Phase ordering of hard needles on a quasicrystalline substratePhilipp Kählitz and Holger Stark Citation: J. Chem. Phys. 136, 174705 (2012); doi: 10.1063/1.4711086 View online: http://dx.doi.org/10.1063/1.4711086 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v136/i17 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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THE JOURNAL OF CHEMICAL PHYSICS 136, 174705 (2012)

Phase ordering of hard needles on a quasicrystalline substratePhilipp Kählitza) and Holger Starkb)

TU Berlin, Institut für Theoretische Physik, Hardenbergstr. 36, D-10623 Berlin, Germany

(Received 16 February 2012; accepted 18 April 2012; published online 4 May 2012)

Quasicrystals possess long-range positional and orientational order. However, they cannot be pe-riodic in space due to their non-crystallographic symmetries such as a 10-fold rotational axis. Weperform Monte Carlo simulations of two-dimensional hard-needle systems subject to a quasiperiodicsubstrate potential. We determine phase diagrams as a function of density and potential strength fortwo needle lengths. With increasing potential strength short needles tend to form isolated clustersthat display directional order along the decagonal directions. Long needles create interacting clustersthat stabilize the nematic phase. At large potential strengths the clusters position themselves on twointerwoven Fibonacci sequences perpendicular to the cluster orientation. Alternatively, one obtainsextended domains of needle clusters which are aligned along all decagonal symmetry directions.© 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4711086]

I. INTRODUCTION

The discovery of quasicrystals and their publication in1984 (Ref. 1) raised a lot of scientific interest in the pastdecades. Quasicrystals possess long-range positional order,however, their non-crystallographic rotational symmetry isincompatible with any periodic repetition of a unit cell inspace.2–5 Therefore, quasicrystals constitute are third solidstate between amorphous and crystalline materials. Since thefirst observation of quasicrystals a huge number of artificiallyproduced quasicrystalline materials has been reported.6, 7

While they are mostly solid, quasicrystalline structures insoft matter also exist.8, 9 Only recently, a natural quasicrystalhas even been found.10 Quasicrystals have exceptional mate-rial properties,11 such as a low coefficient of friction,12 cat-alytic qualities,13 and the possibility of producing photonicbandgaps.14 Hence, a growing topic in the last years hasbeen the question how atoms order on quasicrystalline sur-faces with the ultimate goal to controllably grow quasicrys-talline materials.11, 15–24 In particular, research groups havestudied the interaction of quasicrystalline surfaces with dif-ferent types of atomic adsorbates, which usually form con-ventional crystalline phases. As a result, they have identifiednew self-assembled structures of adatoms on surfaces of qua-sicrystalline alloys in experiments15, 18, 25 and in theoreticalsimulations.23, 24

A quite recent research line in connection with qua-sicrystals uses dispersions of micrometer sized colloidal par-ticles that are commonly used to model atomic and molecu-lar systems.26, 27 With the help of interfering laser beams onecan create a quasicrystalline substrate potential for colloidalparticles.28 Under the influence of this light pattern, colloidsorder in new phases such as a Archimedian-like tiling or aphase with 20-fold bond-orientational order that have beenidentified both in experimental and theoretical studies.28–34

The novel phase behavior arises from the competition of the

a)Electronic mail: kaehlitz@itp.tu-berlin.de.b)Electronic mail: holger.stark@tu-berlin.de.

natural hexagonal ordering of colloidal particles and their in-teractions with the quasicrystalline substrate. Colloidal dis-persions offer the possibility of studying generic features ofadsorbates on quasicrystalline substrates without dealing withthe subtle chemical properties of individual materials. An-other example in this direction is the diffusion of colloids inquasicrystalline potentials.35

In this article, we address how hard needles as the sim-plest form of elongated particles order on a quasicrystallinesubstrate. Due to their elongated shape particles or moleculesexhibit a wealth of different liquid-crystalline phases thatare characterized by long-range orientational ordering.36 Thesimplest is the nematic phase for which Onsager introducedthe hard-rod model and presented a first theoretical treat-ment on the molecular level.37 Besides conventional ther-motropic and lyotropic liquid crystals,36 nature providesmany elongated particles or molecules that display phaseswith orientational ordering. For example, alkane moleculeson graphite form two-dimensional molecular patterns remi-niscent to smectic phases.38 Mosaic tobacco viruses with alength of 300nm and a width of 18nm approximate hard nee-dles very well. At higher packing densities they form severalliquid-crystalline phases including the nematic phase.39–42

Even higher aspect ratios exist for natural clay or Boehmiterods.43–45

In this article, we investigate the phase behavior of hardneedles in a quasicrystalline substrate potential with decago-nal symmetry by varying density and potential strength. Be-sides the nematic phase we will identify a surface-induceddirectional phase and at large density and potential strengtha phase region where an initial nematic ordering can befrozen in. The decagonal surface potential possesses one ma-jor length scale aV . To illustrate how the structure of the sur-face potential influences the phase ordering of the needles, westudy two needle lengths L: the “short” needles with L = aV

explore details of the substrate potential and form nearly iso-lated needle clusters for increasing potential strength. “Long”needles with L = 3aV average over such details and stabilizenematic order. The needle clusters interact with each other

0021-9606/2012/136(17)/174705/10/$30.00 © 2012 American Institute of Physics136, 174705-1

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174705-2 P. Kählitz and H. Stark J. Chem. Phys. 136, 174705 (2012)

and form domains with uniform orientation. Note that in thefollowing we frequently speak about surface-induced order-ing and thereby always mean the ordering in a monolayer ofneedles induced by the substrate potential.

The quasi-nematic phase of hard needles in two di-mensions was first investigated numerically by Frenkeland Eppenga.46 For the density-controlled transition to theisotropic phase they suggest a continuous transition of theKosterlitz-Thouless type.47 Kosterlitz evaluated the relevantcritical exponents of the XY model,48 which were confirmedfor the hard-needle model just recently by Vink using MonteCarlo simulations.49 Orientational correlations of the nee-dles in the quasi-nematic phase decay algebraically. Thus thequasi-nematic phase does not exhibit a true but only a quasi-long-range orientational order.46 In our simulations, we rec-ognize a slight decay of nematic order with system size butour systems are sufficiently small so that the algebraic decaydoes not play any role here.

The article is organized as follows. In Sec. II we intro-duce the hard-needle model, discuss relevant properties of thesubstrate potential, and mention details of our Monte Carlosimulations. Section III presents the phase behavior of shortand long needles and discusses relevant features in detail. Weclose with a summary and conclusions in Sec. IV.

II. MODEL

A. Hard needles

Hard needles are particles with an aspect ratio needlelength to diameter of L/D = ∞. They interact through hardcore repulsion only. This means they are not allowed to crosseach other which is a severe limitation for their positions andorientations at high densities. The two-dimensional nematicorder parameter S of a N-needle system is defined as

S = N−1

⟨N∑

i=1

cos(2θi)

⟩, (1)

where θ i is the angle of the ith needle with the nematic di-rector. In practice, we introduce the orientational tensor orderparameter for one configuration of N needles,

Qab = N−1N∑

i=1

[2ua(i)ub(i) − δab], (2)

where −→u (i) = [u1(i), u2(i)] is a unit vector indicating the di-rection of needle i. Since Qaa = 0, the two eigenvalues of Qab,Q, and −Q add up to zero and the nematic order parameter isthe ensemble average S = 〈Q〉 over the positive eigenvalue ofQab.46

Under the influence of a substrate potential, the needleswill align along the symmetry directions of the potential. Todescribe this type of ordering, we introduce the directionalorder parameter for m-fold order,

�m = 〈ψm〉 with ψm = N−1

∣∣∣∣∣∣N∑

j=1

eimαj

∣∣∣∣∣∣ , (3)

where αj is the angle of the needle j with respect to an arbi-trary axis. In the case of m = 2 this order parameter is iden-tical to the nematic order parameter S = �2. To track thealignment of the needles along the ten symmetry directionsof the decagonal substrate, we use the tenfold directional or-der parameter �10. For perfect nematic order angles α andα + π occur with the same frequency and also �10 is one.Non-perfect nematic order strongly decreases the decagonaldirectional order parameter since m = 10 in the exponent of�m “amplifies” any deviations from the perfect alignment. Asa result, in our simulations without a substrate potential �10

is always well below 0.2 even at the highest densities wherenematic order exists. We note that this is just an artificialcontribution to �10.

Often it is difficult to locate a phase transition by juststudying the relevant order parameter. To better determinethe position of the phase transition, it helps to look at thefluctuations of the order parameter which become maximalat a phase transition and, in particular, show critical behav-ior in the vicinity of second-order phase transition.50 Thefluctuations of the nematic order parameter are characterizedby the variance with respect to S, which is also known assusceptibility,51, 52

χS = βN (〈Q2〉 − S2). (4)

In the same way, we define the variance for fluctuationsaround the directional order parameter,

χ� = βN(⟨ψ2

m

⟩ − �2m

). (5)

Furthermore, we also look at the specific heat capacity, whichis connected to fluctuations in the energy E of a needle con-figuration

c = β2

N(〈E2〉 − 〈E〉2). (6)

B. Substrate potential

In recent experiments five laser beams were used to cre-ate a quasicrystalline interference pattern in which the two-dimensional ordering of colloidal particles was studied.28 Weuse this pattern to investigate the phase ordering of hard nee-dles on a simple substrate potential with decagonal symmetry.It is easy to describe in theory,29, 30, 35

V (�r) = −V0

25

∑4

j=0

∑4

i=0cos[( �Gi − �Gj ) · �r]. (7)

The vectors �Gi are the projection of the wave vectorsof the five laser beams onto the plane of the substrate andaV = 2π/| �Gi | defines a characteristic length scale of the po-tential. In the following, we give all lengths in units of aV . Thefive wave vectors �Gi point to the vertices of a pentagon andgenerate a star of vectors with an angle φ = 2π /5 between twoneighboring vectors ( �Gi, �Gi+1) (see Fig. 1). In particular, wechoose �G0 = (0, 2π/aV ). The parameter V0 determines thedepth of the deepest minima in the potential located at theorigin. In our simulations, we give V0 in units of the ther-mal energy kT, where k is the Boltzmann constant and T is

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174705-3 P. Kählitz and H. Stark J. Chem. Phys. 136, 174705 (2012)

FIG. 1. Wave vectors pointing along the vertices of a pentagon for setting upthe substrate potential in Eq. (7) with decagonal symmetry.

temperature. Figure 2 shows the substrate potential and alsothe characteristic length scale aV (short red line).

The needles interact with the substrate potential V (�r) byaveraging it over the full needle length. This leads to a poten-tial VN (�r, �u) that depends on both the needle center at �r andits orientation �u,

VN (�r, �u) = 1

L

∫ 1/2

−1/2V (�r + lL�u) dl. (8)

In the following, we will concentrate on two needle lengths:Needles with L = 1aV , which we call short needles, and nee-dles with L = 3aV , which we call long needles. In the il-lustration of the substrate potential V (�r) in Fig. 2 their pre-ferred positions and orientations are such that they connectthe potential minima of the substrate potential, so their usualpositions will be between the minima. Therefore, the loca-tion of the minima on straight lines is important for a bet-ter understanding of the ordering of the needles. For each ofthe five equivalent directions, the minima define straight lines

FIG. 2. Illustration of the substrate potential with decagonal symmetry fromEq. (7). White means V = 0 and the darkest shading corresponds to the mostnegative potential values. Two possible locations and orientations of needleswith lengths L = aV and L = 3aV are drawn in red. Solid lines connect thedeepest minima and the more shallow minima lie on the dotted lines. Theydefine two Fibonnaci sequences as discussed in the text.

which we indicate by the solid green lines in Fig. 2. Simi-lar to Refs. 53 and 54, the decagonal substrate potential inFig. 2 can be covered by a Penrose tiling of fat and skinnyrhombic unit cells. The Penrose tiling is a quasicrystallinetiling of the plane with a fivefold rotational symmetry. A spe-cial decoration of the two unit cells with line segments leadsto a set of parallel lines in each of the five symmetry direc-tions of the tiling. These lines are also called Ammann bars.Parallel Ammann bars follow a Fibonacci sequence55, 56 (to beexplained below). From mappings of our substrate potentialon the Penrose tiling,57–59 where the potential minima appearin the decoration of the rhombic unit cells, we expect that thedistances L and L + S between the full green lines follow aFibonacci sequence as well. The Fibonacci sequence definesa typical one-dimensional quasicrystalline structure. In gen-eral, one generates it iteratively from a short (s) and a long(l) length scale by introducing a sequence Sn with the ruleSn = Sn − 1Sn − 2 which starts at S0 = l and S1 = ls. Besidesthe lines predicted by the Penrose tiling, the weaker min-ima of the potential lie on the dotted green lines in Fig. 2.One half of them together with the full green lines defines aFibonacci sequence with the short (S = (τ − 1)aV ) and thelong (L = aV ) length scales, where τ = 1+√

52 is the golden

mean. To predict the positions of the lines connecting all min-ima of the potential along one of the symmetry direction, oneneeds two Fibonacci chains. Of course, in each of the fivesymmetry directions one has the same two Fibonacci chains.The crossing points of all the lines give the locations of thepotential minima.

In the integrated needle potential VN (�r, �u) of Eq. (8) aminimum occurs, when the needles connect minima so theyare oriented along the green lines in Fig. 2. Therefore, the cen-ter of the needle defines the location of a minimum VN (�r, �u)with a unique needle orientation. There are no positions in theplane, where a needle sees a minimum potential energy withtwo possible orientations.

C. Monte Carlo simulation

We use a Monte Carlo NVT algorithm with periodicboundary conditions.60, 61 Since the quasicrystalline potentialis not periodic, discontinuities at the boundaries of the period-ically repeated simulation boxes occur. To minimize the dis-continuities, we choose special box sizes following Ref. 62.For the decagonal potential the edge lengths of the simula-tion box sizes have to be X = 2naV and Y = maV /sin(π/5),where n and m are Fibonacci numbers. Since we fix the boxsizes to discrete values, we vary the particle number to real-ize different densities. Most of our simulations are performedwith n = 3, m = 5 for short needles and n = 13, m = 21for long needles. The particle number varies between 232 and654 for the short needles and between 331 and 886 for thelong needles to realize an appropriate density range. The lim-itations on the possible box sizes also makes it difficult to per-form finite-size analysis. The next larger box size for the longneedles needs 776 needles at the lowest and 2318 at the high-est simulated density. Because of the very long computationtime, we performed a search for finite-size effects only for

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174705-4 P. Kählitz and H. Stark J. Chem. Phys. 136, 174705 (2012)

two densities of the two needle lengths. We confirmed resultsfrom Ref. 46 that in the two-dimensional needle system theposition of the isotropic-to-quasi-nematic transition dependson the system. However, for the onset of the decagonal di-rectional order, which is mainly determined by the substratepotential, we don’t find a finite-size effect. In the following,we measure the needle density ρ in units of the square of theneedle length, 1/L2. In this way, the isotropic-nematic phasetransition always occurs at the same value of the reduced den-sity ρ independent of needle length L.

Each Monte Carlo cycle for one particle is divided intothree steps. We first perform a Monte Carlo step, where weonly change its position followed by a separate rotational stepfor its orientation. The advantage is that we can decouple theacceptance rate for both steps from each other and choose foreach of them the optimal maximal value, separately. How-ever, we also need a third combined positional and rotationalstep to avoid very high potential barriers. They occur whenwe shift and rotate the needles separately. Positional and ro-tational sifts are chosen from a constant probability distribu-tion with a maximum distance and a maximum rotation angle.We adjust the combined acceptance rate for all three steps to-gether to 0.5 by varying the maximum distance and the max-imum rotation angle. About every 100th step these maximumvalues are set to three times the needle length and to 180◦.Such moves have a very low acceptance rate but help the sys-tem to dissolve locked cluster states of the needles. To equili-brate the system, we use a few 105 Monte Carlo sweeps at lowdensities and low potential strengths up to a few 106 sweepsfor high densities and high potential strengths. One MonteCarlo sweep consists of the number of particles times a sin-gle Monte Carlo cycle for each particle. As initial conditionswe choose two different needle configurations. The first con-sists of a random distribution of both the needle positions andorientations. With such isotropic starting conditions one canstudy whether the hard needles are able to build up nematicorder. In the second configuration the positions of the needlesare randomly distributed but they all align along an arbitrarilychosen common direction. Starting with such an ideal nematicorder, we investigate how stable the nematic phase is. At least10 simulation runs with independent initial conditions are per-formed for each initial condition and each density.

III. RESULTS

We present the simulation results for short and long nee-dles in Subsections III A and III B. At the beginning of eachsection, we briefly introduce the phase diagram and then dis-cuss its different regions in detail.

A. Short needles

The phase diagram of the short needles in Fig. 3 can bedivided into four regions. The most important transition sep-arates surface-induced directional or decagonal order at highsubstrate strength from a region at low or zero strength. Here,the substrate does not influence the typical phase behaviorwhere below a density of ρ ≈ 5.9 we observe an isotropic

FIG. 3. Phase diagram for the short-needle system.

phase followed by the quasi-nematic phase as already re-ported for this system size by Frenkel and Eppenga.46 Abovethe main transition line also two regions exist. In the low-density region a pure decagonal phase exists without any ne-matic order. In the region at high densities, the realized order-ing depends on the initial condition. In particular, it is possibleto freeze in a starting configuration with nematic order in ad-dition to the surface-induced decagonal order. Now, we give adetailed account of our results.

In Fig. 4(a), we plot the decagonal order parameter�10 and its susceptibility χ� as a function of the potentialstrength V0. The maximum of the susceptibility coincideswith the inflection point of the decagonal order parameter

FIG. 4. (a) Decagonal order parameter �10 and its susceptibility χ� for ashort-needle system plotted versus potential strength V0 at a density of ρ

= 8.1. (b) Decagonal order parameter �10 versus potential strength V0 fordifferent densities ρ.

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174705-5 P. Kählitz and H. Stark J. Chem. Phys. 136, 174705 (2012)

FIG. 5. Snapshots of short-needle systems at a density of ρ = 8.1. Left:Potential strength V0 = 10, right: V0 = 40.

at a value of �10 ≈ 0.2. We find this behavior at all sim-ulated densities. Therefore, we let the decagonal directionalphase of the needles start at �10 ≈ 0.2. Figure 4(b) showsthe decagonal order parameter versus V0 for different den-sities. In the short-needle system, all these curves look sim-ilar and there is no pronounced density dependence. Onlyfor low densities, decagonal ordering needs a higher poten-tial strength to develop. Accordingly, the transition line inthe phase diagram 3 slightly bends upwards at low densities.The minimum potential strength for observing the decago-nal phase is about V0 = 15 at high densities. Despite theclear maximum in the susceptibility, we do not observe amaximum in the heat capacity at the same position in thephase space. This is reminiscent to the work of Frenkel andEppenga.46 They only observed a weak maximum in the heatcapacity shifted against the actual transition from the isotropicto the nematic phase.

The snapshots of Fig. 5 show the short-needle systemat a density ρ = 8.1. At V0 = 10 (left) the system is stillin the nematic phase and the substrate potential did not en-force any decagonal ordering. At V0 = 40 the needles formdensely packed clusters with the length of one needle and amuch smaller width that are well separated from each other.As explained in Subsection III B, the clusters connect twolocal minima in the quasicrystalline potential. They form ahighly ordered decagonal phase with directional order param-eter �10 ≈ 0.8. The pair correlation function for V0 = 40 inFig. 6 shows the dense packing within the clusters throughthe large first maximum very close to r = 0. One recognizesthe isolated clusters by the deep and broad first minimum atr = 0.5. Finally, we note that the profiles for the decagonalorder parameter in Fig. 4(b) do not change by increasing thenumber of needles from ∼600 to 1200. So there is no size de-pendence. This is in agreement with the fact that the decago-nal order is due to the local values of the substrate potential.

We now discuss the transition from the nematic phaseinto the region with surface-induced directional order for in-creasing V0. Figure 7(a) plots the nematic order parameterprofile for different densities. The nematic order first de-creases slowly until a potential strength of about V = 18where the surface-induced decagonal order sets in. Now,

FIG. 6. Pair correlation function g(r) for short needles at a density ofρ = 8.1 for different potential strengths V0.

local needle clusters form that isolate the needles against eachother. This leads to a sharp drop of the order parameter toS ≈ 0.2 and nematic order vanishes. So the loss of nematicorder is strongly correlated with the appearance of decago-nal directional order. The nematic susceptibility plotted inFig. 7(b) for various densities indicates the loss of nematicorder with a pronounced maximum at the transition line. Forcomparison the nematic susceptibility in the isotropic phaseat ρ = 4.5 does not exhibit such a maximum.

FIG. 7. Nematic order parameter S (a) and nematic susceptibility χS (b) ofthe short-needle system as a function of potential strength V0 for differentdensities ρ.

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174705-6 P. Kählitz and H. Stark J. Chem. Phys. 136, 174705 (2012)

FIG. 8. Orientational correlation function for nematic order, gS(x, y), forshort needles at a density ρ = 4.5 and V0 = 60.

We use the orientational correlation function for nematicorder,

gS(x, y) = 〈cos[2( (0) − (x, y)]〉, (9)

to illustrate the orientational order of the clusters at high po-tential strength. Figure 8 displays a central spot surrounded by10 red spots which indicate directions in space along whichclusters assume the same orientation as the central cluster.In between, the blue spots give directions with perpendicularorientation. The pattern in Fig. 8 displays the same decago-nal symmetry and positional order as the substrate potential.Therefore, the needle clusters exhibit the same long-range po-sitional and orientational order as the substrate potential.

Figure 9 illustrates the fourth region in the phase dia-gram in Fig. 3 named frozen initial configuration. At densityρ = 8.1 we plot the nematic order parameter versus V0 for twoinitial configurations. Starting with random orientations of allneedles, the system does not develop any nematic order at

FIG. 9. Nematic order parameter versus potential strength at ρ = 8.1 forshort needles and different initial configurations.

FIG. 10. Snapshot of a short-needle system with frozen nematic order at adensity ρ = 8.1 and a potential strength V0 = 40. The nematic order param-eter is S ≈ 0.5.

high potential strengths. However, an initial nematic order re-mains or freezes in beyond V0 = 36. We have confirmed thisbehavior by doubling the simulation time normally neededfor equilibrating the system. The snapshot of Fig. 10 showsthe short-needle system with frozen nematic order. Needleclusters with the same orientation order along the Fibonaccilines as expected from the structure of the substrate potential(Fig. 2). In between needle clusters with different orientationoccur. In our simulations, the order parameter never exceedsS ≈ 0.7 since between clusters oriented along neighboringFibonacci lines always clusters with different orientations canbe inserted. Therefore, in a frozen nematic state we alwaysobserve at least two of the five possible cluster orientationsand the nematic order is never perfect.

In the region of frozen initial configuration, the energy ofthe system, the decagonal order, and the heat capacity do notdepend on the degree of nematic ordering. So, configurationswith different frozen nematic order just seem to constitute dif-ferent possible realizations of the same decagonal order whichcorresponds to a highly degenerate ground state.

B. Long needles

The phase diagram of the long needles in Fig. 11 can bedivided into five regions and exhibits pronounced differencescompared to the short-needle system and its phase behaviorin Fig. 3. The most important division line marks again theonset of surface-induced decagonal order with �10 > 0.2 forincreasing substrate strength V0. However, whereas for shortneedles this line is more or less horizontal, it now tilts towards

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174705-7 P. Kählitz and H. Stark J. Chem. Phys. 136, 174705 (2012)

FIG. 11. Phase diagram for the long-needle system.

smaller V0 when density ρ increases. Below the decagonaltransition line, one observes again the isotropic and quasi-nematic phase with the transition located at ρ ≈ 6.2 for thesimulated system size. Interestingly and in contrast to shortneedles, the transition at ρ ≈ 6.2 extends beyond the maindecagonal transition line to larger V0, where now three differ-ent phase regions exist. At densities below ρ ≈ 6.2 the systemassumes pure decagonal order without any nematic ordering.At densities larger than ρ ≈ 6.2 a phase with both nematic anddecagonal order exists up to a substrate strength of V0 ≈ 35.In the short-needle system, such a phase does only occur ina very narrow region of V0. For ρ > 6.2 and V0 > 35, thestarting configuration again freezes in. Now, we describe thephase behavior in more detail and try to explain it.

We first discuss the decagonal ordering of the long nee-dles as illustrated by the decagonal order parameter �10 plot-ted versus V0 for several densities in Fig. 12. Most propertiesof �10 are the same as in the short-needle system. The max-imum of the susceptibility χ� when plotted as a function ofV0 occurs again when the decagonal order parameter assumesthe value �10 = 0.2 and the heat capacity does not show anymaximum at this position. However, in contrast to short nee-dles, the potential strength V0 necessary to induce decagonal

FIG. 12. Decagonal order parameter �10 for the long-needle system plottedversus potential strength V0 at different densities ρ.

FIG. 13. Snapshot of the long-needle system at a density of ρ = 8.6 and apotential strength of V0 = 40.

ordering strongly decreases with increasing density (see alsoFig. 11). We understand such a behavior qualitatively. Withincreasing V0, short needles tend to form compact clusters ofthe size of one needle length when they connect two potentialminima. The clusters are well separated from each otherregardless their density. In contrast, long needles connect sev-eral minima and even share one or two of them. Now, clusterswith the same orientation form elongated domains which havea length equal to several needle lengths (Fig. 13). The widthsof the domains oriented along one of the decagonal directionsalso extend beyond one needle length since geometrically it issimpler to align clusters. For larger densities, we expect suchdomains to form more easily which explains the behaviorof the decagonal transition line. Figure 14 plots the paircorrelation function for the center of mass of the needles. Thegraph illustrates the formation of needle clusters by the sharppeak close to r = 0 which builts up for increasing V0. Further

FIG. 14. Pair correlation function for the center of mass of long needles at adensity of ρ = 8.6.

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174705-8 P. Kählitz and H. Stark J. Chem. Phys. 136, 174705 (2012)

FIG. 15. Nematic order parameter versus potential strength V0 in the long-needle system. (a) For different densities ρ when the strength V0 graduallyincreases. (b) At ρ = 8.6 for isotropic and nematic starting configurations ateach V0.

maxima at r ≈ av and larger radii indicate characteristicdistances between the needle clusters. In particular, theyare visible within the domains perpendicular to the needleorientation.

At low densities ρ < 6.2 the needles show surface-induced decagonal order without any nematic ordering. Sim-ilar to Fig. 13 the needle clusters form aligned domains thatare equally distributed in all 10 decagonal directions. Aboveρ = 6.2 the formation of the needle clusters does not destroynematic order since the clusters overlap with each other asexplained in the previous paragraph. As a result, the phaseregion with both stable nematic and decagonal order in thephase diagram of Fig. 11 occurs. Figure 15(a) demonstratesthat for each density the nematic order parameter is nearlyconstant as a function of V0. One recognizes a slight increasewhen the decagonal order is established and a decrease be-yond V0 = 40 in the region of frozen initial configuration. InFig. 16(a) we show a typical needle snapshot of the combinednematic and decagonal order at ρ = 8.6 and V0 = 18. Oneclearly recognizes an average direction of the needles alongthe director, which points along one of the decagonal direc-tions. The single needle orientations fluctuate around the di-rector. The corresponding orientational distribution functionof the needles is plotted in Fig. 16(b). Besides the orienta-

FIG. 16. (a) Snapshot of the long-needle system at a density of ρ = 8.6and a potential strength of V0 = 18, where both nematic and decagonal orderexists. (b) Corresponding orientational distribution function for the needles.The angle α is measured with respect to the horizontal.

tion of the director at α = 2π /5, two weaker maxima ap-pear at the neighboring decagonal directions at α = π /5 and3π /5. Increasing the density, these maxima become weakerin agreement with the increasing nematic order parameter S.All three maxima become sharper when V0 increases restrict-ing the needles more and more to the decagonal directionsof the substrate potential. For a large substrate strength ofV0 = 60 this is illustrated in Fig. 17. The needles form againclusters which are mostly aligned along one decagonal direc-tion. Some needle clusters deviate from the nematic directorand point along other decagonal directions reducing the ne-matic order parameter below S = 1. Still the directional orderparameter indicates decagonal ordering. The blow-up of oneregion of the snapshot in Fig. 17 reveals that the positions ofthe needle clusters possess one-dimensional quasicrystallineorder perpendicular to the nematic director. The order is char-acterized by the two Fibonacci chains, which we identified inthe substrate potential as illustrated in Fig. 2.

Finally, in the region termed frozen initial configura-tion in the phase diagram of Fig. 11, the mobility of theneedles is so small that the system is not able to changean initial configuration. Like short needles, long needlesare able to freeze in an initial nematic order. However, ifthe simulation starts without any orientational order, nematicordering does not develop during equilibrating the system.Figure 15(b) shows how the nematic order parameter S de-pends on the starting configuration. The energy, heat capacity,and decagonal order are the same whether the system freezesin the nematic or isotropic state. In the isotropic system, nee-dle clusters are aligned within domains the orientations ofwhich are distributed equally on all ten decagonal directions.A typical needle snapshot was already discussed in Fig. 13.The ordering in the nematic system was already introduced inthe last paragraph and Fig. 17.

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174705-9 P. Kählitz and H. Stark J. Chem. Phys. 136, 174705 (2012)

FIG. 17. Snapshot of the long-needle system at a density of ρ = 8.6 and a po-tential strength of V0 = 60 in the region of frozen initial configuration. Blow-up: One-dimensional quasicrystalline positional order of the needle clusterson two Fibonacci chains.

IV. SUMMARY AND CONCLUSIONS

We have determined phase diagrams of a two-dimensional hard-needle system under the influence of adecagonal substrate potential. In the short-needle system thequasi-nematic order is destroyed with increasing potentialstrength. The system exhibits directional order where the nee-dles gradually form disconnected clusters located betweentwo potential minima and oriented along the symmetry di-rections of the decagonal potential. As a result, the needleclusters exhibit the same long-range positional order as thesubstrate and their relative orientations also display long-range order. Finally, at sufficiently high densities and poten-tial strengths it is possible to freeze in nematic order up to anorder parameter of S = 0.7.

Long needles tend to connect several potential minimawith increasing potential strength and to form clusters that in-teract with neighboring clusters. In contrast to short needles,now extended domains of uniformly oriented clusters alongthe decagonal directions form. At larger densities the inter-action between needles enforces directional order to set in atlower potential strengths compared to the short-needle systemand 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 ori-ent themselves along lines defined by the potential minima.These lines follow a one-dimensional quasicrystalline orderthat is described by two interwoven Fibonacci chains. The ef-fect becomes very pronounced for large potential strengths,where one can again freeze in nematic order with any valueof the order parameter S.

The combination of a hard-needle system, which tendsto form a quasi-nematic phase in two dimensions, and a qua-sicrystalline substrate potential leads to interesting patterns of

clustered needles especially for large potential strengths. Itwould be interesting to perform experiments with the help ofquasicrystalline light patterns as in Refs. 28, 30, and 33 usingappropriate colloidal needle systems. Especially with interfer-ing laser beams it should be possible to tune the characteristiclength aV of the resulting decagonal potential and to explorehow the needle system reacts to varying this length scale. Intheory, we currently explore needles with a finite thicknessinstead of ideal needles. We already recognize differencesin the phase diagram. It is also appealing to use the result-ing two-dimensional needle adsorbates as templates to buildthree-dimensional structures and to explore how well the qua-sicrystalline cluster phases extend into the third dimension.

ACKNOWLEDGMENTS

We thank M. Schmiedeberg and K. E. Gubbins for help-ful discussions and the International Research Training GroupIGRTG 1524 funded by the German Science Foundation forfinancial support.

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