons83. Here, a set of integers (n1.n2.n3. · · · ) denotes a tiling of

a vertex type in the way that n1-gon, n2-gon, and n3-gon, · · · , meet consecutively on each

vertex. Superscripts are employed to abbreviate when possible.

If the interactions between ABC polymer components are equally strong, and only if one

molecular environment is allowed in each tiling, only (63), (4.82), and (4.6.12) (Fig. 8(c)-(e))

belonging to the single junction class (SJC) can be obtained as direct patterns, where each

polygon in the Archimedean tiling directly corresponds to each polymeric microdomain. It

is firstly because only three polygons corresponding to ABC microdomains should meet on

a vertex, and secondly because only even polygons should appear, the fact that is called

even polygon theorem85. Three phases were firstly confirmed by simulation85, then by

experiment86.

The first breakthrough was an experiment of an ABC star-shaped polymer alloy com-

posed of polyisoprene (I), polystyrene (S) and poly(2-vinylpyridine) (P) revealed a complex

Archimedean tiling phase (Figures 8(f) and 9) denoted as (32.4.3.4)20, or it is called σ phase

in the Frank-Kasper family84. The tiling is composed of equilateral triangles and squares,

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whose edge length is about 80 nm. The tiling is more complex than the SJC because molec-

ular environment splits; however, the skeleton structure is the (32.4.3.4) Archimedean tiling.

Since the (32.4.3.4) phase has the triangle/square ratio 2 and thus is akin to dodecagonal

quasicrystals, it was strongly suggested that dodecagonal quasicrystals were expected to

exist in polymeric systems.

Other study concerning tiling structures for supramolecules is found in Tschierske’s

review87.

Simulations of Polymeric Quasicrystals

In this subsection, we describe simulation study of polymeric quasicrystals motivated by

the aforementioned (32.4.3.4) experiment. A simple extension of the bead-and-bond lattice

polymer MC method called diagonal bond method was used85,88. A bead occupies only one

lattice point to ensure excluded volume interactions. The bond length can be 1,√

2, or√

3

in the unit of lattice spacing. One ABC star block copolymer consists of NA A-type beads,

NB B-type beads, NC C-type beads, and one Y-type bead (junction), which are connected

by N -1 bonds, where N=NA+NB+NC+1. To represent energetics that drives the system

to microphase separation, unit contact energies are imposed only between pairs of different

species within the body diagonal distance√

3: We consider the Hamiltonian as H =∑

εij,

where εij = 1 when i 6= j, and i and j stand for A, B, C. The MC procedure is the following:

We select one bead randomly and choose a trial move randomly out of possible moves; if the

trial is a vacancy, we determine move or not according to the Metropolis algorithm.

Here we chose NA=9, NB=7, and NC=12-18. In order to get broad two-dimensional struc-

tures, whole simulations were carried out in a quasi-two-dimensional box with Lx=Ly=128

and Lz=10 subjected to periodic boundary conditions. The C component of a few star

polymers can interact with themselves across Lz, leading to quick formation of cylinders

parallel to the z-direction. The number of polymers in the system was determined such that

the occupation ratio of beads in the lattice points was 0.75: 4237 (NC=12), 3964 (NC=14),

3724 (NC=16) or 3511 (NC=18). The system was prepared as totally randomized at the

infinite temperature, and then quenched at β = 1/kBT=0.071 to wait ordering, where kB is

the Boltzmann constant and T is absolute temperature. This temperature was selected low

15

enough to retain ordering processes, but high enough to get sufficient entropy. To form a

structure, we performed a run of 107 Monte Carlo steps (MCS) at β = 0.071.

In all simulations, cylindrical structures developed. We obtained a phase sequence:

(4.82) → (32.4.3.4) → DDQC → (4.6.12), (13)

with increasing C component. See Figure 10. In Figure 10(c) , we obtained a quasicrystal.

Results are summarized as follows: (i) The MC averaged structure function shown in the inset

of Figure 10(c) is almost 12-fold. (ii) We find local 12-fold wheel patterns in Figure 10(c).

The centers of the wheels form a Stampfli self-similar lattice. (iii) Cells are dynamically

rearranged and deformed, and consequently the wheels change their positions. This collective

dynamics can be viewed as a rearrangement of the square-triangle tiling. (iv) Contrary to the

usual square-triangle tiling, there is no six-fold node. Rather, the simulation resembles the

density wave pattern of C component obtained from a Landau theory72. A similar situation

occurred in the simulations of water and silicon bilayers77,78.

The phase behavior can be regarded as a transition from square tiling to triangle tiling via

square-triangle tiling. In other word, from fourfold to sixfold. See superimposed solid lines

in Figure 10 (a), (b) and (d). In Figure 10(a), C domains form square tiling as displayed

in solid lines, while C domains form triangle tiling in Figure 10(d). Between them, exact

five-fold (72◦) is desirable, but it is incompatible with crystalline structures. To compromise,

one remedy is to deform five-fold nodes. In fact, a little flattened five-fold node (60◦-60◦-

90◦-60◦-90◦) is the elemental structure in (32.4.3.4) in Figure 10(b). Another way is to

introduce new degrees of freedom producing additional entropy. In the case of A9B7C16,

we frequently find that neighboring C domains are dynamically connected and disconnected

(Fig. 11), meaning that the five-fold nodes and the wheel patterns are highly mobile. The

collective dynamics on a large scale can be viewed as a type of move given in Figure 4(b).

Concerning the thermodynamic effect of phason degrees of freedom, Edagawa and Kajiyama

have carefully measured an unusual increase in specific heat at high temperatures only in

quasicrystalline samples (Al-Pd-Mn and Al-Cu-Co); the authors have attributed its origin

to the phason modes89. We have compared 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.

16

APPLICATION: PHOTONIC CRYSTAL

The lattice constants of block copolymer and colloidal structures can reach the wave-

length of visible light. Hence, they may open the possibility of constructing novel photonic

band gap (PBG) devices, such as waveguides or dielectric mirrors, where the propagation of

electromagnetic waves or the spontaneous emission of light is forbidden90–94.

It is worthwhile to mention that dodecagonal quasicrystals are thought to be promising

candidates for PBG structures because of their high degree of rotational symmetry95–103. It

was suggested that a complete PBG opens with low dielectric contrasts in a dodecagonal

structure95. Furthermore, we mention a remarkable study by Zhang et al. that the nega-

tive refraction of electromagnetic waves and the superlens imaging were demonstrated103,104.

They have observed that the high-symmetric photonic quasicrystals can exhibit an effective

refractive index close to -1 in a certain frequency window. The index shows small spatial

dispersion, consistent with the nearly homogeneous geometry of the quasicrystal. Figure 12

displays that the incident wave from the left-hand side enters horizontally and the refractive

wave goes out upward of the surface normal.

Finally, we call attention that with high dielectric contrasts, band structures are deter-

mined by local structures, implying that long-ranged quasiperiodicity is not necessary. The

effect of randomness of quasicrystals may be problematic; if it is, the approximant structures,

which can be easily controlled in experiments, are useful. Using plane wave method, Ueda et

al. found that three types of the (32.4.3.4) structures open complete PBGs and that both

dielectric and air cylinders with the same shape have PBGs105.

SUMMARY AND OUTLOOK

Base on the result obtained by the polymer simulations, we searched polymeric quasicrys-

tals in the composition range between those of (32.4.3.4) and (4.6.12) phase, and finally found

as shown in Figure 1. Simple strategy exists that quasicrystals are found between the both

ends: triangular and square lattices. Then one need to look for the (32.4.3.4) phase or other

approximant phases106. The study of Frank-Kasper phases is still a sound approach even in

clathrate hydrates107. This scenario has repeated in metallic alloys, chalcogenide, dendrons,

nanoparticles, and mesoporous systems. As shown in simulations, a high temperature or

17

a full of solvent is needed to acquire molecular mobility necessary for phason dynamics, or

rapid quenching may provide another possibility to keep randomness.

We have seen that (i) solids can exist with 5-, 8-, 10-, 12-fold, or icosahedral symmetry. (ii)

δ-function Bragg peaks are for periodic and quasiperiodic structures. (iii) thermodynamically

stable solids can be crystalline and quasicrystalline. (iv) soft quasicrystals exist. It has

been shown that dodecagonal quasicrystals and related rational approximants cover a wide

variety of materials, demonstrating that two-dimensional quasicrystalline order is universal

over different length-scales: The edge lengths of polygons are 0.5 nm for metallic alloys, 2

nm for chalcogenide, 10 nm for liquid crystalline organic dendrons and mesoporous silica, 20

nm for binary nanoparticles, 50 nm for colloids, and 50-80 nm for ABC star terpolymers.

To encounter new complex phases, we have discussed two approaches: multi-components

or complex potential. Surprisingly enough, Hayashida et al. demonstrated a TEM image of

ABC star terpolymers and Talapin et al. 15 showed a TEM image of binary alloys of surfactant

capped metallic nanoparticles as shown in Figure 13. These system are alloy systems as hard

quasicrystals. To the contrary, one component spherical systems can form quasicrystals: This

direction is very new in soft quasicrystalline materials, and it should be further explored both

theoretically and experimentally: core-shell particles such as dendrons, copolymers, colloidal

particles with electric double layers108, surfactant or polymer coated metallic nano particles

or silica particles are possible candidates. Future study includes other quasicrystals such

as decagonal and icosahedral quasicrystals. A long search for the Boron quasicrystals109,

and the construction ideas of the zincblende lattices may be helpful110,111. Randomness in

known soft quasicrystalline phases should be investigated to make each formation mechanism

clear. In addition, the most recent mysterious discovery is the 18-fold diffraction pattern

from a colloidal system18,112 and the origin of the system should be clarified. Finally, the

experimental realization of a photonic soft quasicrystal is awaited.

ACKNOWLEDGMENTS

The author is grateful to Y. Matsushita, A. Takano, K. Hayashida, A. Hatano, T. Gemma,

J. Matsuzawa, K. Ueda, T. Oshiro, Y. Nakanishi, O. Terasaki, N. Fujita, Y. Sakamoto,

K. Edagawa, T. Ishimasa, A. P. Tsai, Y. Ishii, M. Matsumoto, P. J. Steinhardt, H. C. Jeong,

18

C. L. Henley, M. de Boissieu, and P. Ziherl for collaborations, several discussions and com-

munications on this topic. This work was supported by a Grant-in-Aid for Scientific Research

(C) (No.22540375) from JSPS, Japan.

1. Shechtman, D.; Blech, I.; Gratias, D.; Cahn, J. W. Phys. Rev. Lett. 1984, 53, 1951.

2. Levine D.; Steinhardt, P. J. Phys. Rev. Lett. 1984, 53, 2477.

3. Steinhardt P. J., Ostlund S. Eds. The Physics of Quasicrystals; World Scientific, Singapore,

1987.

4. Bendersky, L. Phys. Rev. Lett. 1985, 55, 1461.

5. Wang, N.; Chen, H.; Kuo, K. H.; Phys. Rev. Lett. 1987, 59, 1010.

6. Ishimasa, T.; Nissen, H.-U.; Fukano, Y. Phys. Rev. Lett. 1985, 55, 511.

7. Chen, H.; Li, D. X.; Kuo, K. H. Phys. Rev. Lett. 1988, 60, 1645.

8. Tsai, A. P.; Inoue, A.; Masumoto, T. Jpn. J. Appl. Phys. 1987, 26, L1505.

9. Tsai, A. P. Acc. Chem. Res. 2003, 36, 31.

10. Takakura, H.; Gomez, C. P.; Yamamoto, A.; de Boissieu, M.; Tsai, A. P. Nature Mater. 2007,

6, 58.

11. Lifshitz R.; Diamant, H. Phil. Mag. 2007, 87, 3021.

12. Conrad, M.; Krumeich, F.; Harbrecht, B. Angew. Chem. Int. Ed. 1998, 37, 1383.

13. Conrad M.; Harbrecht, B. Chem. Eur. J. 2002, 8, 3094.

14. Zeng, X.; Ungar, G.; Liu, Y.; Percec,V.; Dulcey, A. E.; Hobbs, J. K. Nature 2004, 428 157.

15. Talapin, D. V.; Shevchenko, E. V.; Bodnarchuk, M. I.; Ye, X.; Chen, J.; Murray, C. B. Nature

2009, 461, 964.

16. Xiao, C.; Miyasaka, K.; Fujita, N.; Sakamoto, Y.; Che S.; Terasaki, O. private communication.

17. Terasaki, O.; Ohsuna, T.; Liu, Z.; Sakamoto, Y.; Garcia-Bennett, A. E. Mesoporous Crystals

and Related Nano-Structured Materials; Terasaki, O. ed., Elsevier, Amsterdam, 2004.

18. Fischer, S.; Exner, A.; Zielske, K.; Perlich, J.; Deloudi, S.; Steurer, W.; Lindner, P.; Forster,

S. PNAS. 2011, 08, 1810.

19. Hayashida, K.; Dotera, T.; Takano, A.; Matsushita, Y. Phys. Rev. Lett. 2007, 98, 195502.

20. Takano, A.; Kawashima, W.; Noro, A.; Isono, Y.; Tanaka, N.; Dotera T.; Matsushita, Y. J.

19

Polym. Sci. Part B: Polym. Phys. 2005, 43, 2427.

21. Lee, S.; Bluemle, M. J.; Bates, F. S. Science 2010, 330, 351.

22. Mikhael, J.; Roth, J.; Helden, L.; Bechinger, C. Nature 2008, 454, 501.

23. Mikhael, J.; Schmiedeberg, M.; Rausch, S.; Roth, J.; Stark, H.; Bechinger, C. PNAS 2010,

107, 7214.

24. Roichman, Y.; Grier, D. G. Optics Express 2005, 13, 5434.

25. Janssen, T.; Chapuis, G.; de Boissieu, M. Aperiodic Crystals: From Modulated Phases to

Quasicrystals; Oxford Univ Press, Oxford, 2007.

26. Stampfli, P. Helv. Phys. Acta 1986, 59, 1260.

27. Yang, Q. B.; Wei, W. D. Phys. Rev. Lett. 1987, 58, 1020.

28. Elser, V. Phys. Rev. Lett. 1985, 54, 1730.

29. Oxborrow, M.; Henley, C. L. Phys. Rev. B 1993, 48, 6966.

30. Dotera, T.; Nakanishi, Y. to be published in Phys. Rev. B.

31. Socolar, J. E. S.; Lubensky, T. C.; Steinhardt, P. J. Phys. Rev. B 1986, 34, 3345.

32. Ishii, Y. Phys. Rev. B 1989, 39, 11862.

33. Edagawa, K.; Suzuki, K.; Takeuchi, S. Phys. Rev. Lett. 2000, 85, 1674.

34. Coddens, G.: Lyonnard, S.; Hennion, B. Phys. Rev. B 2000, 62, 6268.

35. Abe, E.; Pennycook, S. J.; Tsai, A. P. Nature 2003, 421, 347.

36. Francoual, S.; Livet, F.;de Boissieu, M.; Yakhou, F.; Bley, F.; Letoublon, A.; Caudron, R.;

Gastaldi, J. Phys. Rev. Lett. 2003, 91, 225501.

37. Abe, H.; Saitoh, H.; Ueno, T.; Nakao, H.; Matsuo, Y.; Ohshima, K.; Matsumoto, H. J. Phys.

Condens. Matter 2003, 15, 1665.

38. Kawamura, H. Prog. Theor. Phys. 1983, 70, 352.

39. Widom, M. Phys. Rev. Lett. 1993, 70, 2094.

40. Henley, C. L. In Quasicrystals: The State of the Art, Chap. 8; Steinhardt, P.J.; Divincenzo,

D. P., Eds.; World Scientific: Sigaplore, 1999; 2 edition.

41. Tang, L.-H.; Jaric, M. V. Phys. Rev. B 1990, 41, 4524.

42. Blunt M. O.; Russell, J. C.; Gimenez-Lopez, M.; Garrahan, J. P.; Lin, X.; Schroder, M.;

Champness, N. R.; Beton, P. H. Science 2008, 322, 1077.

43. Dotera, T.; Steinhardt, P. J. Phys. Rev. Lett. 1994, 72, 1670.

20

44. Dotera, T. Material Science Forum 1994, 150-151, 375;

45. Dotera, T. Mater. Sci. Eng. A 1994, 181-182, 758.

46. Jeong H.-C.; Steinhardt, P. J. Phys. Rev. B 1993, 48, 9394.

47. Nelson, D. R. Defects and Geometry in Condensed Matter Physics; Cambridge University

Press, Cambridge, 2002.

48. Chaikin P. M.; Lubensky, T. C. Principles of condensed matter physics; Cambridge University

Press, Cambridge, 1995.

49. Murray, C. A.; Sprenger, W. O.; Wenk, R. A. Phys. Rev. B 1990, 42, 688.

50. Marcus, A. H.; Rice, S. A. Phys. Rev. Lett. 1996, 77, 2577.

51. Zahn, K.; Lenke, R.; Maret, G. Phys. Rev. Lett. 1999, 82, 2721.

52. Segalman, R. A.; Hexemer, A.; Kramer E. J. Phys. Rev. Lett. 2003, 91, 196101.

53. Steinhardt, P. J.; Nelson, D. R.; Ronchetti, M. Phys. Rev. B 1983, 28, 784.

54. Levine D.; Steinhardt, P. J. Phys. Rev. B 1986, 34, 596.

55. Lancon, F.; Billard L.; Chaudhari, P. Europhys. Lett. 1986, 2, 625.

56. Widom, M.; Strandburg, K. J.; Swendsen, R. H. Phys. Rev. Lett. 1987, 58, 706.

57. Leung, P. W.; Henley, C. L.; Chester, G. V. Phys. Rev. B 1989, 39, 446.

58. Dzugutov, M. Phys. Rev. Lett. 1993, 70, 2924.

59. Roth J.; Gahler, F. Euro. Phys. J. B 1998, 6, 425.

60. Keys, A. S.; Glotzer, S. C. Phys. Rev. Lett. 2007, 99, 235503.

61. Quandt, A.; Teter, M. P. Phys. Rev. B 1999, 59, 8586.

62. Skibinsky, A.; Buldyrev, S. V.; Scala, A.; Havlin, S.; Stanley, H. E. Phys. Rev. E 1999, 60,

2664.

63. Engel, M.; Trebin, H.-R. Phys. Rev. Lett. 2007, 98, 225505.

64. Rechtsman, M.; Stillinger, F. H.; Torquato, S. Phys. Rev. E 2007, 75, 031403.

65. Cho, B.-K.; Jain, A.; Gruner, S. M.; Wiesner, U. Science 2004, 305, 1598.

66. Ziherl P.; Kamien, R. D. J. Phys. Chem. B 2001, 105, 10147.

67. Jagla, E. A. J. Chem. Phy. 1999, 110, 451.

68. Glaser, M. A.;Grason, G. M.; Kamien, R. D.; Kosmrlj, A.; Santangelo C. D.; Ziherl, P.

Europhys. Lett. 2007, 78, 46004.

69. Barkan, K.; Diamant, H.; Lifshitz, R. Phys. Rev. B 2011, 83 172201.

21

70. Alexander, S.; MacTague, J. Phys. Rev. Lett. 1974, 41, 702.

71. Mermin, N. D.; Troian, S. M. Phys. Rev. Lett. 1985, 54, 1524.

72. Dotera, T. Phil. Mag. 2007, 87, 3011.

73. Lifshitz, R.; Petrich, D. M. Phys. Rev. Lett. 1997, 79, 1261.

74. Dotera, T.; Gemma, T. Philos. Mag. 2006, 86, 1085.

75. Dotera, T. Philos. Mag. 2008, 88, 2245.

76. Haji-Akbari, A.; Engel, M.; Keys, A. S.; Zheng, X.; Petschek, R. G.; Palffy-Muhoray, P.;

Glotzer, S. C. Nature 2009, 462, 773.

77. Johnston, J. C.; Kastelowitz, N.; Molinero, V. J. Phys. Chem. Phys. 2010, 133, 154516.

78. Johnston, J. C.; Phippen, S.; Molinero, V. J. Phys. Chem. Lett. 2011, 2, 384.

79. Leibler, L. Macromolecules, 1980, 13, 1602.

80. Hamley, I. W. The Physics of Block Copolymer, Oxford Univ. Press, Oxford, 1998.

81. Bates, F. S. Fredrickson, G. H. Phys. Today, 1999, 52, 32.

82. Hadjichristidis, N.; Pipas, S.; Floudas, G. Block Copolymers: Synthetic Strategies, Physical

Properties, and Applications; Wiley, Inc. Hoboken, NJ, 2003.

83. Grunbaum, B.; Shephard, G. C. Tilings and Patterns; Freeman, NewYork, 1986.

84. Frank, F. C.; Kasper, J. S. Acta. Cryst. 1959, 12, 483.

85. Gemma, T.; Hatano, A.; Dotera, T. Macromolecules, 2002, 35, 3225.

86. Takano, A.;Wada, S.; Sato, S.;Araki, T.; Hirahara, K.; Kazama, T.; Kawahara, S.; Isono, Y.;

Ohno, A.; Tanaka, N. ; Matsushita, Y. Macromolecules 2004, 37, 9941.

87. Tschierske, C. Chem. Soc. Rev. 2007, 36, 1930.

88. Dotera T.; Hatano, A. J. Chem. Phys. 1996, 105, 8413.

89. Edagawa, K.; Kajiyama, K. Mater. Sci. Eng. A 2000, 294-296, 646.

90. Joannopolous, J. D.; Meade, R. D.; Winn, J. N. Photonic Crystals; Princeton University,

Princeton, 1995.

91. Sakoda, K.; Optical Properties of Photonic Crystals; Springer-Verlag, Berlin, 2001.

92. Steurer, W.; Sutter-Widmer, D. J. Phys. D: Appl. Phys. 2007, 40, R229.

93. Park, C.; Yoon, J.; Thomas, E. L. Polymer 2003, 44, 6725.

94. Fink, Y.; Urbas, A. M. Bawendi, M. G. Joannopoulos, J. D.; Thomas, E. L. J. Lightwave

Technol. 17, 1999, 1963.

22

95. Zoorob, M. E. Charlton, M. D. B.; Paker, G. J.; Baumberg, J. J. Netti, M. C. Nature (London)

2000, 404, 740.

96. David, S. Chelnokov, A. Lourtioz, J.-M. IEEE J. Quantum Electron. 37, 2001, 1427.

97. Hiett, B. P.; Beckett, D. H.; Cox, S. J.; Generowicz, J. M.; Molinari, M. Thomas, K. S. J.

Mater. Sci.: Mater. Electron 2003,14, 413.

98. Zhang, X.; Zhang, Z.-Q.; Chan, C. T. Phys. Rev. B 2001, 63, 081105(R).

99. Gauthier R. C.; Mnaymneh, K.; Opt. Express 2005, 13, 1985.

100. Wang, Y.; Wang, Y.; Feng, S.; Li, Z. Y. Europhys. Lett. 2006, 74, 49.

101. Notomi, M.; Suzuki, H.; Tamamura, T. Edagawa, K. Phys. Rev. Lett. 2004, 92, 123906.

102. Man, W.; Megens, M.; Steinhardt, P. J.; Chaikin, P. Nature (London) 2005, 436, 993.

103. Feng, Z; Zhang, X; Wang, Y; Li, Z.-Y.; Cheng, B; Zhang, D.-Z. Phys. Rev. Lett. 2005, 94,

247402.

104. Pendry, J. B.; Smith, and D. R. Scientific American 2006, 295, 60.

105. Ueda, K.; Dotera, T.; Gemma, T. Phys. Rev. B 2007, 75, 195122.

106. Iga, H.; Mihalkovic, M.; Ishimasa, T. Phil. Mag. 2011, 91, 2624.

107. Matsumoto, M.; Tanaka, H. J. Phys. Chem. B, 2011, 115, 8257.

108. Denton, A. R.; Lowen. H. Phys. Rev. Lett. 1998, 81, 469.

109. Miyazaki, Y.; Okada, J.; Abe, E.; Yokoyama, Y.; Kimura, K. J. Phys. Soc. Jpn. 2010, 79,

073601.

110. Kalsin, A.M.; Fialkowski, M.; Paszewski, M.; Smoukov, S. K. Science 2006, 312, 420.

111. Hayashida, K.; Takano, A.; Dotera, T.; Matsushita, Y. Macromolecules, 2008, 41, 6269.

112. Glotzer, S. C.; Engel, M. Nature 2011, 471, 309.

23

FIG. 1: Polymeric quasicrystalline square-triangle tiling by Hayashida et al19. Reproduced from

Ref.19, with permission from American Physical Society.

24

FIG. 2: Self-similar transformation of a dodecagonal square-triangle tiling.

25

A

A A

A

A

B

B

B

Physical space

Perpendicular space

FIG. 3: Fibonacci lattice points obtained as the projection of the square lattice inside the window

whose width is that of a square shown between broken and solid lines. A and B are arranged in a

Fibonacci sequence, ABAABABA... obtained by substitution rules: A→AB, B→A.

26

⇔

⇔

(a)

(b)

FIG. 4: Tile rearrangement process by phason modes: (a) Penrose rhombus tiling case, (b) do-

decagonal case.

27

x = r

x Physical space

Perpendicular space

Phasonfluctuation

QuasicrystalPhason v

4d

Phonon u

k1

k2

k3

k4

(a)

k2

k1

k3

k4

(b)

FIG. 5: (a) Phonon and phason. (b) Reciprocal vectors in physical and perpendicular spaces for

dodecagonal quasicrystals.

28

FIG. 6: Random tiling in a two-dimensional molecular network by Blunt et al.42. STM images,

molecular networks and the corresponding tiling representations. Reproduced with permission.

Copyright 2008, Science.

29

(a) (b)

FIG. 7: (a) Coreshell particles. (b) Two lengths whose ratio is (√

6+√

2)/2 = 1.932... in dodecago-

nal quasicrystals. The ratio is favorable to form (32.4.3.4) (σ) vertices.

30

FIG. 8: Archimedean tiling phases from ABC star polymers: (a) ABC star block terpolymer; (b)

and (c) (63), (d) (4.82), (e) (4.6.12), (f) (32.4.3.4) phases. The first three direct patterns constitute

the single junction class, whereas the (32.4.3.4) net gives the skeleton of the real structure.

31

FIG. 9: Transmission electron micrograph for an ISP star-shaped block terpolymer molecule,

I1.0S1.0P1.320. Archimedean tiling pattern, (32.4.3.4), is drawn as thin solid lines.

32

FIG. 10: Archimedean tiling phases (a), (b), (d) and a dodecagonal quasicrystal (c) for ABC

starblock copolymers: A (transparent), B (pink) and C (blue). (a) (4.82) phase for A9B7C12.

Squares connecting centers of C regions are superimposed. (b) (32.4.3.4) phase for A9B7C14. A

regular (32.4.3.4) graph is displayed. (c) Dodecagonal quasicrystal for A9B7C16. A two-periodic

unit cell made up of four replicas is displayed. Wheel patterns construct a magnified (32.4.3.4)

lattice superimposed (solid line), known as the Stampfli scaled lattice. (d) (4.6.12) phase for

A9B7C18. Triangle tiling is superimposed.

33

FIG. 11: Dynamic phason modes in the A9B7C16 system. Superimposed lines represent tiling

rearrangement from broken lines to solid lines.

34

FIG. 12: Negative refraction on a dodecagonal quasicrystal wedge103. Reproduced with permission.

Copyright 2005, American Physical Society.

35

FIG. 13: Dodecagonal quasicrystals self-assembled from binary nanoparticles by Talapin et al.15.

Reproduced with permission. Copyright 2009, Nature.