Self-assembly of nanorods on
quasicrystalline substrates
vorgelegt von
Diplom Physiker
Philipp Kählitz
Rüdersdorf
von der Fakultät II - Mathematik und Naturwissenschaften
der Technischen Universität Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
Dr. rer. nat.
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr. Martin Schoen
1. Gutachter: Prof. Dr. Holger Stark
2. Gutachter: PD Dr. Thomas Gruhn
Tag der wissenschaftlichen Aussprache: 20.12.2012
Berlin 2013
D 83
ii
Abstract
Quasicrystals possess long-range positional order with a non-crystallographic ro-
tational symmetry. Quasicrystalline order leads to interesting surface properties.
Adsorbates on quasicrystalline surfaces can form new self-assembled structures.
In this work we study by computer simulations the phase behavior and mobility
of hard rods in a quasicrystalline substrate potential. The quasicrystalline sub-
strate is derived from the interference pattern of ve laser beams and possesses
a decagonal rotational symmetry. We take two dierent particle models into ac-
count namely the hard needles and the hard spherocylinders. Hard needles are the
simplest form of elongated particles which form liquid crystalline phases. They
do not exhibit an excluded volume. To take excluded volume eects into account,
we also studied the hard spherocylinder model. In two dimensions the hard rods
are known to undergo a density driven phase transition from an isotropic phase to
a quasi-nematic phase. When the substrate potential is present, we nd dierent
phase behavior for two rod lengths. Short rods can connect only two minima of
the substrate potential. Long rods are able to connect many potential minima.
Under the inuence of the substrate, the short needles form disconnected clus-
ters located between two potential minima. The orientations of the clusters are
aligned with the symmetry directions of the potential. Through the formation of
clusters the quasi-nematic order gets destroyed. At high densities and high poten-
tial strengths a nematic order can be frozen in. Long needles also form clusters
under the inuence of the substrate potential. In contrast to the short needles the
clusters are not disconnected but are able to share a few minima. The clusters
form lines which are oriented along the symmetry directions of the potential. In
this way, a nematic phase can be stabilized. The distances between the lines fol-
low two interwoven Fibonacci sequences. At low densities the needles form small
regions of clusters oriented along dierent symmetry directions of the substrate.
This non-nematic decagonal phase can also be frozen in at high densities and high
potential strengths.
Similar to the short needles the short spherocylinders form disconnected clusters
under the inuence of the potential at low densities. Due to their nite width the
system becomes crowed with increasing density. The directional order decreases
iii
signicantly and nally the preferred directions of the clusters shift against the
symmetry directions of the potential at high densities. At suciently high poten-
tial strengths the long spherocylinders order themselves onto lines. The positional
order is very weak and gets lost at high densities. The long spherocylinders re-
main in a decagonal directional order even for high densities. We also investigate
the mobility of the spherocylinders with kinetic Monte Carlo simulations. With
increasing potential strength the short spherocylinders get trapped at their min-
imum positions. In more dense systems the mobility of the short rods rises. In
contrast to short spherocylinders the long spherocylinders can slide along the lines
connecting the potential minima. This results in a high mobility of the long rods
even at high potential strengths.
iv
Zusammenfassung
Quasikristalle besitzen eine langreichweitige Positionsordnung mit einer nicht kri-
tallograschen Rotationssymmmetrie. Die quasikristalline Ordnung führt zu in-
teressanten Eigenschaften ihrer Oberächen. In dieser Arbeit untersuchen wir
das Phasenverhalten und die Mobilität von harten Stäbchen auf einem quasicrys-
tallinen Substrat mittels Computersimulationsn. Das quasicrystalline Substrat
wird durch die Inteferenz von fünf Laser Strahlen erzeugt und besitzt eine dekag-
onale Rotationssymmetrie. Wir untersuchen zwei verschiedene Teilchenmodelle,
das der harten Nadeln und das der harten Spherozylinder. Die harten Nadeln
sind das simpelste Teilchenmodell welches üssigkristalline Phasen aufweist. Sie
besitzen aber kein Volumen. den Einuss eines Teilchenvolumens untersuchen wir
im Modell der harten Spherozylinder. In einem zweidimensionalen System ndet
unter Erhöhung der Dichte der Stäbchen ein Phasenübergang von der isotropen in
eine quasi-nematische Phase statt. Auf dem Substrat zeigen Stäbchen zweier ver-
schiedener Längenskalen unterschiedliches Phasenverhalten. Die kurzen Stäbchen
können nur zwei Minima des Substrates verbinden. Die langen Stäbchen können
mehrere Minima verbinden.
Unter dem Einuss des Substrates nden sich die kurzen Nadeln zwischen
den Minima des Potenzials zu Clustern zusammen, die jeweils von einander ge-
trennt liegen. Die Cluster sind nach den Symmetrierichtungen des Substrates
ausgerichtet. Durch diesen Prozess wird die quasi-nematische Phase zerstört. Für
hohe Dichten und Potentialstärken kann eine nematische Phase eingefroren wer-
den. Auch die langen Nadeln bilden auf dem Substrat Cluster. Im Gegensatz zu
den kurzen Nadeln sind diese Cluster miteinander verbunden und können sich Min-
imapositionen teilen. Mehrere Cluster können sich zu Linien zusammen setzen,
die in Richtung der Symmetrieachsen des Substrates liegen. Auf diese Weise kann
die quasi-nematische Phase stabilisiert werden. Die Abstände zwischen den Linien
folgen zwei verochtenen Fibonacci Sequenzen. Bei niedrigen Dichten formieren
sich die Cluster in unterschiedlich nach den Symmetrieachsen des Potentials aus-
gerichtete Regionen. Eine solche nicht nematische dekagonal ausgerichtete Phase
kann auch bei hohen Dichten und Potentialstärken eingefroren werden.
Ähnlich den kurzen Nadeln bilden die kurzen Spherozylinder unter Einuss
v
des Substratets getrennte nach den Symmetrieachsen des Potentials ausgerichtete
Cluster. Aufgrund ihres Volumens füllt sich die Fläche bei einer Erhöhung der
Dichte. Dabei schwächt sich die dekagonale Ausrichtung deutlich ab und bei
sehr hohen Dichten verschieben sich die bevorzugten Richtungen der Stäbchen
relativ zu den Symmetrieachsen des Potentials. Die langen Spherozylinder ordnen
sich auf dem Substrat entlang von Linien. Die räumliche Ordnung der Stäbchen
ist sehr schwach und wird letztendlich durch sehr hohe Dichten völlig zerstört.
Die dekagonale Ausrichtung hingegen bleibt selbst bei hohen Dichten erhalten.
Zusäzlich haben wir die Monbilität der Stäbchen bestimmt. Unter Erhöhung
der Potentialstärke werden die kurzen Stäbchen zwischen den Minimapositionen
gefangen. Die Mobilität steigt wieder, wenn die Dichte erhöht wird. Die langen
Spherozylinder können entlang der Symmetrielinien des Potenzials entlang gleiten.
Dadurch bleiben sie selbst bei hohen Potentialstärken mobil.
vi
Contents
1 Introduction 1
2 Quasicrystals 5
2.1 Introduction to quasicrystals . . . . . . . . . . . . . . . . . . . . . 5
2.2 Quasicrystalline surfaces . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 Atomicsurfaces........................ 6
2.2.2 Colloidal systems . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.3 Colloids on quasicrystalline surfaces . . . . . . . . . . . . . 10
2.3 Mathematical concepts of quasicrystals . . . . . . . . . . . . . . . 15
2.3.1 Fibonacci sequences . . . . . . . . . . . . . . . . . . . . . 16
2.3.2 Tilings............................. 19
2.3.3 Penrosetiling......................... 22
3 Nematic liquids 29
3.1 Liquid-crystalline order . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Phase transitions in two dimensions . . . . . . . . . . . . . . . . . 31
3.3 Orderparameters........................... 33
3.4 Particlemodels ............................ 37
3.4.1 Hardneedles ......................... 37
3.4.2 Hard spherocylinders . . . . . . . . . . . . . . . . . . . . . 38
3.5 Rod - substrate interaction potential . . . . . . . . . . . . . . . . 39
4 Computational methods 45
4.1 Canonicalensemble.......................... 45
4.2 Monte Carlo simulation . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3 KineticMonteCarlo ......................... 48
4.3.1 Brownianmotion....................... 49
4.3.2 Simulation scheme . . . . . . . . . . . . . . . . . . . . . . 51
vii
Contents
4.4 Wang Landau Monte Carlo . . . . . . . . . . . . . . . . . . . . . . 54
4.5 Simulationdetails........................... 56
Details of hard needle simulations . . . . . . . . . . . . . . 57
Details of hard spherocylinder simulations . . . . . . . . . 58
5 Hard needles 61
5.1 Shortneedlesystem.......................... 61
5.1.1 Phasediagram ........................ 61
5.1.2 Decagonal directional order . . . . . . . . . . . . . . . . . 61
5.1.3 Nematicorder......................... 64
5.1.4 Positional and Bond-orientational order . . . . . . . . . . . 69
5.2 Longneedlesystem.......................... 70
5.2.1 Phasediagram ........................ 70
5.2.2 Decagonal directional order . . . . . . . . . . . . . . . . . 71
5.2.3 Nematicorder......................... 72
5.2.4 Positional order . . . . . . . . . . . . . . . . . . . . . . . . 77
6 Hard spherocylinders 79
6.1 Short-spherocylinder system . . . . . . . . . . . . . . . . . . . . . 79
6.1.1 Phasediagram ........................ 79
6.1.2 Bond-orientational order . . . . . . . . . . . . . . . . . . . 80
6.1.3 Decagonal directional order . . . . . . . . . . . . . . . . . 83
6.2 Long-spherocylinder system . . . . . . . . . . . . . . . . . . . . . 88
6.2.1 Phasediagram ........................ 88
6.2.2 Positional order . . . . . . . . . . . . . . . . . . . . . . . . 89
6.3 Dynamics of hard spherocylinders . . . . . . . . . . . . . . . . . . 93
6.3.1 Short spherocylinders . . . . . . . . . . . . . . . . . . . . . 93
6.3.2 Long spherocylinders . . . . . . . . . . . . . . . . . . . . . 95
7 Conclusion and outlook 97
Appendix : Decomposition of the potential into a tiling 101
List of publications 107
List of Figures 109
viii
1 Introduction
Quasicrystals are solids which exhibit a long range positional order. The or-
der is aperiodic and it can possess rotational symmetries which are forbidden
for crystalline structures. Quasicrystals can possess symmetry axes with non-
crystallographic eight-, ten- and twelve-fold rotational symmetry. In many qua-
sicrystals physical properties are found to be dierent from usual crystals but also
dierent from unordered amorphous materials[94,196]. It is not surprising that
since the publication of their rst discovery[177], quasicrystals attracted a lot of
scientic interest.
The surface properties of quasicrystals are of particular interest. Quasicrys-
talline surfaces show a low adherence[57,146,147] which can be useful for non-
sticky coatings but is also an obstacle for epitaxial applications. A detailed under-
standing of the interfacial lms between crystalline and quasicrystalline structures
is important for the usage of quasicrystals, e.g. to enhance the adhesion between
quasicrystals and simple metal substrates [52]. Monolayers of adatoms can adopt
the quasicrystalline structures or arrange themselves in new fascinating structures.
In experiments with quasicrystalline alloys adatoms formed new self-assembled
structures[56,60,62,132,183]. The investigations are also directed towards the
understanding of the growth of quasicrystalline structures from quasicrystalline
templates.
Colloidal systems provide a model system for atomic order. They exists in dif-
ferent size and shapes and their interactions are easy to control in experiments[164,
165,216]. The length scales and time scales of the dynamic of colloidal particles
allow to study the phase behavior of colloidal systems on the level of single parti-
cle trajectories. Colloidal particles can be controlled by external elds like intense
laser beams[73,96]. The patterns of interfering laser beams can create two di-
mensional crystalline and quasicrystalline structures[171]. These patterns serve as
1
1 Introduction
substrates which mimic the structural properties of quasicrystalline surfaces. One
can investigate new structures and phase transitions on quasicrystalline surfaces
with colloidal particles conned to such two dimensional laser substrates[170].
Experimental and theoretical studies with spherical particles on a decagonal sub-
strate found new interesting phase behavior[71,137,138,139,169].
We extend the well studied setup of colloids on a quasicrystalline substrate from
spherical colloids to rodlike particles. Hard rodlike particles exhibit a long and
a short axis. In two dimensions this anisotropic shape introduces an additional
orientational degree of freedom[70]. At suciently high densities rodlike parti-
cles align their orientations along a common director and form a nematic phase.
In a nematic phase, a long-ranged order of the orientations of the rods is estab-
lished while the center of mass positions of the particles stay liquid like disordered.
The nematic phase is therefore the simplest example of a liquid crystal. In two
dimensions the nematic order is called quasi-long-ranged because orientational
correlations decay algebraically and the nematic director can only be dened lo-
cally in a nite radius around each particle[65].
Hard rods can also eectively model organic molecules on quasicrystalline sur-
faces like alkenes or aromatic hydrocarbons. The most common application of rod-
like particles are liquid crystal displays[91]. In such devices the liquid-crystalline
particles are placed into conned geometries. The interactions of particles with
the interfaces play an important role for their phase ordering in the bulk and the
possibility of switching between dierent states[116].
We identify new phases of hard rods under the inuence of a quasicrystalline
decagonal substrate potential. The orientational degree of freedom of hard rods
leads to interesting directional order along the decagonal symmetry directions of
the substrate potential in combination with pronounced cluster formation. At high
densities this ordering competes with the quasi-nematic phase. The interaction of
the rods with the potential is strongly dependent on the length of the rods with
respect to the typical length scale of the substrate potential. The nematic order
can be enhanced as well as destroyed by the substrate potential. In this work,
we investigate this interesting phase behavior and the mobility of hard rods on a
quasicrystalline substrate with computer simulations.
2
The outline of the work is as follows. In chapter 2we present the properties
of quasicrystals and quasicrystalline surfaces. The chapter also introduces our
decagonal substrate created from the interference pattern of ve laser beams. In
the following chapter 3, we introduce the properties of nematic liquids in general
and our particle models in particular. We briey summarize the phase transi-
tions of the hard rod models we analyzed in two dimensions. Furthermore, we
explain the quantities to investigate the phase transitions and the phase regions
in detail. At the end of chapter 3we present the interaction of the hard rods
with the substrate potential and its structural characteristics. In chapter 4, we
give the details of the simulation techniques used in this work. In chapter 5we
present the results of our study of the hard-needle model. We display the phase
diagram for short and long needles and give a detailed account of the new phases
and structures. In chapter 6we summarize the results of our investigations of the
hard-spherocylinder model. The phase behavior of spherocylinders signicantly
diers from those of the needles. In addition, we have also investigated the dy-
namics of the spherocylinders with kinetic Monte Carlo simulations. We conclude
our ndings in chapter 7and give an outlook of future topics which may arise
from the results of this work.
Parts of this work have been published in [A] and in [B].
3
1 Introduction
4
2 Quasicrystals
In this chapter we introduce rst the quasicrystals in general. Thereafter we
present the properties of atomic quasicrystalline surfaces. Afterwards we give an
introduction to colloidal systems in general and two dimensional systems under the
inuence of laser elds in particular. An explanation of the experimental setup of
colloids on a quasicrystalline substrate follows and we present our quasicrystalline
decagonal substrate potential. In the last past of the chapter we explain the
mathematical concepts of quasicrystals and the Penrose tilings in particular.
2.1 Introduction to quasicrystals
Before the discovery of the quasicrystalline matter only two types of ordered solid
matter has been known. On the one hand unordered structures like amorphous
materials. On the other hand crystalline materials with a periodic long-range spa-
tial order. The dierent species of crystals are classied by their rotational and
translational symmetry. The possible point groups to build a crystal in two or
three dimensions are well known. A lattice belonging to a crystalline point group
possesses
n
-fold rotational symmetry if it is invariant under a rotation of an angle
of
2π/n
with respect to a well dened rotational axis. The only possible values for
n
for a periodic lattice are
n= 1,2,3,4,6
. The distinct properties of crystalline
matter arises from this well ordered structure. Until the discovery of quasicrystals
the crystalline periodic structures believed to be the only possibilities of long-
ranged ordered matter. Quasicrystals possess at least one non-crystallographic
rotational symmetry. They were discovered rst in 1982 by D. Shechtman when
he was investigating alloys of
Al
and
Mn
. This rst quasicrystal has an icosa-
hedral symmetry consisting of
6
dierent rotational axes where each has a
5
-fold
rotational symmetry which is forbidden for any periodic crystalline structure. In
the same year A. Mackay published the diraction pattern of a Penrose tiling[128].
At this time the Penrose tiling was known only as a purely mathematical long-
5
2 Quasicrystals
range ordered structure. Mackay showed that the diraction pattern of atoms
arranged in such a tiling shows a non-crystallographic
5
-fold rotational symmetry.
Shechtman and the crystallographic community were not aware of the ndings by
Mackay. As a result Shechtman faced tough resistance against his interpretation
of an ordered phase with non-crystallographic rotational symmetry. He was able
to publish his ndings nally in 1984 [177], two years after his discovery. After this
publication the scientic community directed a lot of attention at this topic but
still encountered a lot of opposition, in particular by the double Nobel prize winner
L. Pauling [150]. The quasicrystalline alloy found by Shechtman was metastable
and could be produced only by rapid quenching of the melt. The small grains of
a few micrometer in size were dicult to study in detail. This situation changed
in 1986 when the rst stable quasicrystalline phase was found by Dubost et al.
in an alloy of
Al6Li3Cu
[51]. At the end of the 80s the number of quasicrystalline
phases in dierent alloys rapidly increased. As a reaction to the discovery of the
quasicrystals the International Union of Crystallography changed its denition
of a crystal to any solid having an essentially discrete diraction diagram[36].
This denition is much wider than necessary for the incorporation of the rst
quasicrystals because it discards also the need for any rotational symmetry. Now
quasicrystals are a commonly accepted particular state of matter. Shechtman has
been rewarded the Nobel prize for chemistry for his discovery in 2011.
The physical properties of quasicrystals turned out to be dierent from conven-
tional crystals as well as from disordered glass phases[94,196]. Most quasicrystal
materials are found in alloys of
Al
. But this could be an artifact of history of the
discovery of quasicrystals. The
Al
rich quasicrystals are brittle, hard and poor
conductors of heat and electricity. Because of their brittleness the search for tech-
nical applications concentrates on their usage as coatings or composites. For this
purposes the surface properties of the quasicrystals are very important.
2.2 Quasicrystalline surfaces
2.2.1 Atomic surfaces
As the quality and size of the quasicrystalline samples has improved, the surface
properties of such materials came in reach of scientic investigations[45,56,108].
In the beginning it was not clear whether quasicrystalline bulk structure also
6
2.2 Quasicrystalline surfaces
appears at their surfaces. Fortunately one is now able to produce high quality
quasicrystalline surfaces which can be used as templates for the growth of thin
lms or nanostructures[59,132,183,184,198]. The sizes of the samples vary
between a few millimeters to 10cm. The main goal of forming such thin lms
on quasicrystalline surfaces is to force the atoms in the lm into a quasicrys-
talline structure. This can result in quasicrystalline structures composed only
from one single chemical element in contrast to natural quasicrystals which are
composed of at least two dierent elements. Noble gases have a low chemical
reactivity and were used to study the physorption properties of various quasicrys-
talline surfaces[55,200]. The experiments were accompanied by computational
simulations of the noble gases[38,46,47,174,175]. The rare gases can retain
the quasicrystalline structure if the length scale of the atomic bonds in their
crystalline ordered phase matches the typical length scale of the quasicrystalline
structures of the substrate like
Xe
on a quasicrystalline
AlNiCo
alloy. Experi-
ments have been performed with metal elements like
Au
[181] and
Ag
[60] but also
thin lms of
Pb
[1,42],
Bi
and
Sb
[63] lead to interesting results. Simulations of
metal lms helped to understand the experimental results like quasicrystalline
clusters and locked crystalline domains[18,140]. The competition of the qua-
sicrystal template structure and the solid crystalline state of the lm material
can lead to fascinating new structures, e.g. the step structure of a
Cu
lm on a
i−AlPdMn
interface follows a one-dimensional quasicrystalline sequence known
as the Fibonacci sequence[115,155].
C60
molecules placed onto such
Cu
lms
show an unusual low mobility[185]. Chemically more reactive metals like
Fe
can
also penetrate into the surface structure[213]. The low friction coecient of qua-
sicrystalline surfaces[57,146,147] raises questions about the interaction of the
surfaces with lubricants commonly made from carbohydrates[39,48,133]. Qua-
sicrystalline interlayers also may be a possibility to connect two crystalline mate-
rials with incommensurate periodicity[64]. Recently, also special adsorption sites
on quasicrystalline surfaces have been identied. Due to the aperiodic nature of
the surface multiple chemical decorations of a quasicrystalline surface are possible
[202].
7
2 Quasicrystals
2.2.2 Colloidal systems
Colloidal systems are omnipresent in biological systems as well as in industrial
applications. The motivation to understand the behavior of colloidal particles is
not limited to those applications. They are also a model system for the under-
standing of atomic and molecular structures and dynamics[12,216]. Due to their
size they are easy to observe with microscopic devices working in the visible light
spectrum. The time scales of their dynamical behavior are much longer than for
atomic systems. This results in a good time resolution of dynamical processes
where it is even possible to follow single particle trajectories. Colloidal particles
can have almost arbitrary shapes and sizes[164]. It is possible to mix dierent
colloidal particle species to investigate phenomena such as phase separation[102].
The interaction strength between particles is tunable over a wide range e.g. in
systems witch electrostatic interactions with dierent salt concentrations of the
solution[218]. A very important feature of colloidal particles is their sensitivity to
external elds introducing even more possibilities to force a colloidal system into
a physical situation of interest. Colloidal particles are not only easy to probe in
experiment but also easy to treat in theory and computer simulations[10]. This
makes them a perfect model to study phase transitions or glass and gel formation.
Two-dimensional systems are particularly interesting. The phase behavior of
particles conned to two dimensions is very dierent from the 3D case[195]. The
KTNHY theory [110,141,220] is able to explain the two-dimensional melting
as a disclination unbinding transition leading to a continuous phase transition in
contrast to one rst-order phase transition observed in 3D systems. In particular,
the theory predicted a two stage melting in a system of spherical particles. First,
a phase transition from the solid phase to a uid hexatic phase can be observed
where the bond orientational order of the solid is preserved only locally and the
phase possesses no long-ranged order. Afterwards, in a second phase transition,
the hexatic uid phase melt and the system exhibits an isotropic phase. This two
stage process could be investigated in an experiment with colloids in a 2D con-
ned geometry[221]. Also simulations of conned systems show velocity[3,50,54]
and angular momentum[207] autocorrelations with long positive tails. In early
experiments with conned colloids[152,153] the conning walls were structureless
glass plates. In practical applications the substrates for a 2D colloidal lm are
8
2.2 Quasicrystalline surfaces
usually structured. Therefore, the substrates are able to induce an order in the
colloidal monolayers[43]. That can be used to build template patterns which are
able to not only order the rst monolayer but also inuence the order of the bulk in
a technique named after its atomic counterpart as colloidal epitaxy[4,92,197,203].
To model a wide range of dierent substrate patterns we make use of the sen-
sitivity of colloidal particles to laser elds. Laser eld experiments exploit the
dierent dielectric constants of the solvent and the colloidal particle. The inverse
of the laser frequency is much smaller than the relevant time scale of the motion
of the particle. From the point of view of the particle the laser eld is therefore
of a static intensity. The particle is subject to two dierent contributions of the
laser eld. Those are easily derived for the force of the induced dipole moment of
the particle [73,96]:
−→
FL= 0.5α∇E2+∂
c∂t(−→
E×−→
B)
(2.1)
with
E
the electric eld,
B
the magnetic eld and
α
the polarizability of the parti-
cle. The rst term is the gradient force of the laser eld and the second term is the
scattering force describing the momentum transfer between the laser beam and the
particle. In a two-dimensional system the scattering force points perpendicular to
the plane and usually leads to a higher eective mass of the particle. To gener-
ate a substrate potential for colloidal particles the gradient force is the important
part. This force is proportional to the gradient of the intensity. Depending on the
particle shape the prefactors may be more complicated [9,124,199].
The most common applications of the inuence of intense laser elds on colloidal
particles are optical tweezers[77]. They are widely used instruments to conne and
manipulate single particles. This tool is of a high scientic importance and its
applications are the basis of many scientic developments of the past decades. In
our focus, the usage of laser elds should not be limited to a single particle but
to a system of interacting particles. Laser elds serve as a substrate potential in
a two-dimensional system. Ordered structures of colloids which form under the
existence of light elds are named optical matter[28]. The rst investigated laser
substrates where one-dimensional, periodic intensity proles. The interference
pattern of two laser beams consisted of parallel stripes. The colloids move to
the high density regions and eventually crystallize to a state which is referred
9
2 Quasicrystals
Figure 2.1: a) the Archimedian tiling
(33,42)
b) quasicrystalline modulated Archi-
median tiling, the distances
S
and
L
between the double triangle rows follow a
Fibonacci sequence.
to as laser induced freezing(LIF)[33,124]. Later, it was found in theory and
numerical simulations that such a system can undergo a second phase transition
at even higher intensities. The crystalline phase melts again and this reentrant
phase transition is called laser induced melting (LIM)[14,30,96,212]. With the
LIF and LIM transition the one dimensional periodic substrate shows a rich and
surprising behavior.
With two-dimensional laser patterns one can form a periodic lattice of potential
minima. At suciently high intensities the colloids freeze and occupy every poten-
tial minima with one particle. A special new phenomenon occurs if the number of
colloids exceeds the number of potential minima. Two or three colloids are conned
in one potential minimum to form a colloidal molecule[2,23,24,53,159]. One
colloidal molecule is not spherical anymore but has an orientation. For interaction
lengths longer than the distance between two potential minima the orientations
between colloidal molecules of the whole system are able to couple and new forms
of orientational order arises in a system of spherical particles.
2.2.3 Colloids on quasicrystalline surfaces
Laser elds could also used to create quasicrystalline substrates. The interference
pattern of the laser beams can produce substrates belonging to arbitrary classes
10
2.2 Quasicrystalline surfaces
of rotational symmetry[171]. The resulting intensity patterns are identical to
the density functions in the phase eld crystal model[163]. Most attention has
been payed to the decagonal systems with a
10
-fold rotational symmetry[170].
The decagonal symmetry is the most common symmetry found at quasicrystalline
surfaces. On decagonal light substrates spherical colloids can order themselves
into a quasicrystalline phase with
10
-fold or
20
-fold bond orientational symmetry.
The most interesting discovery in the investigation of the phase diagram of the
colloids is the occurrence of a phase with quasicrystalline Archimedian tiling[71,
137,138,139,169] as shown in Fig. 2.1. This phase arises if the length scale of
the bonds of the preferred hexagonal crystal is close to the typical length scale of
the quasicrystalline decagonal substrate. The normal Archimedian tiling
(33,42)
consist of a periodical repetition of alternating rows of triangles and squares.
The Archimedian tiling
(36)
consists of triangles only and the vertices belong
to a triangular lattice. At high densities and without any substrate structure
the colloidal particles form a hexagonal phase. If one draws the positions of
the colloids as vortices and the bonds between nearest neighbors as edges, an
Archimedian tiling
(36)
of triangles appears. A new phase emerges if one place
the colloidal hexagonal phase onto a decagonal substrate with a length scale and
an interaction strength matching the length scale and interaction strength of the
bonds of the hexagonal phase. In this new phase, the bonds between the particles
dene a pattern of triangles and squares like in the Archimedian tiling
(33,42)
.
But in contrast to the normal Archimedian tiling the sequence of rows of triangles
and squares is not periodic but follows a one dimensional quasicrystalline sequence
known as Fibonacci sequence. To do so, the rows of triangles are doubled and the
resulting tiling can be described as an alternating
(36)
and
(33,42)
Archimedian
tiling.
There also have been investigations of the dynamical features of colloids on a
decagonal substrate potential. Colloids on a quasicrystalline substrate nd their
potential minima similar to colloids on amorphous surfaces[168]. On a dierent
decagonal substrate a directional locking of driven particles has been seen[158].
Quasicrystals do not only have translational modes. The dynamics of so called
phasonic drifts which have no counterpart in periodical crystals have been inves-
tigated as well[111].
The experimental setup necessary for the creation of quasicrystalline substrates
11
2.2 Quasicrystalline surfaces
is shown in Fig. 2.2. The setup consists of a laser beam which is split up into par-
tial beams. The wave vectors of the laser beams are arranged along the
n
edges
of a prism that is made of a perfect
n
-sided polygonal base. The laser beams
interfere again in the plane of the sample which contains the colloidal suspension.
The number and wave vectors of the beams determine the structure of the in-
terference pattern. The wave vectors of the laser beams are arranged to form a
rotational symmetric star. The rotational symmetry of the star is equal to the
rotational symmetry of the interference pattern if the number of beams
n
is even.
The rotational symmetry of the pattern is
2n
if the number of wave vectors is odd.
As shown in Eq.(2.1), the potential acting on a colloid is proportional to the
intensity of the electric eld. For the average potential strength at a given point
we have to integrate the interference of
j
laser beams over one period
T
dened
by the inverse of the oscillation frequency of the laser beam
ω
:
V(−→
r)∝ −
T
Z
0
dt{
n−1
X
j=0
Ejcos(−→
Gj·−→
r+ϕj+ωt)}2
∝ −
T
Z
0
dt{
n−1
X
i=0
n−1
X
j=0
EiEjcos(−→
Gj·−→
r+ϕj+ωt) cos(−→
Gi·−→
r+ϕi+ωt)}
∝ −
T
Z
0
dt
n−1
X
i=0
n−1
X
j=0
EiEj{cos((−→
Gj−−→
Gi)·−→
r+ϕj−ϕi)
+ cos((−→
Gi+−→
Gj)·−→
r+ϕi+ϕj+ 2ωt)}
Where
Ej
is the amplitude and
ϕj
the phase and
Gj
is the wave vector of the laser
beam. After integration over the time we get:
V(−→
r)∝
n−1
X
i=0
n−1
X
j=0
EiEj{cos((−→
Gj−−→
Gi)·−→
r+ϕj−ϕi)
(2.2)
We are not interested in dierent phase shifts expressed by the term
ϕj−ϕi
. The
general form of the potential reads then:
V(−→
r) = −V0
n2
n−1
X
j=0
n−1
X
i=0
cos[(−→
Gi−−→
Gj)·−→
r].
(2.3)
13
2 Quasicrystals
Figure 2.3: Star of wave vectors (left) and the resulting substrate potential (right)
created from a) 5 laser beams and b) 3 laser beams.
The characteristic length scale of the structures of the potential is given by
aV=
2π/|−→
Gi|
. Therefore we express all lengths in units of this length scale. The
parameter
V0
gives the depth of the deepest minima. In our simulations we give
V0
in units of the thermal energy
kT
, where
T
is the temperature and
k
the
Boltzmann constant. In Fig. 2.3 two dierent choices of a star of wave vectors
−→
Gi
for
n= 5
and
n= 3
are shown. In Fig. 2.3 a) The ve wave vectors
−→
Gi
point to the vertices of a pentagon and generate a star of vectors with an angle
φ= 2π/5
between two neighboring vectors
(−→
Gi,−→
Gi+1)
the resulting potential has
a decagonal quasicrystalline symmetry. In Fig. 2.3 b) the three wave vectors point
to the vertices of an equilateral triangle. Right next to it the resulting triangular
crystalline potential is shown. In the following we use the decagonal potential as
our model for a quasicrystalline substrate potential.
14
2.3 Mathematical concepts of quasicrystals
2.3 Mathematical concepts of quasicrystals
Unlike crystalline matter the connection of the quasicrystalline order to its math-
ematical description is not straight forward[95,119,187]. We start with the three
basic properties found in real material. Afterwards we explain quasicrystalline
geometrical patterns in one and two dimensions.
The property of a long-ranged positional order leads to sharp peaks in the elec-
tron diraction patterns of quasicrystals similar to crystalline matter. The most
obvious property of quasicrystalline materials is the long range orientational order
of the bonds of nearest neighbors. One can order the bond angles along a star of
axis. The set of star axis can have a rotational symmetry. In most quasicrystals
the rotational symmetry is non-crystallographic. That means that no crystalline
order is able to have the same rotational symmetry. It is easy to picture this in two
dimensions. Crystals with a unit cell shape of triangles, rectangles or hexagons
have a
3
,
4
or
6−
fold rotational symmetry respectively. The quasicrystalline Pen-
rose tiling consists of two dierent rhombic unit cells which edges are orientated
onto a
5−
fold rotational symmetry.
The second property is connected to the translational order of the atoms and
atomic clusters. The density function can be decomposed into a sum of periodic
functions. If the ratio of the periods of some of the functions is an irrational num-
ber those functions are called incommensurate. If a decomposition of a distribution
into periodic functions exhibits also incommensurate functions the distribution is
quasiperiodic. Quasicrystals possess a quasiperiodic translational order.
These two properties can be deduced from the electron diraction patterns of
the quasicrystals. One basic property is left to avoid unphysical mass distribu-
tions which can not be obtained in real materials. In real materials the atoms
can not come arbitrary close to each other. Therefore there must be a minimal
spacing between two atoms. There should also be a maximum spacing between
two atomic sides. Otherwise the material can consist of arbitrary huge holes and
gaps.
In the following we will use a further restriction for the quasicrystals mentioned
15
2 Quasicrystals
in this work. We always use quasicrystalline structures with non crystallographic
rotational symmetries. This is a large restriction in mathematical terms but since
nearly every known real quasicrystal material possesses such a rotational sym-
metry, the reduction of the mathematically possible quasicrystals leads to almost
no constraints of the physical scope. This reduction gives us another important
feature for most of the known quasicrystalline structures. Quasicrystalls possess-
ing rotational symmetry are self-similar. That implies the ability to construct
quasicrystals with self-similarity transformations like ination and deation rules
what is not possible for general aperiodic structures. In the next section we want
to present some examples of quasicrystals and demonstrate the mentioned prop-
erties in detail.
2.3.1 Fibonacci sequences
A very important one-dimensional quasicrystalline set is the Fibonacci sequence[94,
119,187]. It serves as an easy to understand toy model for more complicated cases
of quasicrystalline structures. The Fibonacci chain is also very closely related to
the Penrose tilings which we will discuss in the next section. In one dimension an
rotational symmetry has no meaning so the construction of the Fibonacci sequence
is only dened by the other two properties. The Fibonacci sequence consists of
a quasiperiodic repetition of two spacings with dierent lengths. We denote the
short length scale
(s)
and the long length scale
(l)
. The spacings can be seen as the
unit cells of the quasicrystalline structure which are repeated in a quasiperiodic
order. In this way we satised the two conditions of an aperiodic translational
order and a minimum and a maximum distance identical to the length scales of
the unit cells. There are many ways to create this sequence exploiting dierent
properties of an one dimensional quasicrystal. In general, one generates the se-
quence iteratively with construction or deation rules. Both methods start with
an initial sequence of spacings which grows with each step by a nite number of
additional spacings. After each step, every obtained sequence is a nite patch
of the innite Fibonacci sequence which would be obtained for the step number
n→ ∞
.
A construction rule describes how to add congurations of the previous step to
obtain a new longer sequence in the next step. A start conguration with the step
16
2.3 Mathematical concepts of quasicrystals
Figure 2.4: Subdivision of the upper box marked
L
into smaller parts using the
deation rule
S→L
and
L→LS
at each step. The fraction of the two distances
always obey
L/S =τ
. The step number is shown on the left and the number of
sub boxes at each step is shown on the right.
numbers
n= 0, n = 1
is dened by
S0=l
and
S1=ls
. The construction rule
for the Fibonacci sequence reads then
Sn=Sn−1Sn−2
. The recurrence relation for
a Fibonacci number series is
Fn=Fn−1+Fn−2
with the seed values
F0= 0
and
F1= 1
. With this construction rule the similarity between the Fibonacci chain
Sn
and the Fibonacci series
Fn
become apparent. In particular the number of
spacings in the patch of the Fibonacci chain
Nn
grow with
Nn=Nn−1+Nn−2
.
The seed numbers of the Fibonacci sequence
N0= 1
and
N1= 2
are identical to
the Fibonacci series steps
F2= 1
and
F3= 2
. Therefore the number of spacings
in every nite patch of
Sn
is the Fibonacci number
Nn=Fn+2
.
The deation rules uses a given set of spacings and subdivide them into a new
set of more spacings. In this way the lengths
(s)
and
(l)
cannot be preserved.
In every step new lengths
(s0)
and
(l0)
are dened in a way such that the ratio
between them stays constant and they dier from the previous lengths by only a
constant factor. Because we are only interested in the sequence of the short and
long lengths and not in their absolute value we will leave out the subscripts in
the following. The easiest deation rule substitutes in every step the spacing with
s→l
and
l→ls
. In Fig. 2.4 the deation is shown. The number of spacings after
each step follows again the Fibonacci number series like in the construction rule
above. The deation rules show another very interesting symmetry if we introduce
a dot as a marker in the initial step as
l.s
. The rst 4 substitutions are shown in
Fig. 2.5.
The sequence is despite of the rst
s
and
l
exactly symmetric around the central
dot. After each step the order of the inner asymmetric
s
and
l
is interchanged.
17
2 Quasicrystals
l.s
ls.ll
lsll.slsl
lsllsls.llsllsl
lsllslsllsll.slsllslsllsl
Figure 2.5: First 4 deations of the middle-C sequence.
This Fibonacci sequence around a central dot is called middle-C sequence [78].
This sequence plays an important role in the cartwheel tiling which we introduce
in the next sections.
The deation rule can reveal another connection of the Fibonacci chain to the
Fibonacci number series. If we assign the probability of nding an
l
after step
n
at a certain position with
pn(l)
and a short distance with
pn(s)
we can express the
probabilities of step
n
through the probabilities of the previous step
n−1
using
the deation rules. For the long distance the probability
pn(l)
reads :
pn(l) = pn−1(l) + pn−1(s)
2pn−1(l) + pn−1(s)
(2.4)
With
pn(l) = 1 −pn(s)
follows for the limit
n→ ∞
:
p∞(l) = 1
1 + p∞(l)
(2.5)
p∞(l) = √5−1
2= 1 −τ
(2.6)
and therefore:
p∞(l)
p∞(s)=τ
(2.7)
The number
τ
is known as the golden ratio and is also the limit
n→ ∞
for the
fraction of Fibonacci numbers
Fn+1/Fn
. Therefore the ratio of the number of long
distances and the number of short distances in a innite Fibonacci sequence is the
golden ratio
τ
.
The Fibonacci sequence can also be obtained by a projection method. One
can project a periodic regular square lattice onto a line with a slope of
1/τ
. The
18
2.3 Mathematical concepts of quasicrystals
Figure 2.6: Construction of the Fibonacci chain via the projection method. The
central black line has a slope of
1/τ
. The dotted lines denotes the projection strip.
projection is done only for close points within a small environment around the
line called the projection strip. There are only two dierent distances between the
projected points which can be marked again as short
s
and long
l
. The sequences
of the distances then follows the Fibonacci sequence. The size of the projection
strip satises the condition of a minimum and maximum distance between two
points in the Fibonacci sequence. The projection line don't have to cross the
origin of the coordinate system and also the projection strip can be displaced as
long as the width of the strip is preserved. Choosing the projection in the same
way as shown in Fig. 2.6 we nd again the exact sequences from the construction
rules above. The sequence obtained with the deation method is identical to the
sequence starting from the origin of the coordinate system going into the negative
direction. The symmetry center of the middle-C sequence is the cross section of
the upper acceptance boundary with the Y axis.
2.3.2 Tilings
In the scope of this work we want to study a system of rod like particles on a qua-
sicrystalline surface. Therefore the properties of two-dimensional quasicrystalline
19
2 Quasicrystals
Figure 2.7: a) the tiles of a Girih tiling[37], b) a patch of a Girih tiling[126].
structures are important. As a mathematical model system we present the charac-
teristics of quasicrystalline tilings of the plane. Tilings are well known geometrical
structures based on a nite subset of topological discs which can be arranged to
ll a two-dimensional space by periodic or aperiodic repetition. The rst qua-
sicrystalline tiling was found in ancient Islamic art about 500 years ago[126]. The
tiling consist of so called Girih tiles as shown in Fig. 2.7. In Islamic art the Girih
tiles are widely used as decorations until today, e.g. the portal of the Turkish
embassy in Berlin is decorated with such a tiling. Most of these tilings are peri-
odic but with a very large repeating patch. The most important quasicrystalline
tiling related to our work is the Penrose[151] tiling which has a 5-fold rotational
symmetry. The Penrose tiling was the rst quasicrystalline tiling of the plane ex-
plored in modern mathematics. Therefore the features of the tiling are well known.
There are many possibilities to obtain a quasicrystalline two-dimensional struc-
ture by a tiling of the plane[8,11,161]. Quasicrystalline tilings consists of sets
of two or more tiles which serves as unit cells of the tiling[187]. To be able to
cover the whole plane without holes, one need in addition a special construction
rule. There are dierent types of construction rules for a quasicrystalline tilings
which can be classied as in the case of the one dimensional Fibonacci sequence.
Most of them start from a single unit cell building an arbitrary huge patch. An
easy to understood rule is a set of matching rules[120,188] for the edges of the
unit cells. The unit cells are decorated with markers which have to be matched
by the neighboring unit cells. Starting with an arbitrary unit cell one can just
add more cells and grow a patch which can cover the whole plane. Unfortunately
20
2.3 Mathematical concepts of quasicrystals
the matching rules alone are ambiguous. Every patch of a tiling must fulll the
matching rules but not every patch placed correctly with respect to the matching
rules can be completed without defects to ll the entire plane. The importance of
the matching rules arises from the fact that real quasicrystalline materials have
to grow by local interactions between the atoms[34,123]. This local interaction
has to be reected in the possibility to build a perfect quasicrystalline tiling with
local matching rules for each tile[69]. Another problem is the building of two or
more dierent tiles of the correct stoichiometric number out of an initial isotropic
liquid state. This problem could be avoided by using one single prototile with a
set of overlapping rules instead of dierent unit cell tiles with matching rules[196].
We will present these construction methods in the example of the Penrose tiling
in the next section.
A dierent approach to obtain a tiling of the plane are deation rules[119,187].
The deation rules describe how a tile can be subdivided into smaller unit cells.
This method uses the self-similarity of many quasicrystalline tilings and is a two-
dimensional analogon to the deation rules of the Fibonacci sequence. One starts
again with an arbitrary tile and subdivide it into a sub pattern. One repeats this
subdivision until the needed size of the quasicrystalline patch is reached. This
is a very reliable method which always produces perfect patches and is a vital
approach for mathematical treatments. Obviously such a deation can not serve
as a model for the growth of real quasicrystals.
A more general way to obtain a quasicrystalline tiling of the plane is the multi-
grid method[68,107,189]. A grid is a countable, discrete set of parallel lines. The
multigrid is a sum of many grids with dierent line directions. This method is
analogue to the projection method of the Fibonacci sequences. One can identify
the line directions with the symmetry axis of the system and regard them as pro-
jections of a higher dimensional hypercube on the two-dimensional plane. The
crossing points of the lines can be regarded as dual to the positions of the unit
cells of a tiling. Dening for every type of crossing points a corresponding tile,
one can construct aperiodic tilings of the whole plane. This can lead to more gen-
eral aperiodic tilings which can not be constructed by matching rules or deation
methods. This generalization can also introduce tilings without any rotational
symmetry[186].
21
2 Quasicrystals
2.3.3 Penrose tiling
The Penrose tiling is an aperiodic tiling of the plane with a
5−
fold rotational
symmetry[151]. There are at least 3 popular dierent sets of unit cells[78]. We
show them in Fig. 2.8. The Penrose set in Fig. 2.8 a) consist of six dierent
unit cells which are four kinds of pentagons, a pentacle, a rhomboid and a half
pentacle. This tiling has untypical many unit cells but displays the
5−
fold rota-
tional symmetry in the most understandable way. A very common set is the set
of two dierent unit cells of kites and darts shown in in Fig. 2.8 b). The last set
has also two unit cells, a skinny and a fat rhomboid, Fig. 2.8 c). The two unit
cells of the last set turned out to be very useful also for the descriptions of other
tilings with dierent rotational symmetries. The justication to call all of them
a Penrose tiling stems from their equivalence under self-similar transformations.
One can transform a perfect tiling of one set into a tiling of another set and vice
versa. The Penrose tiling can be build from matching rules, deation methods
and a multigrid of 5 periodic grids in the symmetry directions of a pentagon. The
matching rules of the rhomboid set are shown in Fig. 2.9 a). The rhomboids are
marked with arrows. Two rhomboids are allowed to be placed side by side if the
shared edges have the same number of arrows pointing in the same direction. The
matching rules can be obtained from the projection of a
5
dimensional crystalline
grid onto the 2 dimensional plane[40]. In Fig. 2.9 b) the deation rules are shown.
The rhomboids can be subdivided into a small sub patch of smaller rhomboids.
The length of the edges shrinks by a factor of
1/τ
. After each deation one can
inate the obtained tiling by
τ
and obtain a bigger patch of a perfect tiling with
the same edge lengths as in the initial patch.
Patterns with the same properties of an underlying tiling can be derived by
the decoration of the unit cells[94]. One famous decoration of the fat and skinny
rhomboids is the Ammann bar decoration[78,119,187]. The rhomboids are dec-
orated with line segments which can also be used as markings for the matching
rules. The marked tiles composing a Penrose tiling with the line segments merged
into a set of parallel lines in every of the 5 symmetry directions. The Ammann
bar decoration and the resulting tiling is shown in Fig. 2.10. In every set of par-
allel lines there are only two dierent distances, a short
S
and a long distance
L
which are ordered in a Fibonacci sequence. It is also possible to construct the
22
2 Quasicrystals
Figure 2.9: a) matching rules for thick and thin rhomboids. b) deation of thick
and thin rhomboids (red) into a smaller tiling.
Figure 2.10: a) decoration of the tiles with Ammann bars b) resulting Penrose
tiling with Ammann bars.
24
2.3 Mathematical concepts of quasicrystals
grid of parallel lines rst and obtain the tiling stemming from the grid structure
afterwards. If the lines have a sequence of long
L
and short
S
distances obeying
L/S =τ
, following the Fibonacci sequence and the orientation of the lines is
along the
5
-fold symmetry directions of a pentagon, the grid is called Pentagrid.
The Pentagrid is the underlying multigrid for all Penrose tilings. The intriguing
property of the Ammann bar decoration is that it reveals the connection between
matching rules, the multigrid method and the Fibonacci sequence in a Penrose
tiling.
A quite sophisticated kind of deation method for the kites and darts in a Pen-
rose tiling is the cartwheel tiling[78]. Starting with a kite and two darts, a so called
ace, and deate this by the corresponding deation rules, one gets sequentially
growing patches. The patches with even numbers of deations posses very interest-
ing features. All patches have a mirror symmetry like the rst ace. Furthermore,
the even numbered patches consists of sub patches in the shape of a decagon.
These patches are called cartwheels. A
n
th order cartwheel is a cartwheel after
2n
deations starting from the ace. In Fig. 2.11 the initial ace patch a) is shown.
After two deations one reaches the decagon patch b) with the ace in the middle.
This decagon patch is the rst order cartwheel. Two more deations show a bigger
decagon, displayed in Fig. 2.11 c). In the center of the big decagon is again the
ace and the rst order cartwheel. In the center of a cartwheel of
n
th order are
always all cartwheels of the order
m < n
. One can show that every Penrose tiling
consists of an innite composition of cartwheels. Another important feature of
the cartwheel tiling is the sequence of Ammann bars. The distances between the
Ammann bars follow the middle-C sequence of the Fibonacci chain shown in the
previous section.
The decoration of the unit cells is a widely used method to connect a tiling
with actual quasicrystalline materials. The tiling together with a decoration which
stands for the positions of the atoms can serve as an approximant of a real ma-
terial. But the decoration of the tiling of a quasicrystal consisting of many unit
cells remains unsatisfactory. This unit cells fall short in explaining how a real qua-
sicrystall grows, especially how the single atomic bonds know the correct matching
rules for a macroscopic quasicrystalline order without any defects and holes. An
alternative approach inspired by the cartwheel tilings was given by the work of
25
2 Quasicrystals
Figure 2.11: The cartwheel tiling derived by deation rules. a) The central ace
conguration b) the rst order cartwheel c) the second order cartwheel.
26
2.3 Mathematical concepts of quasicrystals
Figure 2.12: a) decagon prototile b) overlap rules for four small overlaps A and a
big overlap B.
Figure 2.13: a) Cartwheel decoration of a Gummelt decagon b) Decoration of the
decagon with a Jack.
Gummelt[66,79,80,97]. The set of a few dierent unit cells is replaced by just
one single quasi unit cell. The quasi unit cell for the Penrose tiling is a decagon.
The decagons are allowed to cover parts of each over by a set of matching rules as
displayed in Fig. 2.12. The benet of this approach is that the matching rules of
the decagon can be explained by the inner structure of the atomic clusters without
the need of explaining how dierent building blocks are forming. A decoration of
a decagon with the corresponding positions of the atoms reveals that the decagon
clusters sharing just groups of atoms with each other[157,192]. One group of
atoms can belong to more than one decagon structure. The geometrical match-
ing rules of the decagons correspond to the sharing possibilities of the atomic
structures. Three dimensional quasicrystals can be build from the stacking of the
two-dimensional clusters[135,194]. One can derive the Penrose tiling from the
Gummelt decagon easily by a decoration of the decagon as shown in Fig. 2.13.
The kites and darts decoration is just the rst order cartwheel tiling as shown
in Fig. 2.11 b). The rhomboids tiling can be derived by adding a so called Jack
conguration as a decoration of the decagon.
27
2 Quasicrystals
28
3 Nematic liquids
In this chapter we introduce rst dierent kinds of liquid crystalline order. Af-
terwards we explain the special properties of phase transitions in two dimensions.
Phase transitions can described quantitatively by order parameters. The order
parameters used in our work are shown in the third section of this chapter. In
section 3.4 we display our particle models. Then we present the interaction of the
hard rods with the substrate potential and choose the lengths of the rods.
3.1 Liquid-crystalline order
The mechanical properties and symmetries of liquid crystal phases are intermedi-
ate between a liquid and a crystalline state. They are therefore often referred to
mesophases. They were rst discovered by Reinitzer[160] and shortly after named
liquid crystals by Lehmann [118] in the 19th century. The components of liquid
crystalline matter are anisotropic building blocks. The size of the building blocks
range from small molecules up to the scale of colloidal particles. The most com-
mon shapes are elongated particles or disc-like particles. There are many dierent
liquid crystalline phases which are categorized with regard to the order found in
these phases[70]. The simplest phase is the nematic phase. In a nematic the par-
ticles are oriented along a common axis called the director. The particles can ow
in every direction like in the liquid phase. The diusion of the particles is usually
dierent in the direction of the director and perpendicular to it. In the cholestric
phases the orientation of the director smoothly changes in space and has a helical
structure around an axis perpendicular to the director. The period of the helix
is called the pitch of the cholestric. The pitch is typically much larger than the
size of one particle. If the pitch is innitely large we obtain the nematic phase.
If the particles are positionally ordered in one dimension and the orientations are
ordered along a common director like in a nematic phase the phase is called smec-
tic. An one-dimensional spatial crystalline order denes two-dimensional layers in
29
3 Nematic liquids
Figure 3.1: Schematic drawing of a nematic , smectic and columnar order.
which the particles can ow like a liquid. There are many dierent possibilities
to build a smectic order so there is a large subgroup of smectic phases. The most
common smectic phases are the smectic A where the layers are perpendicular to
the director and the smectic C where the director is tilted with respect to the
normal of the planes. The most ordered phase is the columnar phase. In this
phase the particles are periodically ordered in two dimensions and liquid only in
the third one. The name stems from the conformation of the particles into parallel
columns. The particles are in a liquid state along the axis of the column only.
Fig. 3.1 displays schematically the dierent phases. Another important classica-
tion of liquid crystals is how the matter approaches the liquid crystalline phase.
Particles which undergo easily a phase transition into a liquid crystalline phase by
a temperature change are called thermotropic. If the phase transition is density
driven, the liquid crystal is called lyotropic.
Liquid crystals are nowadays most widely used in liquid crystal displays (LCD)[91].
The surface interactions play a very important role in the development of LCD
systems. The direct interaction of an anisotropic molecule with a surface is called
anchoring[99,100]. The molecule can prefer to lie planar at the surface or have a
specic contact angle. At the vicinity of the surface the liquid crystal can undergo
a phase transition of the Kosterlitz-Thouless type[182]. The surface monolayer
can induce a liquid crystalline order in the bulk[101,130,193]. The formation of
an interfacial monolayer at a wall is strongly connected with the wetting behavior
of the liquid[41,173,179,191]. In LCDs the liquid crystals are usually conned
between two parallel walls. Therefore the inuence of the walls with and without a
30
3.2 Phase transitions in two dimensions
substrate pattern raised a lot of attention[5,32,49,76,208]. Dierent techniques
of patterning a surface are known, which can induce controlled phase behavior in
the bulk phase[81,106,156,219,222]. The patterns can consist of topological
gratings[26] or chemical structures [22]. Also patterns produced by laser elds are
known[117,127]. It has been shown that patterns with crystallographic symme-
tries can be used to build liquid crystal devices which have bistable or tristable
nematic order[21,105,116]. The most interesting experiments for the scope of
our work are nematogens in contact with crystalline surfaces. Some systems have
already been investigated experimentally like alkene on graphite[88] or pentacen
on
SiO2
[31], and through computer simulations[145].
Lyotropic hard core nematics are easy to model in computer experiments in
the most general fashion. There are less lyotropic liquid crystals known than
thermotropic but they have become more and more important also for practical
applications. Many of the lyotropic nematics are anorganic rods[67,165,227]
such as bohemite rods[27,204,205,206,215] sepiolite clay[217,223]
LaPO4
rods
[104], rodlike silica [112] or
β−FeOOH
needles[129]. One of the oldest lyotropic
liquid crystals is the tobacco mosaic virus[29,144]. Because of its monodispersity
it served as an ideal experimental model system[61,74,214] for a long time. The
application of the self-organization of liquid crystals on crystalline surfaces raises
scientic interest in other elds as well. For example, the self-assembly of nano
rods can be used to build nanostructures on semiconductors[121,122].
In the following we want to restrict ourself to the simulation of a two-dimensional
system. The phase behavior of two-dimensional systems is quite dierent from the
three dimensional bulk systems. We take two dierent models for the particles
into account namely the hard needles and the hard spherocylinders.
3.2 Phase transitions in two dimensions
A phase transition occurs when a system changes its order from one state to an-
other. The phase transition can be tracked quantitatively by a properly chosen
order parameter. One can distinguish two basic types of phase transitions with
respect to the behavior of the free energy function at the transition temperature
31
3 Nematic liquids
TC
. The transition is called rst order if the rst derivatives of the thermodynamic
potential, e.g. the free energy, are discontinuous at the transition temperature.
The size of the discontinuity is unimportant for this classication and can be ar-
bitrary small or large in dierent systems. The transition is called continuous if
the second or higher-order derivative of the thermodynamic potential exhibit a
discontinuity.
Onsager predicted in 1949 a rst order isotropic to nematic phase transition of
a system of innitely thin rods in three dimensions[143]. The phase transition is
driven purely by the shape of the particles without the need of additional interac-
tion potentials. This transition is well understood and experimentally conrmed.
If one connes such a system between two plates, the rst order phase transition
gets lost as the distances between the two plates get narrower and the system
becomes quasi two-dimensional. The phase behavior of a two-dimensional system
is quite dierent from the three-dimensional case. The Mermin-Wagner theo-
rem states that there is no long-range order in one or two-dimensional systems
with short-ranged interactions between the particles[134]. This rigorous result can
be extended to more complicated long-ranged potentials as shown by Bruno[25].
Nevertheless, phase transitions still exsist, namely between an isotropic phase
and a phase with long-ranged correlations which decay algebraically. In a two-
dimensional system the nematic phase has no common director in the thermody-
namic limit satisfying the Mermin-Wagner theorem. One can dene the nematic
director only locally for nite regions around every particle. Therefore the phase
is called quasi-nematic. The nature of those phase transitions is of the KTNHY
type[82,110,220]. Of special interest for our simulation of rods conned on a
two-dimensional plane is the universality class of the two-dimensional XY model
of freely rotating spins on a lattice. In this system the isotropic to quasi-nematic
phase transition is continuous. Kosterlitz derived the critical exponents for this
model quite early[109] but the simulation of this particular model turned out to be
rather complicated and has been done more recently [58,85]. In our simulations
the long-ranged orientational correlations lead to a strong system size dependence
of the the isotropic to quasi-nematic phase transition.
32
3.3 Order parameters
3.3 Order parameters
Order parameters are a vital tool to investigate the dierent states of matter
and their phase transitions. In our system of rods conned to a plane in a qua-
sicrystalline substrate several order parameters are necessary to describe its phase
behavior. In the following we present the order parameters in detail.
We measure the existence of a nematic phase with the nematic order parameter
S
which is dened as
S=N−1h
N
X
i=1
cos(2θi)i,
(3.1)
where
N
is the particle number and
θi
is the angle of the
i
th rod with the nematic
director.
Figure 3.2: Rodlike particle with an length
L
and the attached orientational unit
vector
−→
u(i)
.
In practice, it is more convenient to use the orientational tensor order parameter.
For a conguration consisting of
N
rods it is dened as
Qab =N−1
N
X
i=1
[2ua(i)ub(i)−δab],
(3.2)
where
−→
u(i) = [u1(i), u2(i)]
is a unit vector indicating the direction of rod
i
as
illustrated in Fig 3.2. Since
Qaa = 0
, the two eigenvalues of
Qab
,
Q
and
−Q
add
up to zero and the nematic order parameter is the ensemble average
S=hQi
over
the positive eigenvalue of
Qab
.
Under the inuence of the substrate potential the rods can form additional
phases. They can be characterized by a bond-orientational or directional ordering
in the plane. To describe a directional order where the rods are aligned along
33
3 Nematic liquids
the symmetry direction of the substrate we have introduced the directional order
parameter for
m−
fold order :
Φm=hϕmiwith ϕm=N−1
N
X
j=1
eimαj
(3.3)
where
αj
is the angle of the rod
j
with respect to an arbitrary axis.
Figure 3.3: The decagonal directional order parameter
Φ10
measures whether the
angles
α
of all particles with with respect to an arbitrary axis match onto a
10−
fold
rotational symmetric star.
In our decagonal substrate the important directional order is along the
10
sym-
metry directions of the substrate. To measure the occurrence of this directional
order, we use
Φ10
as displayed in Fig. 3.3. Like the nematic order parameter the
directional order parameter measures only the existence but not the orientation
of an ordered phase. One can obtain the orientation by taking the phase of the
complex sum in Eq(3.3) despite of the absolute value. We note that this phase
can also be measured if the system has a very low directional order which slightly
prefers the symmetry directions.
Another kind of order observed in our system is a bond-orientational order of
the rod centers. We measure this order quantitatively with the bond orientational
order parameter which reads for a
m−fold
order :
Ψm=hψmiwith ψm=N−1
N
X
j=1
n−1
j
nj
X
k=1
eimθjk
(3.4)
where
θjk
is the angle of a bond between particle
j
and one of its
nj
nearest neigh-
bors
k
with respect to an arbitrary reference direction, see Fig. 3.4. The nearest
34
3.3 Order parameters
Figure 3.4: The bond orientational parameter uses the angle of a bond
θjk
between
the particle
j
and its neighbor
k
with respect to an arbitrary axis.
neighbors are determined with the help of a Voronoi tessellation [167,180]. At
high potential strengths the rods form clusters. We dene such clusters by the
conditions that all nearest neighbors
j
of each particle
i
within a cluster stay be-
low a maximum distance of
|ri−rj|<0.4L
where
L
is the length of the particle
and in addition have a maximum dierence in their orientation of
|αi−αj|<0.2
.
We tested dierent values for the denition of the clusters. The cluster structure
turned out to be robust against changes. For each cluster one can dene its posi-
tion
rC
by the center of mass and a orientation
αC
by its director. We can dene
the order parameters for the clusters as before and denote the directional order
parameter of the clusters
ΦCm
and the cluster bond orientational order parameter
ΨCm
respectively. We are interested in the
10−
and
20−
fold bond orientational
order of the particles and the clusters. We name the bond orientational order of
a system after the bond orientational parameter with the highest value.
The localization of a phase transition by studying the relevant order parameter
can be a dicult task. It is helpful to take a look at the uctuations of the
order parameter instead of the order parameter itself. The uctuations become
maximal at a phase transition and even show critical behavior in the vicinity
of second-order phase transition [7,190]. The uctuations of the nematic order
parameter is known as susceptibility, which is dened as the variance with respect
to
S
[20,142],
χS=βN(hQ2i−S2)
(3.5)
with
β= 1/kT
. In the same way, we dene the variance for uctuations around
35
3 Nematic liquids
the directional order parameter and the bond orientational order parameter,
χΦ=βN(hϕ2
mi−Φ2
m)
(3.6)
χΨ=βN(hψ2
mi−Ψ2
m)
(3.7)
Furthermore, we also look at the specic heat capacity, which is connected to
uctuations in the energy
E
of a rod conguration
c=β2
N(hE2i−hEi2).
(3.8)
Another very useful tool to investigate the structure of a system are correlation
functions. Of particular interest are the spatial distribution functions. The radial
pair correlation function
g(r)
gives the probability of nding two particles having
a distance of
r
normalized to the probability of nding two particles with the same
distance in a system with the same density but perfect random positions of the
particles[6]. It is an important quantity since it can be also measured in scattering
experiments.
g(r)
can be determined in an ensemble average:
g(r) = 1
2πrρN h
N
X
i
N
X
i6=j
δ(r+|−→
ri−−→
rj|)i
(3.9)
where
−→
ri
,
−→
rj
are the positions of the center of mass of the rods and
N
is the
number of particles in the system. In our system not only the radial distribution is
of particular interest but the correlations in two dimensions. The two dimensional
correlation can be measured with respect to dierent coordinate frames. We use
the frame of the simulation box:
gR(x, y) = 1
ρN h
N
X
i
N
X
i6=j
δ(x+ (xi−xj))δ(y+ (yi−yj))i
(3.10)
where
(xi, yi)
and
(xj, yj)
are the coordinates of the rods with respect to the coor-
dinate system of the simulation box. Sometimes the choice of a the local frame of
each rod is more illuminating. In the pair correlation function of the local frame
gR(rk, r⊥)
, the axes of the local frame are oriented parallel and perpendicular to
the director of the rod.
We also calculate the radial orientational correlation function:
gS(r) = h
N
X
i
N
X
i6=j
δ(r+|ri−rj|) cos(2(αi−αj))
δ(r+|ri−rj|)i
(3.11)
36
3.4 Particle models
Figure 3.5: Left: Snapshot of a hard-needle system in the isotropic regime Right:
Needles in a quasi-nematic phase
The orientational correlations can also be dened in two dimensions in the frame
of the simulation box
gS(x, y)
and in the local frame of each rod
gS(rk, r⊥)
as
dened for the spatial correlation functions before.
3.4 Particle models
3.4.1 Hard needles
Hard needles are particles with a nite length but without any lateral extension.
The aspect ratio of the needle length to diameter is
L/D =∞
. The needles
interact with each other by a hard core repulsion potential
VI
only. They are not
allowed to cross each other that means
VI=∞
if they cross and
VI= 0
otherwise.
Without any substrate the system was rst investigated numerically by Frenkel
and Eppenga[65]. The isotropic to quasi-nematic phase transition takes place
at a number density of
ρ≈71
L2
. In the quasi-nematic phase the orientational
correlations decay algebraically. In computer simulations, the position of the
phase transition and the value of the nematic order parameter strongly depend
on the size of the simulation box. In our simulations the transition density is at
ρ≈61
L2
and the maximum value of the order parameter is between
0.8< S < 0.9
.
The critical exponents of the phase transition were investigated numerically by
Vink[209]. They turned out to be consistent with the XY model. In Fig. 3.5
37
3 Nematic liquids
Figure 3.6: Hard spherocylinder with an aspect ratio of
L/d = 10
.
two snapshots from simulations of the isotropic and the quasi-nematic phase are
shown.
3.4.2 Hard spherocylinders
In contrast to the hard needle model hard spherocylinders exhibit a nite aspect
ratio. In two dimensions they consist of a rectangular rod with the diameter
d
and a length
L
with two half circular caps at each end. The diameter of the cap
circles is also
d
. In Fig. 3.6 a spherocylinder with an aspect ratio of
L/d = 10
is
shown.
The phase behavior of the spherocylinders strongly depend on the aspect ratio.
The model was investigated numerically rst by Bates and Frenkel [13]. For
low aspect ratios with
L/d < 7
the rods exhibit only an isotropic phase for low
densities and a solid phase at high densities. More slender rods with higher aspect
ratios exhibit also a low density isotropic phase and a high density solid phase but
at intermediate densities a quasi-nematic phase emerges. The properties of the
isotropic to quasi-nematic phase transition are the same as for the hard needle
model. The nature of the solid phase is still unknown. It looks almost smectic but
it is very dicult to perform computer simulations in this density regime. In our
simulations we stay at densities well below the transition to the solid phase. We
use an aspect ratio of
L/d = 10
to be able to simulate a system which possesses a
nematic phase. At this aspect ratio the isotropic to quasi-nematic phase transition
is expected to be at an area fraction of
η≈0.5
. In our simulations, the actual
position of this phase depends again on the size of the simulation box. In Fig. 3.7
two systems of spherocylinders are shown in an isotropic and quasi-nematic phase
38
3.5 Rod - substrate interaction potential
Figure 3.7: Left: Snapshot of a system of hard spherocylinders in the isotropic
regime Right: Hard spherocylinders in a quasi-nematic phase
respectively.
3.5 Rod - substrate interaction potential
In this section we present the interaction of the rods with the substrate potential.
We display the structural features of our substrate potential and choose the length
of the rods. The substrate potential derived in Eq.(2.3) reads for a decagonal
symmetry:
V(−→
r) = −V0
25
4
X
j=0
4
X
i=0
cos[(−→
Gi−−→
Gj)·−→
r].
(3.12)
with the wave vectors
−→
Gi
pointing to the vertices of a pentagon as shown in Fig.
2.3. The particles interact with the substrate by averaging the potential over their
full length
L
:
VR(−→
r , −→
u) = 1
LZ1/2
−1/2
V(−→
r+lL−→
u)dl .
(3.13)
The potential
VR
depends on the choice of the length
L
. We dene the mini-
mal potential energy of a rod at a given position
−→
r
with respect to all possible
orientations
−→
u
of the rod:
VM(−→
r) = min
∀−→
u(VR(−→
r , −→
u))
(3.14)
Fig. 3.8 displays
|VM|
for dierent choices of
L
. For
L= 0
the substrate potential
39
3 Nematic liquids
Figure 3.8: Maximum strength of the potential
|VM|
for dierent rod lengths
L
.
40
3.5 Rod - substrate interaction potential
from Eq. (3.12) is recovered. If
L < 1aV
the particles are shorter than the typical
distance of two potential minima. Therefore positions of the minima in
VM
are
the same as for
V
. Very short particles behave like point like particles despite
averaging over a small length along the substrate potential. At
L= 1aV
the po-
sitions of the deepest minima change and the minima of
VM
are located between
two minima of the bare substrate potential
V
. The minima exhibit a elongated
shape in contrast to the minima in the substrate potential
V
which possess a
circular shape. The orientation of the long axis of the elliptical shaped minima is
identical to the particle orientation with the lowest energy at this position. At a
length of
L= 3aV
a pattern appears which consists of long lines of low potential
energy connecting the minima. Because of the averaging in the denition of
VR
the
deepest minima of the resulting potential become shallower with increasing length
of the particles. The depth of the deepest minima decreases from
Vmin =−1
for
L= 0
to
VRmin =−0.38
for
L= 5aV
.
In the following we choose two lengths for our particles. The length of the short
particles is
L= 1aV
and the length of the long particles is
L= 3aV
.
Restricting the orientation to one of the symmetry directions, one can recognize
a line pattern connecting all potential minima for both particle lengths. In Fig.
3.9 the orientation of the particles points along the
X
-axis. In the middle panel
the projection of the potential of such aligned particles onto the
Y
-axis is shown.
The projection onto an axis perpendicular to the orientation of the particles is
independent of the particle length. One can identify characteristic distances be-
tween lines connecting the potential minima also seen in the distribution of the
maxima in the projected potential. The distances between the lines of the deepest
minima follow a Fibonacci sequence with a mirror symmetry at the center of the
coordinate system. Taking also lines of shallower minima into account the Fi-
bonacci sequence decomposes into two interwoven Fibonacci sequences of smaller
distances which possess the same symmetry around the origin of the coordinate
system as the Fibonacci sequence we have obtained by the projection method
shown earlier in Fig. 2.6. We present a more through investigation of the origin
of these line pattern in the Appendix of this work.
From the structure of
VM
, we can derive the bond orientational order of the
41
3 Nematic liquids
Figure 3.9: The potential
|VR|
for a particle which is oriented along the direction of
the
X
-axis. Left panel for
L= 1
. Right panel for
L= 3
. In the middle panel the
projection
V∗(Y)
of both substrate potentials onto the
Y
-axis is shown together
with the Fibonacci sequences which are dened by the distances between lines of
deep minima.
42
3.5 Rod - substrate interaction potential
Figure 3.10: Bond orientational order parameter for the positions of the substrate
minima for both length scales.
minima. The values of the bond orientational order parameter
Ψ10
are shown
in Fig. 3.10. For the calculation of
Ψ10
we determined
≈20000
positions of
the minima of
VM
. For both particle lengths the bond orientational order is
quite weak. The short particles have a slightly higher bond orientational order of
Ψ10 = 0.28
compared to
Ψ10 = 0.26
for long particles. At every minimum position
of the potential
VM
there is only one rod orientation for which the potential
VR
exhibit this minimum. The rod orientations of every minima are aligned along
one of the symmetry directions of the potential. If we divide the minima positions
into groups of minima with the same orientation, the bond orientational order
parameter of the minima in each group is signicantly higher as for the sum of all
minima. The bond orientational order parameter of every single group is for both
particle lengths
Ψ10 ≈0.75
.
43
3 Nematic liquids
44
4 Computational methods
In this chapter we present the basics of statistical mechanics necessary for our
simulations and give the details of the algorithms used by our simulations.
4.1 Canonical ensemble
In statistical mechanics we characterize a macroscopic thermodynamic system by
the weighted averages obtained from a complete set of its microscopic realizations[201].
We address the sets of microstates together with their statistical weights as en-
sembles. The weights depend on the interaction of the system with a reservoir
and the ensembles are named accordingly. In our simulations we use the canonical
ensemble. The canonical ensemble is described by its number of particles
N
its
volume
V
and its temperature
T
. One can imagine such a system as a box with
a xed size in equilibrium in a heat bath with which it exchanges thermal energy
to stay at a xed temperature. We are interested in rods on a plane. Let
−→
ri
be the coordinate of the
i−
th particle,
−→
ui
a unit vector pointing in the direc-
tion of its long axis and
−→
pi
its momentum and
−→
li
the angular momentum. The
Hamiltonian for this system is given by:
H(−→
r , −→
p , −→
l) =
N
X
i
−→
p2
i
2+
N
X
i
−→
l2
i
2I+
N
X
i
N
X
j
V
pair
(−→
rij,−→
ui,−→
uj)+
N
X
i
V
substrate
(−→
ri,−→
ui)
(4.1)
where
I
is the moment of inertia,
V
pair
is the pair potential and
V
substrate
is the
substrate potential. With the notation
dΓ = 1
h3NN!drdpdl
(4.2)
the partition function can be written as:
Z=ZdΓe
−H(Γ)
kT
(4.3)
45
4 Computational methods
Where
k
is the Boltzmann constant and
h
is Planck's constant.
It is an important feature of the partition function that one can split the function
into the congurational parts and the kinetic parts[75].
Z=ZKZRZV
(4.4)
ZK
is the translational kinetic term,
ZR
the rotational kinetic term and
ZV
the
potential part. The integration of the dierent parts of the partition function can
be done separately. The integral over the translational momenta can be carried
out to:
ZK=VN
N!Λ2N
(4.5)
where
Λ
the thermal de Broglie wavelength is given by:
Λ = h
√2πmkT
(4.6)
The rotational kinetic part reads:
ZR=1
ΛN
IN
mN
(4.7)
The congurational part of the partition function is independent of the dynamics
of the system. The Monte Carlo simulation takes only the congurational part
into account. The translational and rotational kinetic part of the partition func-
tion add only as a prefactor. In the following we denote by
Z
the congurational
part of the partition function only.
We want to derive the ensemble average of static properties in the canonical
ensemble[19,142]. Let
wµ(t)
be the probability to nd a discrete state
µ
in the
system at a given time
t
. If we are interested in a quantity
Q
which is dened for
every state the expectation value is :
hQi=X
µ
Qµwµ(t)
(4.8)
In equilibrium the probability of nding a state
µ
is not time dependent, that is
dwµ
dt = 0
. For clarity, we denote the time independent
wµ
by
pµ=wµ(0) = wµ(t)
.
The value of the equilibration probabilities in a canonical ensemble are well known:
pµ=1
Ze−Eµ/kT
(4.9)
46
4.2 Monte Carlo simulation
The expectation value
hQi
now reads:
hQi=P
µ
Qµe−Eµ/kT
Z
(4.10)
4.2 Monte Carlo simulation
In a Monte Carlo simulation we create a sample of congurations which is chosen
in a way to be representative to the whole thermodynamic ensemble. We generate
the new congurations with a Markov chain. If a system is in a given state
µ
it has
the transition probability
P(µ→ν)
to move from this state to another state
ν
.
The measurement of a quantity
Q
would then become an average over the visited
states of the system. To generate a representative sample of a canonical ensem-
ble in equilibrium, the transition probabilities have to satisfy certain conditions.
The condition of ergodicity means that every possible state is part of the innite
Markov chain. The condition of detailed balance reads:
pµP(µ→ν) = pνP(ν→µ)
(4.11)
The total probability of being in state
µ
and visiting state
ν
is the same as the
probability to be in the state
ν
and going to the state
µ
. These conditions are
necessary to generate an equilibrium distribution. To guarantee that the equilib-
rium distribution is a Boltzmann distribution, the transition probabilities have to
satisfy:
P(µ→ν)
P(ν→µ)=e−β(Eµ−Eν)
(4.12)
We have still a lot of freedom for an algorithm which implements those conditions.
The main focus of an algorithm is therefore to create a good sample of states in
the most computational ecient way. The most popular and common Monte
Carlo algorithm is the Metropolis algorithm. In this algorithm a small change is
introduced in a given system for every simulation step. The transition probability
is split in two parts:
P(µ→ν) = g(µ→ν)A(µ→ν)
(4.13)
g(µ→ν)
is the selection probability to chose the new state
ν
when the system is
in the state
µ
.
A(µ→ν)
is the probability that the change is accepted and the
47
4 Computational methods
new state of the system is state
ν
. If the change is rejected the system stays in the
state
µ
. The ratio between the trials and the accepted changes is the acception
ratio. The rules to perform a change in the system satisfy the condition of detailed
balance. The selections probabilities have to fulll:
g(µ→ν) = g(ν→µ)
(4.14)
The information about the equilibration distribution is in the acceptance condi-
tion:
A(µ→ν) = min{e−β(Eµ−Eν),1}
(4.15)
In our simulations of rodlike particles the algorithm is implemented as follows.
In every simulation step a random particle is chosen. The position
−→
rN
and angle
αN
of the particle at step
N
are changed going to step
N+ 1
with :
−→
rN+1 =−→
rN+r0Γ−→
e
αN+1 =αN+α0(1 −2Γ)
Γ
is a random number between
0≤Γ≤1
with a at distribution.
−→
e
is a
unit vector with a random orientation. The values of
r0
and
α0
determine the
maximum change in position and angle of the particle. The change in angle and
position is made independent of each other to obtain the acception ratios of each
congurational change separately. In an additional step a change in position and
angle is made simultaneously to overcome energy barriers. The maximum values
of the step sizes
r0
and
α0
are adjusted to a mean acceptance ratio of
hAi ≈ 0.5
.
To help the simulation to resolve locked states in a more ecient way, we set the
maximum step sizes to
r0=L
with
L
is the lengths of a rod and
α0=π
at every
100th
simulation step. Those steps have a very low acceptance ratio but account
for a higher diusion of the system through the conguration space.
4.3 Kinetic Monte Carlo
Monte Carlo simulations turned out to be quite ecient in systems with a com-
plicated energy landscape to obtain the static properties of an thermodynamic
equilibrated system. This high eciency raised the question whether it is also
possible to use the algorithm to get informations about the dynamical properties.
It turned out that in the case of an overdamped Brownian systems it is possible to
48
4.3 Kinetic Monte Carlo
use the Monte Carlo scheme with some minor changes only[103]. First we want to
recall the term Brownian dynamics and than show how to apply the Monte Carlo
scheme to this kind of system.
4.3.1 Brownian motion
The Brownian particles are placed in a surrounding medium with which it is in
a thermal equilibrium[44,131]. The particles interact with each other with an
interaction force
−→
F
int
. The system can be inuenced by an external eld
−→
F
ext
.
The inuence of the medium is due to a friction force
−→
F
fric
and a random thermal
force
−→
F
therm
. The Langevin equation for a particle then reads:
md
dt−→
v=−→
F
fric
+−→
F
ext
+−→
F
int
+−→
F
therm
(4.16)
where
m
is the mass of the particle and
v
is its velocity. Regarding a one particle
system without external forces only the friction force and the thermal force have
to be taken into account. The friction force is proportional to the velocity of the
particle
−→
F
fric
=−γ−→
v(t)
, e.g. for a spherical particle according to Stoke's law
γ= 6πRη
, where R is the radius of the particle. The thermal force
−→
F
therm
has
a average of zero because the total system is at rest without a preferred drift
direction. The components of the force are independent of each other. Both
properties of the thermal force lead to[86]:
hF
therm
,i(t)i= 0
(4.17)
hF
therm
,i(t), F
therm
,j(t0)i= 2Bδijδ(t−t0)
(4.18)
The Langevin equation is:
md
dt−→
v+γ−→
v=−→
F
therm
(4.19)
The solution of this dierential equation can be written as[83,149]:
−→
v(t) = −→
v(0)e−γ
mt+e−γ
mt
t
Z
0
dt01
meγ
mt0−→
F
therm
(t0)
(4.20)
We now can take a look at the mean square velocity given by:
−→
v(t)2=−→
v(0)2e−2γ
mt+e−2γ
mt
t
Z
0
dt0
t
Z
0
dt00 1
m2eγ
m(t0+t00 )D−→
F
therm
(t0)·−→
F
therm
(t00)E
(4.21)
49
4 Computational methods
After integration, we derive in the long time limit
tm/γ
:
−→
v(t)2=nB
mγ
(4.22)
where
n
is the number of dimensions. From the equipartition theorem we know
that
mh−→
v(t)2i=nkT/2
and therefore the strength of the uctuations are related
to the temperature:
B=γkT
(4.23)
The particle diuses with time. To quantify this diusion process we take a look
at the mean square displacement (MSD) in the long time limit:
|−→
r(t)−−→
r(0)|2=1
γ2
t
Z
0
dt0
t
Z
0
dt00 D−→
F
therm
(t0)·−→
F
therm
(t00)E
(4.24)
=2nB
γ2t
(4.25)
= 2nDt
(4.26)
The last equation is the denition of the diusion constant
D
and it follows:
D=B/γ2
(4.27)
In the following we are not interested in the diusion of a spherical particle
but of elongated particles like spherocylinders. The diusion of such a particle
is not isotropic anymore[35,93]. If we dene a local coordinate system which is
xed to the symmetry axis and the center of each particle the diusion process
can be divided into a diusion parallel to the long axis of each particle with a
diusion constant
Dk
and perpendicular to the long axis with a dierent diusion
constant
D⊥
. In addition the particle can have an angular diusion relative to the
resting reference frame of the experiment dened by an angular diusion constant
DR
. In the case of spherocylinders we use the following approximate diusion
coecients[125]:
Dk=D0
2π(ln p−0.207 + 0.980/p −0.133/p2)
(4.28)
D⊥=D0
4π(ln p+ 0.839 + 0.185/p + 0.233/p2)
(4.29)
DR=3D0
πL2(ln p−0.662 + 0.917/p −0.050/p2)
(4.30)
50
4.3 Kinetic Monte Carlo
p
is dened by
p= 1+L/d
where
L
is rod length and
d
is the diameter of the rod.
The diusion coecients
Dk
and
D⊥
can be measured via the MSD in the co-
moving frame:
|rk(t)−rk(0)|2= 2Dkt
(4.31)
|r⊥(t)−r⊥(0)|2= 2D⊥t
(4.32)
The total translational diusion coecient is then:
D0= 0.5(Dk+D⊥)
(4.33)
The mean square rotational displacement is connected to the angular correlation
function :
|−→
u(t)−−→
u(0)|2= 2(1 −e−DRt)
(4.34)
where
−→
u(t)
is the unit vector oriented along the long axis of the particle. In the
case of small simulation steps the angle
α
with respect to an arbitrary xed axis
shows the same diusive behavior like the translational variables:
|α(t)−α(0)|2= 2DRt
(4.35)
A mean square displacement proportional to the time is usually found in sim-
ple systems such as a single particle without any external eld for times much
longer than the time scale of the uctuations of the solvent. Interactions between
particles, high densities and external elds can lead to a MSD
∝tν
with
ν6= 1
.
Those systems possess anormal diusion. A coecient
ν < 1
is called subdiusive
and a coecient
ν > 1
is superdiusive. A special case of superdiusivity is the
ballistic motion with
ν= 2
. In the MSD one can distinguish the dierent time
scales in which dierent kinds of diusion occur. In a typical glassy system one
can recognize a ballistic motion at very short time scales, which correspond to the
mean free path of the particles, a subdiusive motion at intermediate time scales
where the particles are caged by its neighbors, and a long time normal diusion
for time scales greater than the mean caging time of a particle.
4.3.2 Simulation scheme
One usually uses the Brownian dynamics simulation scheme to solve the Langevin
equation. In Brownian dynamics simulations the Langevin equation is discretized
51
4 Computational methods
and the computer solves the time evolution of the system. In such a simulation the
calculation of forces can be the most time consuming part of the whole simulation.
In particular for non-spherical particles which exhibit anisotropic friction forces
and torques the computational eort to calculate one single time step can slow
down a simulation. In a Monte Carlo simulation no forces have to be calculated at
each simulation step. The Monte Carlo simulation takes only the potential energy
into account. As a result a kinetic Monte Carlo simulation can be computation-
ally much faster than a Brownian dynamics simulation. In this section we want to
present the derivation of the kinetic Monte Carlo scheme and give the simulation
details of our implementation.
To illustrate the functionality of the kinetic Monte Carlo scheme we consider a
particle in an one-dimensional system[89,90,103]. The particle is at the position
x(t)
within a potential
V(x)
. The Monte Carlo step chooses a new coordinate of
the particle within an interval
[x−δx, x +δx]
with a at probability distribution.
The potential is only slowly varying over the step interval so we can write the
maximum change in potential energy
±∆V≈ ±Fδx
with an approximate con-
stant force
F
. We consider the potential to increase with
x
. In this way we only
x the direction of the movement. A change of position into the negative direction
decreases the potential energy. The move is therefore always accepted. A step
into the positive direction will be accepted with a probability of
e−β∆V≈e−βF δx
only. The mean displacement can be written as:
h∆xi=
0
R
−δx
xdx +
δx
R0
xe−βF xdx
δx
R
−δx
dx
(4.36)
We can now expand the exponential in the next two orders and integrate:
h∆xi=
δx
R0
(e−βF x −1)xdx
2δx
(4.37)
≈
δx
R0
(−βFx2+ 0.5β2F2x3)dx
2δx
(4.38)
≈βF δx2
6(1 −3βFδx
8+O(δx2))
(4.39)
52
4.3 Kinetic Monte Carlo
The mean displacement in one time step is proportional to the strength of the
force
F
corresponding to a drift of the particle. We can do the same expansion
for the mean square displacement of one Monte Carlo step :
∆x2=
δx
R0
(e−βF x −1)x2dx
2δx
(4.40)
≈
2/3δx2−βF
δx
R0
x3dx +o(δx4)
2δx
(4.41)
≈δx2
3(1 −3βFδx
8+O(δx2))
(4.42)
The MSD is independent of
F
in the order of
δx2
. Comparing the MSD with the
diusion equation one can derive an equation for the size of the time step:
∆x2=δx2
3= 2D0δt
(4.43)
δt =δx2
6D0
(4.44)
The corrections of the order
δx3
are the same in the drift (Eq. 4.39) and in
the MSD (Eq. 4.42). In simulations in more than one dimensions the higher
order corrections can be approximated by the acceptance ratio
A
[166]. The nal
denition of the time step in our simulation is:
δt =Aδx2
6D0
(4.45)
This procedure is straightforward for the isotropic diusion of a spherical particle.
However, a rod has dierent diusion coecients parallel and perpendicular to
the rod axis and also performs rotational diusion. Furthermore, the acceptance
ratios for the dierent Monte Carlo steps within a Monte Carlo cycle can be
dierent[148,162]. Equation (4.45) then introduces three dierent time steps
∆tk
,
∆t⊥
, and
∆tR
for one cycle. To be consistent, they have to be the same.
We therefore use as reference time step the diusion time parallel to the rod axis,
∆t0= ∆tk=r2
kAk/(6Dk)
, and adjust the maximum step sizes for the other degrees
of freedom so that
∆tk= ∆t⊥= ∆tR
. This procedure results in the maximum
step sizes
r⊥=sAkD⊥
A⊥Dk
rkand rR=sAkDR
ARDk
rk.
(4.46)
53
4 Computational methods
4.4 Wang Landau Monte Carlo
Monte Carlo simulations can lose eciency if the system possesses a rough energy
landscape with high energy barriers or in a system which is at the vicinity of a
phase transition. There are many techniques to overcome high energy barriers
but they are not always sucient[113]. The so called Wang Landau Monte Carlo
simulation method[114,210,211] is a quite recent approach to use the Monte
Carlo scheme to derive the density of states (DOS)
g(E)
where
E
is the internal
energy of a state rather then to sample an ensemble at a xed temperature. The
algorithm was rst used for lattice systems but can applied to o lattice systems
as well[154,178,226]. The acceptance criterion of the Monte Carlo simulation is
modied to:
A(µ→ν) = min{g(Eν)
g(Eµ),1}
(4.47)
With this acceptance ratio every state is visited with an equal probability. The
density of states is the result of the simulation and is initially not known. We
have to start with an arbitrary rst guess of the shape of
g(E)
. One usually starts
with histogram for
g(E)
of a at distribution giving every state the same weight,
e.g.
g(E) = 1
. During the simulation every visited state changes the DOS by
multiplying
g(E)
with an constant factor
f > 1
. The more often a state is visited
the larger the corresponding
g(E)
and the less probable is a further visit of the
same state. We record in a second histogram
H(E)
the absolute number of visits
of every state. As soon as the shape of the initial
g(E)
converges to the real
g(E)
the histogram of the number of visits becomes at. The nite value of
f
determines the strength of the uctuations of
g(E)
and is therefore an upper limit
of the accuracy of the obtained DOS. On the other hand a small value of
f
slows
down the convergence of
g(E)
and wastes a lot of computation time. To deal with
this problem one split the simulations into many single runs with dierent values
of
f
. In the rst simulation the modication factor
f1
is chosen to be
f1>1
.
The simulation is done until a reasonable atness in the histogram for the visits
of states is reached. After that a second simulation is started with the outcome of
g(E)
from the previous simulation as the initial shape of the DOS. The next sim-
ulation has a modication factor of
f2=f0.5
1
. Again the simulation is performed
until the histogram of the visited states is suciently at. The scheme is repeated
and the modication factor is converging
fn→1
for
n→ ∞
. One can stop the
simulation when the desired accuracy of the shape of
g(E)
is gained[224].
54
4.4 Wang Landau Monte Carlo
In our simulations we use a renement of this iteration scheme. We do not use
many simulations with xed
fn
but slowly lowering
f(τ)
with increasing simulation
time
τ
. If the modication factor scales down with
f∼1+1/τ
the convergence
time can get shorter and we reach
g(E)
with a higher accuracy[15,16,17,225].
The exact values of the start factor
f
and the timescale
τ
are subject to a ne
tuning which have to be done for every individual system separately. The edges
of the histograms are dened by the energy range we are interested in. To avoid
boundary eects trial moves which lead to congurations outside the preferred
energy range were treated like discarded moves from the acceptance criterion[172].
With the DOS we can calculate the congurational part of the partition func-
tion:
Zconf =X
conf
e−βE =X
E
g(E)e−βE
(4.48)
Because we modify in the simulation scheme only the shape of
g(E)
its absolute
value diers by a constant factor from the real
g(E)
. We still can get a lot of
information from this by normalizing this equation. We get the statistical weight
of the states with a given temperature and a given energy by:
P(E, T) = g(E)e−E
kT
Rg(E)e−E
kT dE
(4.49)
We can record in a simulation run the distribution of any observable as a function
of the energy
O(E)
and get its mean value at a given temperature with:
hOiT=ZP(E, T)O(E)dE
(4.50)
In particular, the internal energy
U
can be obtained by:
U(T) = hEiT=ZP(E, T)EdE
(4.51)
In some systems it is possible that the complex congurational landscape is not
only a function of the energy but also of additional parameters like an order
parameter. We can extent the density of states
g(E)
to a joint density of states
(JDOS) as a two dimensional function of a parameter
O
and the energy that
reads
g(E, O)
. The modication of the simulation scheme is straight forward.
55
4 Computational methods
The histograms for
g(E, O)
and the histogram of visits
H(E, O)
become two
dimensional. The acceptance criterion modies to:
A(µ→ν) = min{g(Eν, Oν)
g(Eµ, Oµ),1}.
(4.52)
We obtain the statistical weight with respect to only one of the variables by
integrating over the other:
P(E, T) = Rg(E, O)e−E
kT dO
RRg(E, O)e−E
kT dEdO
(4.53)
P(O, T) = Rg(E, O)e−E
kT dE
RRg(E, O)e−E
kT dEdO
(4.54)
This can be extended in principle to any number of variables. Because of the high
computational costs most of the Wang Landau simulations are done for only one
or two variables. We use the JDOS Wang Landau simulation with the energy and
the nematic order parameter
S
to obtain an expression for
g(E, S)
.
4.5 Simulation details
In our simulation we use periodic boundary conditions [83,136]. Since the qua-
sicrystalline potential is not periodic, discontinuities at the boundaries of the
periodically repeated simulation boxes occur. To minimize the discontinuities, we
choose special box sizes following Ref. [167]. For the decagonal potential the edge
lengths of the simulation box have to be
X= 2naV
and
Y=maV/sin(π/5)
,
where
n
and
m
are Fibonacci numbers. The periodic repetition of quasicrystalline
subpatches leads to a crystalline structure called a crystalline approximant[72].
Since we x the box sizes to discrete values, we vary the particle number to realize
dierent densities.
As initial conditions we choose two dierent particle congurations. The rst
consists of a random distribution of both the particle positions and orientations.
With such isotropic starting conditions one can study whether the particles are
able to build up nematic order. In the second conguration the positions of the
particles are randomly distributed but the they all align along an arbitrarily chosen
common direction. Starting with such an ideal nematic order, we investigate how
stable the nematic phase is.
56
4.5 Simulation details
Details of hard needle simulations
Most of our simulations of the needle system are performed in simulation boxes
with
n= 3
,
m= 5
for short needles and
n= 13
,
m= 21
for long needles. The
particle number varies between
232
and
654
for the short needles and between
331
and
886
for the long needles to realize an appropriate density range. The limita-
tions on the possible box sizes also makes it dicult to perform nite size analysis.
The next larger box size for the long needles needs
776
needles at the lowest and
2318
at the highest simulated density. Because of the very long computation time,
we performed a search for nite-size eects only for two densities of the two needle
lengths. We conrmed results from Ref. [65] that in the two-dimensional needle
system the position of the isotropic-to-quasi-nematic transition depends on the
system size. However, for the onset of the decagonal directional order, which is
mainly determined by the substrate potential, we do not nd a nite-size eect.
In the following, we measure the needle density
ρ
in units of the square of the
needle length,
1/L2
. In this way, the isotropic-nematic phase transition always
occurs at the same value of the reduced density
ρ
independent of needle length
L
.
To equilibrate the system, we use a few
105
Monte Carlo sweeps at low densities
and low potential strengths up to a few
106
sweeps for high densities and high
potential strengths. One Monte Carlo sweep consists of the number of particles
times a single Monte Carlo cycle for each particle. At least
10
simulation runs
with independent initial conditions are performed for each initial condition and
each density.
We performed Wang landau simulations to derive
g(E, S)
. The Wang Landau
simulations are performed in systems of box sizes with
n= 3
,
m= 5
for short
needles and with
n= 13
,
m= 21
for long needles. The number of short needles
varied between
232
and
403
. Two densities of the long-needle system with
623
and
886
particles have been investigated. For every particle density
6
Wang Landau
simulations are performed in parallel. After
≈107
simulation sweeps the resulting
histograms for
g(E, S)
were averaged. The obtained average
g(E, S)
was used as
an initial shape for the density of states in the following simulation runs. This
simulation scheme was repeated until the modication factor
f
reached
f= 1.0005
for the short needles and
f= 1.0025
for the long needle system. The Wang Landau
57
4 Computational methods
simulations give the resulting order parameter as functions of the temperature
T
of the system for a constant potential strength. Because the interaction with the
potential is the only energy scale in our system, we can transform the results of
the Wang Landau simulation for a given temperature into the results of a system
with a constant temperature but a given potential strength.
Details of hard spherocylinder simulations
For the short-rod system we use a box size with
n= 8
,
m= 13
and a particle
number between
100
and
637
to realize low densities and a box size with
n= 5
,
m= 8
with a particle number between
245
and
776
to realize high densities. The
long-rod system is simulated in a box with
n= 34
,
m= 55
and particle numbers
varying between
229
and
1431
. All particle densities are quantied by the area
fraction
η
which the rods occupy relative to the total area.
To equilibrate the system, we use at least
106
Monte Carlo sweeps for the low
densities and
3·106
sweeps for the high densities. For the short-rod system we per-
form at least ten independent simulation runs at every density, potential strength
and initial condition. For the long-rod systems the number of runs is at least four.
When we perform kinetic Monte Carlo simulations, we use the normal Monte
Carlo scheme to equilibrate the system. Then we start the kinetic Monte Carlo
scheme where we rescale the maximum step sizes until the time steps
tR
and
t⊥
converge to
tk
. One Monte Carlo sweep goes consecutively through all the parti-
cles to avoid unphysical double moves or stops of one rod. The sequence of the
particles is altered randomly between the sweeps.
The step size
rk
should be well below the smallest characteristic length scale
one wants to resolve. In our case, it is given by the diameter of the rod
d
. We
therefore set
rk<0.1D
but also veried that
rk<0.01d
does not change our
simulation results. Our choice of
rk<0.1d
is well suited to resolve the dynamics
of the rod on a time scale it needs to diuse a diameter
d
.
To obtain the single particle diusion we average over
2500
particle trajecto-
ries. The MSD of the rods in an ensemble is an average over at least
7
dierent
58
4.5 Simulation details
simulation runs.
59
4 Computational methods
60
5 Hard needles
In this chapter we show our results of the investigation of the hard-needle model.
First, we start with the short-needle system and then we present our investigation
of the long-needle system. At the beginning of every section we give a short
introduction of the phase diagram and thereafter we explain the dierent phases
in detail. Most results of this chapter are published in [A].
5.1 Short needle system
5.1.1 Phase diagram
The phase diagram of the short needles in Fig. 5.1 can be divided into four regions.
The most important transition separates surface-induced directional or decagonal
order at high substrate strength from a region at low or zero strength. Here the
substrate does not inuence the typical phase behavior where below a density
of
ρ≈5.9
we observe an isotropic phase followed by the quasi-nematic phase
as already reported for this system size by Frenkel and Eppenga [65]. Above
the main transition line also two regions exist. In the low-density region a pure
decagonal phase exists without any nematic order. In the region at high densities,
the realized ordering depends on the initial condition. In particular, it is possible
to freeze in a starting conguration with nematic order in addition to the surface-
induced decagonal order. Now, we give a detailed account of our results.
5.1.2 Decagonal directional order
In Fig. 5.2 we plot the decagonal order parameter
Φ10
and its susceptibility
χΦ
as a function of the potential strength
V0
. The maximum of the susceptibility
coincides with the inection point of the decagonal order parameter at a value
of
Φ10 ≈0.2
. We nd this behavior at all simulated densities. Therefore, we
61
5 Hard needles
Figure 5.1: Phase diagram for the short needles.
let the decagonal directional phase of the needles start at
Φ10 ≈0.2
. Figure 5.3
shows the decagonal order parameter versus
V0
for dierent densities. In the short-
needle system, all these curves look similar and there is no pronounced density
dependence. Only for low densities, decagonal ordering needs a higher potential
strength to develop. Accordingly, the transition line in the phase diagram 5.1
slightly bends upwards at low densities. The minimum potential strength for
observing the decagonal phase is about
V0= 15
at high densities. Despite the
clear maximum in the susceptibility, we do not observe a maximum in the heat
capacity at the same position in the phase space. This is reminiscent to the work
of Frenkel and Eppenga [65]. They only observed a weak maximum in the heat
capacity shifted against the actual transition from the isotropic to the nematic
phase. For comparison a innite dilute system
ρ= 0
is also shown in Figure
5.3. The directional order of such a dilute system is substantially lower than for
higher densities. This can be understood by the dierent mechanism of directional
ordering.
The snapshots of Fig. 5.5 show the short-needle system at a density
ρ= 8.3
. In
the left panel the substrate potential has a strength of
V0= 10kT
just below the
phase transition. The system is in a nematic phase but one can already recognize
a weak modulation of the density of the needles induced by the substrate. The
modulation leads to bundles of needles within the nematic phase. In the right
62
5.1 Short needle system
Figure 5.2: Figure 5.3:
Left: Decagonal directional order parameter
Φ10
and its susceptibility
χΦ
at
ρ=
8.3
.
Right: Decagonal directional order parameter
Φ10
for dierent densities.
Figure 5.5: Snapshot of a short needle system at a density of
ρ= 8.3
. Left : at a
potential strength of
V0= 10kT
, right :
V0= 40kT
63
5 Hard needles
Figure 5.6: Pair correlation function
g(r)
for short needles at a density of
ρ= 8.3
for dierent potential strengths
V0
.
panel of Fig. 5.5 the strength of the substrate potential is
V0= 40kT
. The
needles are conned to clusters at the minima positions of the substrate. The
needles in a cluster are parallel and are ordered along the symmetry directions
of the substrate. The pair correlation function for
V0= 40
in Fig. 5.6 shows the
dense packing within the clusters through the large rst maximum very close to
r= 0
. One recognizes the isolated clusters by the deep and broad rst minimum
at
r= 0.5
. The pair correlation function exhibit no pronounced maximum at
V0= 10kT
but one can identify a weak density modulation in comparison with
the substrate free case
V0= 0kT
. We conclude that the formation of clusters
strongly enhance the directional order through a local nematic phase within every
cluster. The weak density dependence in the simulated range shown in the phase
diagram stems from the minimum potential strength to conne a single needle at
a minimum. Finally, we note that the proles for the decagonal order parameter
in g. 5.3 do not change by increasing the number of needles from approximately
600 to 1200. So there is no size dependence. This is in agreement with the fact
that the decagonal order is due to the local values of the substrate potential.
5.1.3 Nematic order
The isotropic to quasi-nematic transition has already been discussed extensively[65,
209]. We now discuss the transition from the nematic phase into the region with
surface-induced directional order for increasing
V0
. Figure 5.7 plots the nematic
64
5.1 Short needle system
Figure 5.7: Figure 5.8:
Left: Nematic order parameter
S
as a function of the potential strength
V0
for
dierent densities
ρ
.
Right: Nematic susceptibility
χS
as a function of the potential strength
V0
for
dierent densities
ρ
.
order parameter prole for dierent densities. The nematic order rst decreases
slowly until a potential strength of about
V0= 18kT
where the surface-induced
decagonal order sets in. Now, local needle clusters form that isolate the needles
against each other. This leads to a sharp drop of the order parameter to
S≈0.2
and nematic order vanishes. So the loss of nematic order is strongly correlated
with the appearance of decagonal directional order. The nematic susceptibility
plotted in Fig. 5.8 for various densities indicates the loss of nematic order with a
pronounced maximum at the transition line. For comparison the nematic suscep-
tibility in the isotropic phase at
ρ= 4.5
does not exhibit such a maximum. The
fourth region in the phase diagram of the short needles in Fig. 5.1 is named frozen
initial conguration. The plot of the nematic order shown in Figure 5.7 results
from simulations which started with a random distribution of both the needle po-
sitions and orientations. After the equilibration of the system we nd a decagonal
orientated phase where the needle clusters oriented along the symmetry direc-
tions of the substrate potential with an equal probability for each direction. In
the region of frozen initial congurations situated at high potential strengths and
high densities the outcome of the simulation depends on its initial conguration.
When we prepare the initial conguration to be in a nematic phase and there-
65
5 Hard needles
Figure 5.10: Figure 5.11:
Left: Snapshot of a short needle system at a density of
ρ= 8.3
and a potential
strength of
V0= 60kT
with a frozen nematic order of
S≈0.1
.
Right: Snapshot of a short needle system at a density of
ρ= 8.3
and a potential
strength of
V0= 60kT
with a frozen nematic order of
S≈0.6
.
fore exhibit an common director the resulting system remains in a nematic order
after equilibration. We have conrmed this behavior by doubling the simulation
time normally needed for equilibrating the system. Two snapshots of simulations
with dierent nematic order are shown in Fig. 5.10 and Fig. 5.11. In Fig. 5.10
in initial isotropic initial conguration is frozen and as a result the nematic order
parameter is
S≈0.1
. One can recognize a ower like structures of the clusters
which is reminiscent of the overlapping decagons of potential minima in the sub-
strate potential. Fig. 5.11 shows a system with a high nematic order parameter
of
S≈0.6
. The needle clusters are ordered onto lines following the Fibonacci
line structure of the potential. In Figure 5.13 the dierent results of the nematic
order parameter for the dierent initial congurations at a density of
ρ= 8.3
are
66
5.1 Short needle system
Figure 5.13: Figure 5.14:
Left: Nematic order parameter
S
as a function of the potential strength for
isotropic and nematic starting conguration respectively and the decagonal di-
rectional order parameter
Φ10
at a density of
ρ= 8.3
.
Right: Nematic order parameter
S
as a function of the potential strength for
dierent densities
ρ
from nematic initial starting congurations.
shown. Below a potential strength of
V0= 36kT
the results are independent of
the initial conguration. Needle clusters with the same orientation order along
the Fibonacci lines as expected from the structure of the substrate potential (Fig.
3.9). In between needle clusters with dierent orientation occur. In our simula-
tions the order parameter never exceeds
S≈0.7
since between clusters oriented
along neighboring Fibonacci lines always clusters with dierent orientations can
be inserted. Therefore, in a frozen nematic state we always observe at least two
of the ve possible cluster orientations and the nematic order is never perfect. In
Fig. 5.14 the dependency of the frozen nematic order on the density is shown. The
higher the density the lower the potential strengths necessary to freeze an initial
nematic order of the system. At the highest simulated density of
ρ= 12.7
there
is almost no gap of low nematic order between the region of the nematic system
and the region of the frozen initial congurations. The direct transition from a
nematic to a frozen nematic system is accompanied by a very low maximum in
the nematic susceptibility as already shown in Fig. 5.8.
For a further investigation of the nature of the frozen nematic order we per-
formed Wang Landau simulations. A comparison of the Wang Landau results
67
5 Hard needles
Figure 5.16: Figure 5.17:
Left: The nematic order parameter
S
at a density of
ρ= 8.3
as a function of the
potential strength
V0
in a comparison between the Wang Landau MC simulation
and the Metropolis MC scheme.
Right: Distribution of the nematic order parameter
p(S)
at a density of
ρ= 8.3
and a potential strength of
V0= 60kT
for dierent simulations.
with the results from the Metropolis Monte Carlo scheme are plotted in Fig. 5.16
for
ρ= 8.3
. The nematic order derived from the Wang Landau simulation is in
good agreement with the Metropolis simulations. The nematic order decreases
with increasing potential strength. In the region of the frozen nematic congura-
tion the Wang Landau simulations prefer a low nematic order of
S≈0.3
which is
higher than the frozen isotropic conguration but much lower than the maximum
frozen nematic order. In Fig. 5.17 the distribution of the nematic order parameter
at a potential strength of
V0= 60kT
for the dierent simulations is shown. The
peak structure indicates frozen systems with a dierent degree of nematic order.
In the region of frozen initial conguration, the energy of the system, the decago-
nal order, and the heat capacity do not depend on the degree of nematic ordering.
So, congurations with dierent frozen nematic order just seem to constitute dif-
ferent possible realizations of the same decagonal order which corresponds to a
highly degenerate ground state.
68
5.1 Short needle system
Figure 5.19: Figure 5.20:
Left: Two dimensional pair correlation function
gR(x, y)
for a short needle system
with a density of
ρ= 4.5
and a potential strength of
V0= 60kT
.
Right: Two dimensional order correlation function
gS(x, y)
for a short needle
system with a density of
ρ= 4.5
and a potential strength of
V0= 60kT
.
5.1.4 Positional and Bond-orientational order
At high potential strengths the needles are well ordered in clusters at the positions
of the potential minima. The two-dimensional pair correlation function
gR(x, y)
in Fig. 5.19 shows the quasicrystalline order. The two-dimensional orientational
correlation function
gS(x, y)
of the same system is shown in Fig. 5.20. Figure 5.20
displays a central spot surrounded by 10 red spots which indicate directions in
space along which clusters assume the same orientation as the central cluster. In
between, the blue spots give directions with perpendicular orientation. The pat-
tern in Fig. 5.20 displays the same decagonal symmetry and positional order as
the substrate potential. Therefore, parallel needle clusters exhibit the same long-
range positional and orientational order as the substrate potential. The relative
positions of non-parallel clusters can be identied as peaks in the pair correlation
function in Fig. 5.19 which do not match any red areas in the orientational corre-
lation function in Fig. 5.20. The denition of the positions of the clusters at the
vicinity of the transition into the substrate induced order is dicult. The system
undergoes huge uctuations which is indicated by the huge standard deviation of
the number of clusters found in the simulated systems at
V0<20kT
shown in Fig.
69
5 Hard needles
Figure 5.22: Figure 5.23:
Left: Absolute number of clusters
N
in the simulation box and its standard devi-
ation
σN
as a function of the potential strength
V0
at a density of
ρ= 8.3
Right: Decagonal bond-orientational order parameter
ΨC10
for the center of mass
of the clusters as a function of the potential strength
V0
at a density of
ρ= 8.3
for nematic and isotropic initial congurations.
5.22. Therefore, we dene the bond-orientational order of the cluster centers
ΨC10
only from potentials strengths above
V≈20kT
. The bond-orientational order of
a system with
ρ= 8.3
is shown in Fig. 5.23 for the dierent initial congurations.
In the region of the frozen initial conguration the bond-orientational order of the
nematic systems is signicantly higher than for a frozen isotropic order. In this
region a high nematic order is coupled with a high bond-orientational order.
5.2 Long needle system
5.2.1 Phase diagram
The phase diagram of the long needles in Fig. 5.25 can be divided into ve regions
and exhibits pronounced dierences compared to the short-needle system and its
phase behavior in Fig. 5.1. The most important division line marks again the onset
of surface-induced decagonal order with
Φ10 >0.2
for increasing substrate strength
V0
. However, whereas for short needles this line is more or less horizontal, it now
tilts towards smaller
V0
when density
ρ
increases. Below the decagonal transition
70
5.2 Long needle system
Figure 5.25: Phase diagram of the long needle system
line, one observes again the isotropic and quasi-nematic phase with the transition
located at
ρ≈6.2
for the simulated system size. Interestingly and in contrast
to short needles, the transition at
ρ≈6.2
extends beyond the main decagonal
transition line to larger
V0
, where now three dierent phase regions exist. At
densities below
ρ≈6.2
the system assumes pure decagonal order without any
nematic ordering. At densities larger than
ρ≈6.2
a phase with both nematic
and decagonal order exists up to a substrate strength of
V0≈35kT
. In the short-
needle system, such a phase does only occur in a very narrow region of
V0
. For
ρ > 6.2
and
V0>35kT
, the starting conguration again freezes in. Now, we
describe the phase behavior in more detail and try to explain it.
5.2.2 Decagonal directional order
We rst discuss the decagonal ordering of the long needles as illustrated by the
decagonal order parameter
Φ10
plotted versus
V0
for several densities in Fig. 5.26.
Most properties of
Φ10
are the same as in the short-needle system. The maximum
of the susceptibility
χΦ
when plotted as a function of
V0
occurs again when the
decagonal order parameter assumes the value
Φ10 = 0.2
and the heat capacity does
not show any maximum at this position. However, in contrast to short needles, the
potential strength
V0
necessary to induce decagonal ordering strongly decreases
71
5 Hard needles
Figure 5.26: Figure 5.27:
Left: Decagonal directional order parameter
Φ10
as a function of the potential
strength
V0
for dierent densities
ρ
.
Right: Decagonal directional order parameter
Φ10
as a function of the density
ρ
for dierent potential strengths
V0
.
with increasing density. In Fig. 5.27 the order parameter is plotted against the
density. We understand such a behavior qualitatively. With increasing
V0
, short
needles tend to form compact clusters of the size of one needle length when they
connect two potential minima. The clusters are well separated from each other
regardless their density. In contrast, long needles connect several mimima and
even share one or two of them. Now, clusters with the same orientation form
elongated domains which have a length equal to several needle lengths (Fig. 5.29).
The widths of the domains oriented along one of the decagonal directions also
extend beyond one needle length since geometrically it is simpler to align clusters.
For larger densities, we expect such domains to form more easily which explains
the behavior of the decagonal transition line.
5.2.3 Nematic order
At low densities
ρ < 6.2
the needles show surface-induced decagonal order without
any nematic ordering. Similar to Fig. 5.29 the needle clusters form aligned domains
that are equally distributed in all 10 decagonal directions. Above
ρ= 6.2
the
formation of the needle clusters does not destroy nematic order since the clusters
72
5.2 Long needle system
Figure 5.29: Snapshot of a long needle system at a density of
ρ= 8.6
and a
potential strength of
V0= 40kT
.
overlap with each other as explained in the previous paragraph. As a result the
phase region with both stable nematic and decagonal order in the phase diagram
of Fig. 5.25 occurs. Figure 5.30 demonstrates that for each density the nematic
order parameter is nearly constant as a function of
V0
. One recognizes a slight
increase when the decagonal order is established and a decrease beyond
V0= 40kT
in the region of frozen initial conguration. Finally, in the region termed frozen
initial conguration in the phase diagram of Fig. 5.25, the mobility of the needles
is so small that the system is not able to change an initial conguration. Like
short needles, long needles are able to freeze in an initial nematic order. However,
if the simulation starts without any orientational order, nematic ordering does not
develop during equilibrating the system. Figure 5.31 shows how the nematic order
parameter
S
depends on the starting conguration. The energy, heat capacity,
and decagonal order are the same whether the system freezes in the nematic or
isotropic state. In the isotropic system needle clusters are aligned within domains
the orientations of which are distributed equally on all ten decagonal directions.
In Fig. 5.31 the nematic order parameter in the simulations with isotropic
start congurations does not drop immediately but decreases between potential
strengths of
V0= 40kT
till
V0= 60kT
. We believe this is an artifact of the
73
5 Hard needles
Figure 5.30: Figure 5.31:
Left: Nematic order parameter
S
as function of the potential strength
V0
for dif-
ferent densities
ρ
.
Right: Comparison of the nematic order parameter for dierent initial congura-
tions as a function of the potential strength
V0
at a density of
ρ= 8.6
.
Figure 5.33: Distribution of the nematic order parameter at a density of
ρ= 8.6
and a potential strength of
V0= 60kT
obtained from dierent simulation schemes
and initial conditions.
74
5.2 Long needle system
simulation scheme. In the Monte Carlo simulation scheme, the equilibration run
corresponds to a fast cooling of the system. Just above the limit potential strength
of
V0= 40kT
the system can increase its nematic order while equilibrating. This
results in an increased nematic order above
S= 0
even if started with an isotropic
conguration. The nematic order drops with increasing potential strength until
nally the equilibration is fast enough to perfectly freeze in an initial non-nematic
order at
V0= 60kT
. Artifacts of the equilibration run also freeze in when start-
ing in an initial nematic order. Fluctuations of the nematic phase freeze in for
V0>40kT
. In the equilibration run the uctuations are introduced in two dier-
ent ways. The rst source of uctuations is the initial isotropic positional order of
the system. The simulation scheme creates uctuations while nding the proper
minima of the needles. The second source of uctuations are the thermal uctu-
ations of needles around their director which are also usually seen in the case of
zero potential strength. The strength of the uctuations of the substrate free case
are the upper limit for the simulation scheme at high potential strengths and the
nematic order drops to the nematic order of the substrate free system.
A comparison of the results at a density of
ρ= 8.6
and a potential strength of
V0= 60kT
between Monte Carlo simulations with dierent initial nematic order
and the Wang Landau simulations is shown in Fig. 5.33. In contrast to the short
needle system the Wang Landau simulation results favor a high nematic order
of the long needles in the region of frozen initial congurations. In the common
nematic - decagonal directional ordered phase the nematic director is oriented
along one of the symmetry direction of the substrate. In Fig. 5.34 we show a
typical needle snapshot of the combined nematic and decagonal order at
ρ= 8.6
and
V0= 18
. One clearly recognizes an average direction of the needles along the
director, which points along one of the decagonal directions. The single needle
orientations uctuate around the director. The uctuations of the orientations of
the needles with respect to the nematic director prefer the neighboring decagonal
directions of the potential. One can identify such uctuations in the snapshot
Fig. 5.34a) as bundles of needles. The corresponding orientational distribution
function of the needles is plotted in Fig. 5.34(b). Besides the orientation of the
director at
α= 2π/5
, two weaker maxima appear at the neighboring decagonal
directions at
α=π/5
and
3π/5
. Increasing density, these maxima become weaker
in agreement with the increasing nematic order parameter
S
. All three maxima
75
5 Hard needles
Figure 5.34: a) Snapshot of a long needle system at a density of
ρ= 8.6
and a
potential strength of
V0= 18kT
in a nematic and decagonal directional ordered
phase. b) Corresponding orientational distribution function for the needles. The
angle
α
is measured with respect to the horizontal.
76
5.2 Long needle system
Figure 5.35: Figure 5.36:
Left: Pair correlation function for the center of mass of the long needles at a
density of
ρ= 8.6
for dierent potential strengths
V0
.
Right: Two dimensional pair correlation function with respect to the axis parallel
rk
and perpendicular
r⊥
to the director of each long needle at a density of
ρ= 8.6
and a potential strength of
V0= 60kT
.
become sharper when
V0
increases restricting the needles more and more to the
decagonal directions of the substrate potential. In the short needle system the
needles form isolated clusters and therefore the angular correlation is nite and
the eect of directional enhancement is conned to the needles within a cluster. In
the long needle system the nematic phase remains uid. Despite the restriction of
the system to a nite number of directions the phase may be still quasi-nematic.
5.2.4 Positional order
In the vicinity of the transition to the decagonal directional order the positional
order of the long needles is not as pronounced as for the short needles. In Fig.
5.35 the pair correlation function for dierent potential strengths of a dense sys-
tem with
ρ= 8.6
is shown. The growth of rst maximum denotes the creation of
clusters under the inuence of the potential. The maximum stays below
1
even
at a potential strength of
V0= 20kT
which is already in the region of decagonal
directional order. The maximum becomes sharper with higher potential strengths
77
5 Hard needles
Figure 5.38: Snapshot of the long-needle system at a density of
ρ= 8.6
and
a potential strength of
V0= 60kT
in the region of frozen initial conguration.
Blow-up: One-dimensional quasicrystalline positional order of the needle clusters
on two Fibonacci chains.
and the line structure of clusters begin to form. For a large substrate strength of
V0= 60kT
this is illustrated in Fig. 5.38. The needles form again clusters which
are mostly aligned along one decagonal direction. Some needle clusters deviate
from the nematic director and point along other decagonal directions reducing the
nematic order parameter below
S= 1
. Still the directional order parameter indi-
cates decagonal ordering. The blow-up of one region of the snapshot in Fig. 5.38
reveals that the positions of the needle clusters possess one-dimensional quasicrys-
talline order perpendicular to the nematic director. This leads to a pronounced
line structure in the two-dimensional pair correlation function as displayed in Fig.
5.36. The pair correlation function
gR(rk, r⊥)
shows the positional correlations
of the needles center of mass with respect to the coordinate system whose axes
are parallel to the long and short axes of each individual particle. The order is
characterized by the two interwoven Fibonacci chains, which we identied in the
substrate potential as illustrated in Fig. 3.9.
78
6 Hard spherocylinders
In this chapter we discuss the simulation results for the short- and long-spherocylinder
system, separately. We present the phase diagram and then discuss more details of
the phase ordering. In the last section, we show the results from the kinetic Monte
Carlo simulations for the mobility of the rods. Most of our ndings presented in
this chapter are published in [B].
6.1 Short-spherocylinder system
6.1.1 Phase diagram
The phase diagram of the short rods in Fig. 6.1 exhibits several phases which
dier by their bond-orientational, their directional, and orientational order. Be-
low the horizontal black line at a potential strength
V0≈30kT
, the system of
spherocylinders displays the usual phase sequence isotropic-nematic with increas-
ing area fraction
η
which one observes without any substrate potential. With
increasing
V0
the isotropic-nematic phase transition shifts to larger
η
. Above the
potential strength of
V0≈30kT
the phase diagram is divided into regions A-D
with dierent bond-orientational order to be discussed below. In addition three
characteristic density ranges exist. At very low area fractions of
η < 0.12
(re-
gion A and B) the single rods display surface-induced bond-orientational order as
well as pronounced directional order with non-zero order parameter
Φ10
. In the
regime of intermediate area fractions
0.12 < η < 0.35
(regions C an D) the rods
form well separated clusters under the inuence of the surface potential. Now,
the clusters display bond-orientational and directional order similar to the single
rods in the dilute regime. At large area fractions of
η > 0.35
the clusters touch
each other. While the directional order parameter falls below
Φ10 = 0.2
, still a
delicate directional order is observable as we explain below. In particular, in the
gray shaded region the preferred directions of the rods lie between the symmetry
79
6 Hard spherocylinders
Figure 6.1: Phase diagram of the short rods system.
directions of the surface potential. Finally, at area fractions above
η= 0.19
and
suciently large potential strength
V0
, an initial isotropic state or nematic order
remains after equilibration. We now discuss the dierent regions in more detail.
6.1.2 Bond-orientational order
The maximal uctuations in the bond-orientational order parameter occur already
at about
Ψm≈0.1
, so we identify bond order for
Ψm>0.1
. In Fig. 6.2 we plot
bond-orientational order parameters for rods (
Ψ10
,
Ψ20
) and the center of mass of
rod clusters (
ΨC10
,
ΨC20
). Below an area fraction of
η= 0.12
most rods are well
separated from each other and only a few clusters exist. At very low area fractions
η < 0.08
and large potential strengths
V0>60kT
a
20
-fold bond-orientational or-
der dominates (region A in Fig. 6.1). It occurs since the system is so dilute
that nearest neighbors also occupy minima which lie in directions between the 10
symmetry directions of the substrate potential.
10
-fold bond-orientational order
forms at lower potential strengths down to
V0= 40kT
and for area fractions up
to
η= 0.12
(region B).
For
η > 0.08
most of the minima in the rod potential
VR
are occupied and
clusters of rods start to form. This is already visible in Fig. 6.3 at an area frac-
80
6.1 Short-spherocylinder system
Figure 6.2: Bond orientational order parameter of a
10−
fold and
20−
fold symme-
try for the center of mass of rods and the center of mass of clusters of rods versus
area fraction at a potential stength of
V0= 80kT
.
Figure 6.3: Snapshots (left) and 2D pair correlation functions of the center of
mass of the clusters (right) for the short-rod system at
V0= 100kT
and
η= 0.08
in region B.
81
6 Hard spherocylinders
Figure 6.4: Snapshots (left) and 2D pair correlation functions of the center of
mass of the clusters (right) for the short-rod system at
V0= 100kT
and
η= 0.19
in region C.
tion
η= 0.08
. The rods form a pattern of decagonal ower structures and the
sharp maxima in the pair correlation function indicate the long-range positional
order with decagonal symmetry induced by the surface potential. At area frac-
tions above
η= 0.12
the number of rods well exceeds the number of deep minima
in
VR
. Within one cluster two or more rods start to occupy the same minimum
in
VR
or, dierently speaking, they connect two to three minima in the substrate
potential
V(−→
r)
. These clusters behave now like single rods in the very dilute
regime. When the cluster density is low (region C in Fig. 6.1), they exhibit
20
-
fold bond-orientational order as indicated by
ΨC20 >ΨC10
in Fig. 6.2 in the range
η= 0.12
to
0.19
.
Then, further increasing the area fraction
η10
-fold bond-orientational order
dominates in region D which extends to large densities. In Fig. 6.4 we show
the snapshot of a rod system at
η= 0.19
and
V0= 100kT
together with the
pair correlation function of the center of mass of the clusters. As in the single-rod
case, the pair correlation function for the clusters shows sharp maxima. The bond-
orientational order of the clusters decays quite slowly with increasing area fraction
(Fig. 6.2). The 10-fold order parameter
ΨC10
stays above
0.1
for
V0>40kT
and
82
6.1 Short-spherocylinder system
Figure 6.5: Snapshots (left) and 2D pair correlation functions of the center of
mass of the clusters (right) for the short-rod system at
V0= 100kT
and
η= 0.5
in region D.
all simulated area fractions
η > 0.19
.
In Fig. 6.5 we show a high-density system with
η= 0.5
. Even though the clus-
ters are not separated from each other anymore, one can still identify ower-like
structures formed by the densely packed clusters. In these structures, the clus-
ters are oriented more along directions in between the symmetry directions of the
substrate potential, which we will investigate further below. In contrast to the
previous cases, the cluster pair correlation function now displays broader peaks
which we attribute to the following observation. The clusters connect two to three
minima of the substrate potential (for a schematic see Fig. 6.8). This creates a
broad potential well for the rods in which they perform thermal motion resulting
in the broadened peaks.
6.1.3 Decagonal directional order
As in the hard-needle system we nd the maxima of the uctuations of the decago-
nal directional order parameter
χΦ
at an approximate value of
Φ10 ≈0.2
inde-
pendent of the system parameters. Figure 6.6 shows the decagonal directional
order parameter
Φ10
as a function of the potential strength
V0
for dierent area
83
6 Hard spherocylinders
Figure 6.6: Decagonal directional order parameter
Φ10
plotted versus
V0
for dif-
ferent area fractions.
fractions. With increasing
η
decagonal directional order is reduced. Finally, at
η= 0.42
and larger area fractions the directional order parameter
Φ10
always stays
below
0.2
for all simulated potential strengths. However, the substrate potential
still aects the orientations of the rods.
In Fig. 6.7 we plot their full directional distribution function
f(α)
for three
dierent area fractions and potential strengths. At a low area fraction of
η= 0.19
and low potential strength
V0= 26kT
(isotropic phase), the substrate potential
only induces a small modulation of the isotropic distribution. The maxima coin-
cide with the 10 symmetry directions of the decagonal substrate potential. The
directional order parameter stays below
Φ10 = 0.1
. The dierence between max-
ima and minima in
f(α)
grows with increasing
V0
until a directionally ordered
phase with
Φ10 ≥0.2
is established. The locations of the maxima and minima
stay the same. This changes when the area fraction is increased. At
η= 0.35
and
low potential strength
V0= 26kT
, the preferred directions are located between
the 10 symmetry directions. In the phase diagram of Fig. 6.1 the gray shaded
region marks the parameter space where such a shift in the directions occurs. At
the highest simulated potential strength
V0= 90kT
the rod system exhibits di-
rectional order with
Φ10 >0.2
and the preferred directions agree again with the
84
6.1 Short-spherocylinder system
Figure 6.7: Directional distribution function
f(α)
for three area fractions
η
and
three potential strengths
V0
85
6 Hard spherocylinders
Figure 6.8: Single rods and small clusters (left) occupy dierent positions and
directions in the substrate potential than larger clusters (right) which gives rise
to the gray shaded region in the phase diagram of Fig. 6.1 (schematic drawing).
symmetry directions. In dense rod systems with a large area fraction the preferred
directions of the rods remain shifted at large potential strengths
V0
. We illustrate
this in Fig. 6.7 for
η= 0.5
. However, even at
V0= 90kT
, were the maxima are
pronounced does the directional order parameter stay below
Φ10 = 0.2
. Note that
at a potential strength of
V0= 26kT
the rod system is in the nematic phase and
the director aligns along one of the shifted preferred directions. These directions
occur when the size of the rod clusters increases from small clusters with up to
three rods per cluster to larger clusters.
The schematic drawing of Fig. 6.8 illustrates the preferred directions. Single
rods and clusters of up to three rods (Fig. 6.8, left) connect two deep minima of
the substrate potential. However, clusters consisting of four rods and more (right)
connect more shallow minima since they occupy more area which the deep minima
cannot provide. So their preferred directions lie between the symmetry directions
of the substrate potential. The mean energy of a rod in a three-particle cluster is
always below the respective value in larger clusters. So, in the high-density regime
starting at
η= 0.35
the rod system consists of a mixture of small and big clusters
with a rising fraction of big clusters at larger area fractions. Hence, there is no
sharp transition from one set of preferred directions to the other set. Finally, we
nd it remarkable that the gray shaded region in Fig. 6.1 indicating the shifted
preferred directions includes the isotropic and nematic phase, but also regions
with bond-orientational order and with frozen initial conditions. As illustrated in
86
6.1 Short-spherocylinder system
Figure 6.9: Snapshot of a short rod system at
V0= 100kT
and
η= 0.5
with a
frozen nematic order.
87
6 Hard spherocylinders
Figure 6.10: Phase diagram of the long rods.
Fig. 6.9 a certain degree of an initial nematic order can be frozen in. As in the
short needle system the maximum nematic order parameter is limited due to the
distribution of possible cluster positions with a common orientation.
6.2 Long-spherocylinder system
6.2.1 Phase diagram
Figure 6.10 shows the phase diagram of long rods which looks much simpler than
the one for short rods. Below the potential strength
V0= 20kT
, one observes
the typical isotropic-nematic phase transition completely unaected by the sub-
strate potential. Above
V0= 20kT
, spherocylinders with length
3aV
connect
several minima of the surface potential (see Fig. 2.3) and therefore become mainly
oriented along one of the symmetry directions. For this reason, one obtains di-
rectional order with decagonal symmetry that extends along the whole range of
area fractions in contrast to the short-rod system. Figure 6.11 demonstrates that
the directional order parameter plotted versus
V0
only weakly depends on the
area fraction
η
. Since the rods can slide to a certain degree along the minima,
which they occupy, the positional order is not sucient to generate a pronounced
88
6.2 Long-spherocylinder system
Figure 6.11: Decagonal directional order parameter
Φ10
plotted versus potential
strength
V0
for dierent area fractions
η
.
bond-orientational order as observed for short rods. The bond-orientational order
parameter
Ψ
is always below 0.1. Nevertheless, as we will see below, in the pair
correlation function one can identify preferred positions of the rods as dictated by
the substrate potential. Furthermore, rod clusters that form at increasing area
fraction show a delicate short-range order. As in a long-needle system the nematic
phase extends into the region of decagonal directional order. Finally, at higher
potential strengths the initial conguration, for example, a nematic order, can be
frozen in.
6.2.2 Positional order
We now study in more detail positional order with the help of the two-dimensional
pair correlation function which we determine in the local frame of the rod. So we
describe positional correlations of the rod's center of mass with respect to the coor-
dinate system whose axes are parallel to the long and short axes of each individual
rod, respectively. Figure 6.12 shows the snapshot of a low-density system at area
fraction
η= 0.19
. The pair correlation function in the local rod frame exhibits
peaks according to the substrate potential but they are broadened due the motion
of the rods. One also clearly recognizes a stripe pattern parallel to the local rod
on which the peaks lie interrupted by blue bands with low positional correlations.
89
6 Hard spherocylinders
Figure 6.12: Snapshots (left) and 2D pair correlation functions in the local rod
frame (right) for the long-rod system at
V0= 60kT
and
η= 0.19
They belong to the Fibonacci sequence from the substrate potential. However, our
analysis shows that nearest-neighbor minima on this Fibonacci sequence are not
occupied due to thermal motion of the rods so the pronounced stripes occur. At
increasing density clusters of rods form and positional order still exists. Now, the
nearest-neighbor minima are lled up and the pronounced stripe pattern vanishes
as indicated in Fig. 6.13. Unlike short rods, the long rods cannot cluster together
in the same line of potential minima since their diameter approximately equals
the width of the minima. Each rod is trapped in a dierent line of minima. In
the snapshot of Fig. 6.13 one can clearly see spaces between parallel rods. Indeed
the corresponding pair correlation function exhibits a correlation hole between
the excluded volume of the central rod (in white color) and the nearest neigh-
bors. As the relevant maxima indicate, the bond between nearest neighbors is not
along their common short axis but points along the symmetry directions of the
potential at angles
±π/5
relative to the central rod. Hence, the clusters in the
snapshot exhibit their typical rhombic shape. The distance of neighboring rods
is the smallest distance of the Fibonacci sequences associated with the substrate
potential. Compared to short rods, which at comparable area fraction [see Fig.
6.4] form well separated clusters in a close side-by-side conguration, the behavior
of long rods with the same aspect ratio is quite dierent. Increasing the area
fraction further, the whole system is compressed so that the rods start to touch
90
6.2 Long-spherocylinder system
Figure 6.13: Snapshots (left) and 2D pair correlation functions in the local rod
frame (right) for the long-rod system at
V0= 60kT
and
η= 0.44
.
Figure 6.14: Snapshots (left) and 2D pair correlation functions in the local rod
frame (right) for the long-rod system at
V0= 60kT
and
η= 0.57
.
91
6 Hard spherocylinders
Figure 6.15: Snapshots from two systems at an area fraction of
η= 0.57
and a
potential strength of
V0= 90kT
with frozen nematic order. Left: with two main
orientations and
S= 0.85
. Right: Almost perfect nematic order with
S= 0.93
.
each other. This is indicated in the pair correlation function of Fig. 6.14, where
the nearest-neighbor maximum is located at the border of the excluded volume.
Furthermore, the surface-induced positional order has vanished almost completely
since the rods ll the whole space quite uniformly. Partially, they form elongated
clusters by ordering side by side and head to tail. Finally, decagonal directional
order is still very high since the rods strictly align along the symmetry directions.
Frozen nematic states can exhibit very high degrees of nematic order because of
this strict alignment of the rods. In Fig. 6.15 the frozen nematic order is illus-
trated with two snapshots of a system at an area fraction of
η= 0.57
and at a
potential strength of
V0= 90kT
. The rods are aligned along one or two symmetry
directions of the substrate. In contrast to the long needle system the order of the
rods onto the Fibonacci lines of the substrate is lost.
92
6.3 Dynamics of hard spherocylinders
Figure 6.16: Figure 6.17:
Left: Mean square displacement
hr2i
of a single short rod in units of
a2
V
for
dierent potential strengths
V0
. Time is measured in units of
a2
V/Dk
, where
Dk
is
the single-rod diusion coecient parallel to the rod axis [see Eq. (4.28)].
Right: Angular mean square displacement
hϕ2i
and accumulated mean square
displacement parallel (
hr2
ki
) and perpendicular (
hr2
⊥i
) to the rod axis at
V0=
70kT
. The inset shows a snapshot of the rod positions at
t= 500a2
V/Dk
.
6.3 Dynamics of hard spherocylinders
6.3.1 Short spherocylinders
We now present the results of our kinetic Monte Carlo simulations on the mobility
of the rods both for single and multi-particle systems. Figure 6.16 shows the mean
square displacement of a single short rod for dierent potential strengths. At
V0= 10kT
the mean square displacement is nearly the same as without substrate
potential. With increasing
V0
the rod becomes trapped in a pair of deep potential
minima and can only leave them by thermal activation. As a result, a subdiusive
regime develops before the rod performs normal diusion. At
V0= 100kT
the
rod stays trapped in its potential minima and the regime of normal diusion is
not reached within the simulation time. In Fig. 6.17 we plot the angular mean
square displacement
hϕ2i
and the accumulated displacements parallel (
hr2
ki
) and
perpendicular (
hr2
⊥i
) to the rod axis for
V0= 70kT
. Interestingly,
hϕ2i
and
hr2
⊥i
93
6 Hard spherocylinders
Figure 6.19: Mean square displacement of a short-rod system for dierent area
fractions at
V0= 70kT
.
enter the diusive regime at the same time whereas
hr2
ki
becomes diusive more
than two decades later. The reason for this behavior is the following. Whereas
one end of the rod stays trapped in its potential minimum, the other end moves/
rotates into a neighboring minimum like a clock hand by thermal activation. This
motion contributes to both
hϕ2i
and
hr2
⊥i
. To perform a motion parallel to the
rod axis, both ends of the rod have to leave their potential minima and normal
diusion sets in later. This also explains why the total mean square displacement
in Fig. 6.17 follows
hr2
ki
.
The mobility of the short rods changes dramatically when a whole ensemble of
rods is considered. In Fig. 6.19 we plot the mean square displacement at
V0= 70kT
for dierent area fractions. In the dilute regime at
η= 0.06
,
hr2i
behaves similar to
the single-rod system and does not reach normal diusion within the simulation
time. As soon as the rods start to cluster (region C in the phase diagram of
Fig. 6.1), their mobility rises and normal diusion is observed. The outer rods
of a cluster are more weakly bound to the trap and can more easily leave their
potential minima by thermal activation. Nevertheless, the diusion coecient is
much smaller than for free diusion by a factor of ca. 100. Most particles stay
in their potential minima and only a few move from cluster to cluster. Further
94
6.3 Dynamics of hard spherocylinders
Figure 6.20: Figure 6.21:
Left: Mean square displacement
hr2i
of a single long rod in units of
a2
V
for dierent
potential strengths
V0
.
Right: Angular mean square displacement
hϕ2i
and accumulated mean square
displacement parallel (
hr2
ki
) and perpendicular (
hr2
⊥i
) to the rod axis at
V0=
70kT
. The inset shows a snapshot of the rod positions at
t= 500a2
V/Dk
.
increasing the area fraction decreases the mobility again since the space between
clusters becomes more and more crowded.
6.3.2 Long spherocylinders
Compared to short rods, long rods are much more mobile when the substrate
potential is switched on and the mobility is much less aected by the strength
of the potential. The mean square displacement for a single long rod in Fig.
6.20 shows normal diusion at long times even at a potential strength as high as
V0= 100kT
. The subdiusive regime is always quite short. The high mobility
arises from the motion of the rods along the lines of minima. They act as rails along
which the rods can easily slide. To leave a rail, the rod changes its direction and
occupies another rail. This picture is conrmed by Fig. 6.21. Now the accumulated
mean square displacement along the rod axis,
hr2
ki
, drastically exceeds
hr2
⊥i
and
determines translational diusion. The ten symmetry directions for the rails are
clearly visible in the snapshot of Fig. 6.21. Compared to short rods,
hϕ2i
is smaller
95
6 Hard spherocylinders
by a factor of ten. In an ensemble of long rods the usual crowding occurs and in
contrast to the short rods a density-induced mobility enhancement is not visible.
96
7 Conclusion and outlook
We determined the phase behavior of hard rods conned in two dimensions under
the inuence of a decagonal substrate potential. We took two dierent particle
models into account. The hard needle model illustrates the phase behavior of
very slim rods. The inuence of a nite width of the rods is demonstrated using
the hard spherocylinder model. The interaction of the rods with the substrate
strongly depends on the ratio of the length of the rod
L
and the typical length
scale of the substrate potential
aV
. We identied dierent rod - substrate inter-
actions for three dierent rod lengths. If the length of the rod is much smaller
than the length scale of the potential,
L<aV
, the interaction of the rod-like
particle with the substrate does not dier much from the interaction of spherical
particles with the substrate. Rods with a length comparable to the length scale
of the substrate potential,
L≈aV
, can connect two minima of the substrate and
nd the positions of their minimum potential energy in between the minima of the
substrate potential. Rods with a length of a few
aV
can connect more minima at
the same time. With increasing rod length, a grid structure of lines connecting the
potential minima is the most important feature of the quasicrystalline substrate.
The lines are oriented along the symmetry directions of the potential. We could
explain the grid structure from the basic features of an underlying decorated Pen-
rose tiling. In our simulations we took two rod lengths into account. The short
rods have a length of
L= 1aV
. The long rods are three times longer with
L= 3aV
.
In the short-needle system the quasi-nematic order is destroyed with increas-
ing potential strength. The system exhibits directional order where the needles
gradually form disconnected clusters located between two potential minima and
oriented along the symmetry directions of the decagonal potential. As as result,
the needle clusters exhibit the same long-range positional order as the substrate
and their relative orientations also display long-range order. Finally, at suciently
high densities and potential strengths it is possible to freeze in nematic order up
97
7 Conclusion and outlook
to an order parameter of
S= 0.7
. In the region of frozen initial congurations a
high nematic order also results in an increased bond-orientational order.
Long needles tend to connect several potential minima with increasing poten-
tial strength and to form clusters that interact with neighboring clusters. In
contrast to short needles, extended domains of uniformly oriented clusters along
the decagonal directions form. At larger densities the interaction between nee-
dles enforces directional order to set in at lower potential strengths compared
to the short-needle system and to stabilize the nematic phase also in regions of
surface-induced directional order. For densities above the isotropic-nematic phase
transition, the needle clusters position and orient themselves along lines dened by
the potential minima. These lines follow a one-dimensional quasicrystalline order
that is described by two interwoven Fibonacci chains. The eect becomes very
pronounced for large potential strengths, where one can again freeze in nematic
order with any value of the order parameter
S
.
In both systems of hard spherocylinders we observe characteristic positional and
directional order with decagonal symmetry that sets in when the strength of the
surface potential exceeds a threshold value. Short spherocylinders connect two
deep minima and orient along the symmetry directions of the substrate poten-
tial similar to short needles. At low area fractions this enforces directional order
together with 10- and 20-fold bond-orientational order. With increasing area frac-
tion rods form clusters. When the cluster size exceeds three rods, they leave the
deepest minima and connect more shallow minima. This destroys directional order
along the ten symmetry directions. The directional distribution function reveals
a shift by
π/10
for the preferred cluster directions. For the whole range of area
fractions, the pair correlation function shows pronounced positional order induced
by the substrate potential.
Long spherocylinders connect several minima and therefore slide more easily
along their long axis. This results in only weak positional order and precludes any
bond-orientational order for the center of mass of the rods. Pronounced directional
order sets in at lower potential strength compared to the short-spherocylinder sys-
tem. With increasing area fraction, long spherocylinders also cluster but due to
their width they occupy separate lines of minima. Further increase of area fraction
98
compresses the clusters whereby the rods are pushed out of their lines of minima.
Still they stay oriented along the ten symmetry directions and the directional or-
der parameter hardly changes with increasing area fraction. When the clusters
are compressed, the weak positional order vanishes completely.
We have also investigated the mobility of the spherocylinders. The mobility of
short spherocylinders decreases with increasing potential strength since they are
trapped in their pair of minima. They leave this trap by rotating one end into
a neighboring minimum. However, translational diusion is determined by the
hindered mobility along the rod axis. When the rods form clusters at increasing
area fraction, the rod mobility increases drastically since outer rods of the clusters
are more weakly bound so they leave their traps more easily. Long rods can slide
along their lines of minima, therefore their mobility is much less aected by the
substrate potential. Their diusive motion is determined by the sliding while mo-
tion perpendicular to the rod axis is strongly hindered by the substrate potential.
The combination of a hard-rod system, which tends to form a quasi-nematic
phase in two dimensions, and a quasicrystalline substrate potential leads to fasci-
nating patterns of clustered rods especially for large potential strengths. We found
new phases with quasicrystalline directional order and quasicrystalline bond-orientational
order and new fascinating structures which exhibit a nematic and a quasicrys-
talline order at the same time. It would be very interesting to perform experiments
with the help of quasicrystalline light patterns as in Refs. [138,139,169] using
systems of rodlike colloidal particles. The broad diversity of rod-like colloids and
the possibility to tune the characteristic length scales of the substrate potential
should make this experimental setup an ideal testing ground for our ndings.
Another interesting experimental setup for the realization of our ndings are
quasicrystalline atomic surfaces as substrates for monolayers of organic rod-like
molecules of alkenes or aromatic hydrocarbons like pentacene[87]. In atomic sys-
tems it is more dicult to nd substrates and particles with the correct ratio of
length scales than in colloidal systems. First detailed simulations of hexane and
octane have been performed for a quasicrystalline approximant [39,176]. With
growing density an interesting stripe pattern appears in the rst monolayer.
99
7 Conclusion and outlook
It is also appealing to use the resulting two-dimensional rod adsorbate as tem-
plate to build three-dimensional structures and to explore how well the quasicrys-
talline cluster phases extend into the third dimension. The growth of lms on
patterned substrates is the basis for many applications like coatings or electronic
devices from epitaxial overlayers.
In our work we left out the possibility of phasonic defects of the quasicrystalline
potential. Such defects are very common in real quasicrystals and can also be
created with laser elds. A phasonic defect introduces a break and shift in the
Fibonacci line structure of the substrate. Phasonic drifts can initiate fascinating
dynamics of adsorbed particles[111]. A phasonic drift leads to a movement of
the Fibonacci lines of the potential perpendicular to their orientation. Every line
drifts with a dierent velocity depending on the phason mode. That may lead to
locked nematic states of rods placed onto the drifting substrate. Further computer
simulations may reveal a deeper understanding of phasonic defects and drifts of
quasicrystalline structures and its applications.
100
Appendix : Decomposition of the
potential into a tiling
As shown in Fig. 3.9 all deep potential minima can be ordered along straight lines
and the distances of the lines follow two interwoven Fibonacci sequences. The
order of the distances is a hint that there must be a connection of the substrate
potential with the Penrose tiling. We already presented a way to produce decagon
patches of the Penrose tiling called cartwheels[78]. The cartwheel tiling can also
recreated with the overlap rules of the Gummelt decagons[79]. Such a decagon
overlap is shown for a second order cartwheel in the left panel of Fig. .1. In the
right panel of Fig. .1 the kite and darts tiling of a combination of two second
order cartwheels is shown. The two cartwheels are stacked above each other with
the second cartwheel has been rotated by an angle of
180
°
. The resulting double
cartwheel is exhibit a perfect decagonal symmetry except for the inner most cen-
tral part.
If we decorate the Gummelt decagon with a point pattern as shown in the mid-
dle panel of Fig. .2 a the double cartwheel shows a point pattern which is displayed
in the right panel of Fig. .2. In the left panel the corresponding patch of the sub-
strate potential from Eq.(3.12) is shown. The substrate potential is ltered with
a threshold of
V < −0.76
. Comparing the patterns of both patches reveals that
all deep minima in the patch of the potential match positions of the points in the
double cartwheel. The point pattern seems to be not complete for minima which
are close to the threshold. We changed the threshold but there seemed to be no
possibility to create a point pattern that perfectly matches the ltered substrate
potential.
Therefore we change the general approach of using a point pattern as a deco-
ration of the Gummelt decagon to a decagon which takes also the depth of the
101
Appendix : Decomposition of the potential into a tiling
Figure .1: Left: Overlapping Gummelt decagons form a second order cartwheel
tiling. Right : Kites and darts decoration of two second order cartwheels. The
cartwheels are stacked on each other after a rotation of
180
°
building a almost
10−
fold rotational symmetric double cartwheel.
Figure .2: Left: All regions of the substrate potential with a potential strength of
V < −0.76
. Right: Distribution of points from a double cartwheel created from
Gummelt decagons with a point decoration as shown in the middle panel.
102
Figure .3: Left column: Simulation results for the decorations of Gummelt
decagons of three sizes. The edge length of each decagon grows between two
rows with a factor of the golden ratio
τ
starting from the top. Right column: the
corresponding double cartwheel patches from the Gummelt decagons. Deviations
from the real substrate potential are below
1%
.
103
Appendix : Decomposition of the potential into a tiling
potential into account. We dene a decagon shaped patch with a value assigned
to every position inside the patch. We place the decagons in the same way like the
Gummelt decagons with the point decoration to form a double cartwheel struc-
ture with a
10−
fold rotational symmetry. The value at each position in the double
cartwheel is derived by the sum of the values of each decagon contributing to this
position dived by the number of overlapping decagons. The value eld of the
Gummelt decagon can not simply guessed like the point decoration. Therefore,
we derive the eld decoration of the Gummelt decagon numerically. For this pur-
pose the decagon is discretized into a two dimensional data eld. The double
cartwheel is also a discretized data eld with values at each point derived from
the Gummelt decagon overlaps. We dene the energy
E
of the system by the sum
of dierences between the derived values in the double cartwheel and the exact
values of substrate potential. We obtain a perfect match of the substrate poten-
tial with the double cartwheel if the energy is zero
E= 0
. Because of the overlap
structure of decagons, a simple minimum search algorithm for the energy
E
could
produces decagons in locked states with
E > 0
. Therefore, we apply a Monte
Carlo simulation technique to create a Gummelt decagon which produces a pat-
tern matching the substrate potential. In each Monte Carlo step the value of one
randomly chosen point in the Gummelt decagon is altered by a random amount
e0
. Where
e0
is a number randomly chosen from the interval
[−emax, emax]
. Af-
terwards the double cartwheel is build from the new decagon following the given
overlapping rules. Then the energy dierence is calculated and the change of the
decagon patch is accepted with the usual Monte Carlo criterion.
A=min{1, exp(−∆E
kT )}
(.1)
The temperature of the system is lowered slowly to relax the decagon eld to the
absolute minimum dierence to the substrate potential. While doing so,
emax
is
adjusted to keep an acceptance ratio of
a≈0.5
. The Monte Carlo scheme lead
to a decagon decoration which produces a double cartwheel in a very good agree-
ment with the substrate potential. We use two bigger Gummelt decagons where
the lengths of the decagon edges are increased by a factor of
τ
and
τ2
respec-
tively. With the bigger decagons we obtain bigger double cartwheels matching
the corresponding substrate patches. The three Gummelt decagons and the re-
sulting patches are shown in Fig. .3. As a test, we tried to produce similar decagon
patches with a size of the decagon between the small and the medium sized patch.
104
Figure .4: Left : The decoration of the biggest Gummelt decagon obtained by
simulation. Right : The decoration of the biggest Gummelt decagon obtained
from a second order cartwheel made from the smallest Gummelt decagon obtained
by simulation.
The algorithm failed to produce a matching double cartwheel. In this way, we can
be sure that we didn't nd just a general way to produce an arbitrary patch of
potential but a real decomposition of the potential using the basic properties of a
quasicrystalline tiling.
The bigger decagons can be build from deation rules also known for the Gum-
melt decagons. The deation rules for decorated Gummelt decagons are much
more complicated than for the Penrose tiling as shown by Jeong[98]. Two de-
ations of the Gummelt decagon are equivalent to one deation of the Penrose
tiling. Instead using the complicated deation rules of the Gummelt decagon we
take advantage of the properties of the cartwheel tilings. The Gummelt decagon
corresponds to a rst order cartwheel tiling. We know from Fig. .1 how to build
a second order cartwheel from overlapping Gummelt decagons. The second or-
der cartwheel stems from two successive deations of the Penrose tiling or four
deations of the Gummelt decagon tiling. Starting with the Gummelt decagon
decoration displayed in the rst row in Fig. .3 and create a second order cartwheel,
we end up with a double deated decagon of the same size of the Gummelt decagon
displayed at the last row of Fig. .3. In Fig. .4 the Gummelt decagon obtained by
simulation and the Gummelt decagon derived from the deation of the smallest
Gummelt decagon are shown. The similarity of both patches is apparent. With
the deation rules one can produce in principle arbitrary huge patches of the sub-
strate.
105
Appendix : Decomposition of the potential into a tiling
Figure .5: Left : The decoration of the smallest Gummelt decagon with Ammann
bars from the middle-C sequence of the rst order cartwheel. Right : The hori-
zontal Ammann bars (yellow) and the corresponding shifted Ammann bars (red)
from the double cartwheel.
The double Fibonacci sequence found in the substrate is a reminiscence of the
double Penrose tiling as an underlying structure of the decagonal substrate. The
Amman bar decoration of a cartwheel follows a Fibonacci sequence with the sym-
metry of the middle- C sequence[78]. In this tiling, the Ammann bars connecting
the lines of the lowest potential strengths and not the positions of the minima.
The bars connecting the strongest minima follow the Fibonacci sequence of the
projection method. Both Fibonacci sequences are just shifted against each other
by a short distance
S
what we can see in our substrate potential. On the left panel
of Fig. .5 the Ammann bar decoration of the smallest Gummelt decagon is shown
with yellow lines. On the right panel the double cartwheel with the Amman bar
decoration is also displayed with yellow lines for clarity in the horizontal direction
only. The red lines are the shifted Ammann bars which perfectly connect the
minima positions.
106
List of publications
[A] Philipp Kählitz and Holger Stark. Phase ordering of hard needles on a qua-
sicrystalline substrate.
J. Chem. Phys.
,
17
, 174705 (2012)
[B] Philipp Kählitz, Martin Schoen and Holger Stark. Clustering and mobility
of hard rods in a quasicrystalline substrate potential.
J. Chem. Phys.
, accepted
(2012).
107
List of publications
108
List of Figures
2.1 a) the Archimedian tiling
(33,42)
b) quasicrystalline modulated
Archimedian tiling, the distances
S
and
L
between the double tri-
angle rows follow a Fibonacci sequence. . . . . . . . . . . . . . . . 10
2.2 Experimental setup for the generation of a laser eld substrate: a)
Five linear polarized laser beams are focused into a sample cell.
b) The decagonal interference pattern c) Conguration of spherical
colloidal particles exposed to the interference pattern.[139] .... 12
2.3 Star of wave vectors (left) and the resulting substrate potential
(right) created from a) 5 laser beams and b) 3 laser beams. . . . . 14
2.4 Subdivision of the upper box marked
L
into smaller parts using the
deation rule
S→L
and
L→LS
at each step. The fraction of the
two distances always obey
L/S =τ
. The step number is shown on
the left and the number of sub boxes at each step is shown on the
right................................... 17
2.5 First 4 deations of the middle-C sequence. . . . . . . . . . . . . . 18
2.6 Construction of the Fibonacci chain via the projection method. The
central black line has a slope of
1/τ
. The dotted lines denotes the
projectionstrip............................. 19
2.7 a) the tiles of a Girih tiling[37], b) a patch of a Girih tiling[126]. . 20
2.8 The 3 types of a Penrose tiling : a) Pentacles, rhomboids and pen-
tagons, b) kites and darts, c) thick and thin rhomboids from [84] . 23
2.9 a) matching rules for thick and thin rhomboids. b) deation of
thick and thin rhomboids (red) into a smaller tiling. . . . . . . . . 24
2.10 a) decoration of the tiles with Ammann bars b) resulting Penrose
tiling with Ammann bars. . . . . . . . . . . . . . . . . . . . . . . 24
109
List of Figures
2.11 The cartwheel tiling derived by deation rules. a) The central
ace conguration b) the rst order cartwheel c) the second order
cartwheel. ............................... 26
2.12 a) decagon prototile b) overlap rules for four small overlaps A and
abigoverlapB............................. 27
2.13 a) Cartwheel decoration of a Gummelt decagon b) Decoration of
the decagon with a Jack. . . . . . . . . . . . . . . . . . . . . . . . 27
3.1 Schematic drawing of a nematic , smectic and columnar order. . . 30
3.2 Rodlike particle with an length
L
and the attached orientational
unit vector
−→
u(i)
. ........................... 33
3.3 The decagonal directional order parameter
Φ10
measures whether
the angles
α
of all particles with with respect to an arbitrary axis
match onto a
10−
fold rotational symmetric star. . . . . . . . . . . 34
3.4 The bond orientational parameter uses the angle of a bond
θjk
be-
tween the particle
j
and its neighbor
k
with respect to an arbitrary
axis. .................................. 35
3.5 Left: Snapshot of a hard-needle system in the isotropic regime
Right: Needles in a quasi-nematic phase . . . . . . . . . . . . . . 37
3.6 Hard spherocylinder with an aspect ratio of
L/d = 10
. ...... 38
3.7 Left: Snapshot of a system of hard spherocylinders in the isotropic
regime Right: Hard spherocylinders in a quasi-nematic phase . . . 39
3.8 Maximum strength of the potential
|VM|
for dierent rod lengths
L
.40
3.9 The potential
|VR|
for a particle which is oriented along the direction
of the
X
-axis. Left panel for
L= 1
. Right panel for
L= 3
. In
the middle panel the projection
V∗(Y)
of both substrate potentials
onto the
Y
-axis is shown together with the Fibonacci sequences
which are dened by the distances between lines of deep minima. 42
3.10 Bond orientational order parameter for the positions of the sub-
strate minima for both length scales. . . . . . . . . . . . . . . . . 43
5.1 Phase diagram for the short needles. . . . . . . . . . . . . . . . . 62
5.2 Decagonal directional order parameter
Φ10
and its susceptibility
χΦ
at
ρ= 8.3
................................ 63
5.3 Decagonal directional order parameter
Φ10
for dierent densities. . 63
110
List of Figures
5.5 Snapshot of a short needle system at a density of
ρ= 8.3
. Left : at
a potential strength of
V0= 10kT
, right :
V0= 40kT
....... 63
5.6 Pair correlation function
g(r)
for short needles at a density of
ρ=
8.3
for dierent potential strengths
V0
. ............... 64
5.7 Nematic order parameter
S
as a function of the potential strength
V0
for dierent densities
ρ
....................... 65
5.8 Nematic susceptibility
χS
as a function of the potential strength
V0
for dierent densities
ρ
......................... 65
5.10 Snapshot of a short needle system at a density of
ρ= 8.3
and a
potential strength of
V0= 60kT
with a frozen nematic order of
S≈0.1
. ................................ 66
5.11 Snapshot of a short needle system at a density of
ρ= 8.3
and a
potential strength of
V0= 60kT
with a frozen nematic order of
S≈0.6
. ................................ 66
5.13 Nematic order parameter
S
as a function of the potential strength
for isotropic and nematic starting conguration respectively and the
decagonal directional order parameter
Φ10
at a density of
ρ= 8.3
.67
5.14 Nematic order parameter
S
as a function of the potential strength
for dierent densities
ρ
from nematic initial starting congurations. 67
5.16 The nematic order parameter
S
at a density of
ρ= 8.3
as a function
of the potential strength
V0
in a comparison between the Wang
Landau MC simulation and the Metropolis MC scheme. . . . . . . 68
5.17 Distribution of the nematic order parameter
p(S)
at a density of
ρ=
8.3
and a potential strength of
V0= 60kT
for dierent simulations. 68
5.19 Two dimensional pair correlation function
gR(x, y)
for a short needle
system with a density of
ρ= 4.5
and a potential strength of
V0=
60kT
. ................................. 69
5.20 Two dimensional order correlation function
gS(x, y)
for a short nee-
dle system with a density of
ρ= 4.5
and a potential strength of
V0= 60kT
. .............................. 69
5.22 Absolute number of clusters
N
in the simulation box and its stan-
dard deviation
σN
as a function of the potential strength
V0
at a
density of
ρ= 8.3
........................... 70
111
List of Figures
5.23 Decagonal bond-orientational order parameter
Ψ10
as a function of
the potential strength
V0
at a density of
ρ= 8.3
for nematic and
isotropic initial congurations. . . . . . . . . . . . . . . . . . . . . 70
5.25 Phase diagram of the long needle system . . . . . . . . . . . . . . 71
5.26 Decagonal directional order parameter
Φ10
as a function of the po-
tential strength
V0
for dierent densities
ρ
.............. 72
5.27 Decagonal directional order parameter
Φ10
as a function of the den-
sity
ρ
for dierent potential strengths
V0
............... 72
5.29 Snapshot of a long needle system at a density of
ρ= 8.6
and a
potential strength of
V0= 40kT
.................... 73
5.30 Nematic order parameter
S
as function of the potential strength
V0
for dierent densities
ρ
......................... 74
5.31 Comparison of the nematic order parameter for dierent initial con-
gurations as a function of the potential strength
V0
at a density
of
ρ= 8.6
................................ 74
5.33 Distribution of the nematic order parameter at a density of
ρ=
8.6
and a potential strength of
V0= 60kT
obtained from dierent
simulation schemes and initial conditions. . . . . . . . . . . . . . . 74
5.34 a) Snapshot of a long needle system at a density of
ρ= 8.6
and a
potential strength of
V0= 18kT
in a nematic and decagonal direc-
tional ordered phase. b) Corresponding orientational distribution
function for the needles. The angle
α
is measured with respect to
thehorizontal.............................. 76
5.35 Pair correlation function for the center of mass of the long needles
at a density of
ρ= 8.6
for dierent potential strengths
V0
. .... 77
5.36 Two dimensional pair correlation function in respect with the axis
parallel
rk
and perpendicular
r⊥
to the director of a long needle at
a density of
ρ= 8.6
and a potential strength of
V0= 60kT
..... 77
5.38 Snapshot of the long-needle system at a density of
ρ= 8.6
and a
potential strength of
V0= 60kT
in the region of frozen initial con-
guration. Blow-up: One-dimensional quasicrystalline positional
order of the needle clusters on two Fibonacci chains. . . . . . . . . 78
6.1 Phase diagram of the short rods system. . . . . . . . . . . . . . . 80
112
List of Figures
6.2 Bond orientational order parameter of a
10−
fold and
20−
fold sym-
metry for the center of mass of rods and the center of mass of clus-
ters of rods versus area fraction at a potential stength of
V0= 80kT
.81
6.3 Snapshots (left) and 2D pair correlation functions of the center of
mass of the clusters (right) for the short-rod system at
V0= 100kT
and
η= 0.08
inregionB........................ 81
6.4 Snapshots (left) and 2D pair correlation functions of the center of
mass of the clusters (right) for the short-rod system at
V0= 100kT
and
η= 0.19
inregionC........................ 82
6.5 Snapshots (left) and 2D pair correlation functions of the center of
mass of the clusters (right) for the short-rod system at
V0= 100kT
and
η= 0.5
inregionD. ....................... 83
6.6 Decagonal directional order parameter
Φ10
plotted versus
V0
for
dierent area fractions. . . . . . . . . . . . . . . . . . . . . . . . . 84
6.7 Directional distribution function
f(α)
for three area fractions
η
and
three potential strengths
V0
..................... 85
6.8 Single rods and small clusters (left) occupy dierent positions and
directions in the substrate potential than larger clusters (right)
which gives rise to the gray shaded region in the phase diagram
of Fig. 6.1 (schematic drawing). . . . . . . . . . . . . . . . . . . . 86
6.9 Snapshot of a short rod system at
V0= 100kT
and
η= 0.5
with a
frozennematicorder.......................... 87
6.10 Phase diagram of the long rods. . . . . . . . . . . . . . . . . . . . 88
6.11 Decagonal directional order parameter
Φ10
plotted versus potential
strength
V0
for dierent area fractions
η
. .............. 89
6.12 Snapshots (left) and 2D pair correlation functions in the local rod
frame (right) for the long-rod system at
V0= 60kT
and
η= 0.19
.90
6.13 Snapshots (left) and 2D pair correlation functions in the local rod
frame (right) for the long-rod system at
V0= 60kT
and
η= 0.44
.91
6.14 Snapshots (left) and 2D pair correlation functions in the local rod
frame (right) for the long-rod system at
V0= 60kT
and
η= 0.57
.91
6.15 Snapshots from two systems at an area fraction of
η= 0.57
and a
potential strength of
V0= 90kT
with frozen nematic order. Left:
with two main orientations and
S= 0.85
. Right: Almost perfect
nematic order with
S= 0.93
...................... 92
113
List of Figures
6.16 Mean square displacement
hr2i
of a single short rod in units of
a2
V
for dierent potential strengths
V0
. Time is measured in units of
a2
V/Dk
, where
Dk
is the single-rod diusion coecient parallel to
the rod axis (see Eq. 4.28)....................... 93
6.17 Angular mean square displacement
hϕ2i
and accumulated mean
square displacement parallel (
hr2
ki
) and perpendicular (
hr2
⊥i
) to
the rod axis at
V0= 70kT
. The inset shows a snapshot of the rod
positions at
t= 500a2
V/Dk
....................... 93
6.19 Mean square displacement of a short-rod system for dierent area
fractions at
V0= 70kT
......................... 94
6.20 Mean square displacement
hr2i
of a single long rod in units of
a2
V
for dierent potential strengths
V0
. ................. 95
6.21 Angular mean square displacement
hϕ2i
and accumulated mean
square displacement parallel (
hr2
ki
) and perpendicular (
hr2
⊥i
) to
the rod axis at
V0= 70kT
. The inset shows a snapshot of the rod
positions at
t= 500a2
V/Dk
....................... 95
.1 Left: Overlapping Gummelt decagons form a second order cartwheel
tiling. Right : Kites and darts decoration of two second order
cartwheels. The cartwheels are stacked on each other after a rota-
tion of
180
°
building a almost
10−
fold rotational symmetric double
cartwheel. ............................... 102
.2 Left: All regions of the substrate potential with a potential strength
of
V < −0.76
. Right: Distribution of points from a double cartwheel
created from Gummelt decagons with a point decoration as shown
inthemiddlepanel........................... 102
.3 Left column: Simulation results for the decorations of Gummelt
decagons of three sizes. The edge length of each decagon grows
between two rows with a factor of the golden ratio
τ
starting from
the top. Right column: the corresponding double cartwheel patches
from the Gummelt decagons. Deviations from the real substrate
potential are below
1%
......................... 103
114
List of Figures
.4 Left : The decoration of the biggest Gummelt decagon obtained
by simulation. Right : The decoration of the biggest Gummelt
decagon obtained from a second order cartwheel made from the
smallest Gummelt decagon obtained by simulation. . . . . . . . . 105
.5 Left : The decoration of the smallest Gummelt decagon with Am-
mann bars from the middle-C sequence of the rst order cartwheel.
Right : The horizontal Ammann bars (yellow) and the correspond-
ing shifted Ammann bars (red) from the double cartwheel. . . . . 106
115
List of Figures
116
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146
Danksagung
An dieser Stelle möchte ich all jenen danken, die diese Arbeit möglich gemacht
haben.
Zu allerst möchte ich natürlich Prof. Holger Stark danken, ohne dessen Be-
treuung, Unterstützung und vor allem seine Geduld diese Arbeit sicherlich nicht
möglich gewesen wäre.
Des weiteren Dr. Thomas Gruhn, der sich als Gutachter für diese Arbeit zur
Verfügung gestellt hat.
Prof. Martin Schoen möchte ich danken. Nicht nur in seiner Eigenschaft als Leiter
der IGRTG 1524 und als Vorsitzender des Promotionsausschusses, sondern auch
für seinen inspirierenden Optimismus, den er allseits verbreitet.
Bedanken möchte ich mich auch bei der International Graduate Research Training
Group 1524 für die nanzielle und organisatorische Unterstützung, sowie für die
vielen Möglichkeiten, die dieses Graduiertenkolleg geboten hat, mich in anderen
wissenschaftlichen Bereichen weiter zu bilden.
Ein besonderer Dank geht auch an Prof. Keith Gubbins und seiner Arbeitsgruppe
für den angenhemen und produktiven Aufenthalt and der NCSU in Raleigh,NC.
Danken möchte ich auch meinen Mitstreitern im Graduiertenkolleg für die frucht-
baren Diskussionen. Insbesondere seien hier Heiko Schmidle und Gerald Rosenthal
hervor gehoben.
Auch bei Mitgliedern der Arbeitsgruppe möchte ich mich für die Arbeitsatmo-
sphäre und die hilfreichen Diskussionen bedanken.
In besondere zu erwähnen sind Andreas Zöttl und Helge Neitsch die auch Teile
des Skrites gegen gelesen haben.
Abschlieÿend danke ich meiner Familie und meinen Freunden für die Unter-
stützung und Motivation, die sie mir zu Teil werden lieÿen.
147
