Comparing light-induced colloidal quasicrystals with different rotational symmetries

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2012 J. Phys.: Condens. Matter 24 284101

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IOP PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 24 (2012) 284101 (6pp) doi:10.1088/0953-8984/24/28/284101

Comparing light-induced colloidalquasicrystals with different rotationalsymmetries

Michael Schmiedeberg1 and Holger Stark2

1 Institut fur Theoretische Physik 2: Weiche Materie, Heinrich-Heine-Universitat Dusseldorf, D-40204Dusseldorf, Germany2 Institut fur Theoretische Physik, Technische Universitat Berlin, Hardenbergstraße 36, D-10623 Berlin,Germany

E-mail: schmiedeberg@thphy.uni-duesseldorf.de

Received 14 October 2011, in final form 21 November 2011Published 27 June 2012Online at stacks.iop.org/JPhysCM/24/284101

AbstractQuasicrystals are aperiodic structures with long-range orientational order. Unlike crystals,quasicrystals can, in principle, possess any non-crystallographic rotational symmetry.However, only a few of these rotational symmetries have been observed. By usingMonte Carlo simulations of colloidal particles in laser interference patterns withquasicrystalline symmetry, we compare the onset of quasicrystalline order for differentrotational symmetries in two dimensions. We find that quasicrystals with 5-, 8-, 10-, and12-fold rotational axes can be induced with lower laser intensities than quasicrystals withother non-crystallographic rotational symmetries. We relate this finding to the number of localsymmetry centers in the respective interference patterns.

(Some figures may appear in colour only in the online journal)

1. Introduction

While all crystals possess 2-, 3-, 4-, or 6-fold rotationalsymmetry due to their periodicity, quasicrystals, in principle,may exhibit any non-crystallographic rotational symmetry.Quasicrystals are not periodic; however, they still possesslong-range positional and orientational order. Quasicrystalscan be realized in many different systems. For example, theywere found in metal alloys [1] as well as in assemblies ofpolymers [2] or colloidal nanoparticles [3].

Interestingly, almost all observed quasicrystals haveeither 5-, 8-, 10-, or 12-fold rotational symmetry [4]. Tothe best of our knowledge, the only exception found so faris an 18-fold rotational axis in a quasicrystal that consistsof micelles and whose discovery was recently reported byFischer et al [5]. The rotational symmetry of crystals andquasicrystal can be classified according to the so-called rankD, which is the number of incommensurate wavevectors thatspan the reciprocal lattice of a structure with long-rangepositional order [6]. In d dimensions, crystals have rank D =

d, while quasicrystals with 5-, 8-, 10-, or 12-fold rotationalsymmetry have rank D = 2d. For quasicrystals with otherrotational symmetries, D ≥ 3d. Therefore, almost all of theobserved two-dimensional quasicrystals have rank D = 4,while the 18-fold structure found by Fischer et al is theonly quasicrystal that contains two-dimensional layers withD > 4. Note that all quasicrystals with n-fold rotationalsymmetry where n ≥ 7 must consist of periodically stackedlayers with quasicrystalline symmetry [7]. Therefore, a betterunderstanding of the differences between two-dimensionalquasicrystals with rank D = 4 and those with D > 4 mightlead to a deeper insight into the question of why only a fewrotational symmetries have been found for quasicrystals [4,8–10].

Colloidal particles are widely employed as a modelsystem for studying ordering, crystallization, and thedynamics of particles in an external potential [11]. If subjectedto laser fields, colloids are forced into the direction ofhighest light intensity [12, 13]. Therefore, interfering laserbeams can be used to induce complex structures in a charge

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J. Phys.: Condens. Matter 24 (2012) 284101 M Schmiedeberg and H Stark

stabilized colloidal suspension [14]. Since the interferencepatterns created by 5 or n laser beams, where n ≥ 7, possessquasicrystalline symmetry, such light fields act like a substratethat supports quasicrystalline ordering of the colloids[10, 15–20].

Mikhael et al experimentally studied the ordering ofcolloidal particles in laser fields realized with five orseven laser beams [10]. They discovered that for low laserintensities, quasicrystalline order of the colloids only occursclose to local symmetry centers while overall quasicrystallinebehavior is found for sufficiently large laser intensities.Interfering laser beams create a light field with a globalsymmetry center, which is the point where all laser beams areexactly in phase. A local symmetry center is a small region inthe light interference pattern that is similar to the area aroundthe global symmetry center. In the interference pattern offive beams one finds much more local symmetry centers thanin the pattern of seven beams. Therefore, the laser intensityneeded to induce quasicrystalline order is much lower in thecase of five laser beams as compared to that for seven laserbeams. In general, the frequency of local symmetry centersstrongly depends on the rank of a quasicrystal. To be specific,the frequency of symmetry centers in quasicrystals with rankD > 4 is smaller by at least one decade than the frequency inany quasicrystal with D = 4 [10]. Hence, the small numberof local symmetry centers in quasicrystals with D > 4 mightmake it difficult to grow these structures.

In this work we employ Monte Carlo simulations tostudy colloidal ordering on quasicrystalline substrates thatcorrespond to light fields created by n interfering laser beams.For all quasicrystalline patterns with n ≤ 13, we determinehow the amount of quasicrystalline order depends on thesubstrate strength. We find that quasicrystals with rank D = 4occur at much lower substrate strength as compared to thequasicrystalline order on substrates with D > 4. We alsoshow that local symmetry centers act as nucleation pointsfor quasicrystalline order and, therefore, the existence of asufficient number of symmetry centers really is a key elementin the quasicrystalline growth process.

In section 2 we introduce and explain our system in detail.The fluid-to-quasicrystal transition is studied in section 3,while in section 4 we start from triangular colloidal orderingat vanishing substrate strength. In section 5 we relate ourresults to the number of local symmetry centers in thequasicrystalline substrates. Finally, we conclude in section 6.

2. The model

We consider a charged stabilized suspension that isconfined between two glass plates. The repulsive electrostaticinteraction between the colloidal particles is given by thepair potential of the Derjaguin–Landau–Verwey–Overbeektheory [21, 22],

φ(r) =[Z∗e exp(κR)]2

4πε0εr(1+ κR)2exp(−κr)

r, (1)

where r is the distance between two colloids, κ the inverseDebye screening length, R the radius of a colloid, Z∗ its

Figure 1. (a) The crystalline and quasicrystalline substratepotentials are realized by means of interfering laser beams, whichare symmetrically arranged around the direction perpendicular tothe sample. The angle between the normal direction and thedirections of the light beams is assumed to be small. Thepolarization is the same for all beams. (b) Vectors that span thereciprocal lattice of the interference pattern of five laser beams. Thelattice vectors Gj correspond to the wavevectors of the laser beamsprojected onto the sample plane.

effective surface charge, and εr the dielectric constant ofwater. Except for density and screening length, the parametersare chosen as in [18], i.e., R = 1.45 µm, Z∗ = 400 000,εr = 78, T = 300 K. For the results presented in section 3,the screening length is set to κ−1

= 0.2 µm such that thesystem is fluid without an external potential. In section 4 thetransition of a triangular solid into a quasicrystal is studied andtherefore a larger screening length κ−1

= 0.3 µm is chosensuch that the system is in the triangular phase at zero substratestrength. The mean particle distance is set to aS = 6.5 µm andthe length scale of the potential is aV = 2π/|Gj| = 7.5 µm.We therefore avoid additional phases at intermediate substratestrength such as the Archimedean-like tiling, which is foundat special densities [16, 18], or the phase with 20-fold bondorientational order, which occurs on a decagonal substrate atlow densities [17].

The substrate potential corresponds to the interferencepattern of appropriately arranged laser beams as shown infigure 1. The potential then reads [15, 23]

V(r) = −V0

N2

N−1∑j=0

N−1∑k=0

cos[(Gj −Gk) · r], (2)

where Gj are the projections of the wavevectors of the laserbeams in the sample plane (cf figure 1(b)) and V0 is chosensuch that V(r) = −V0 at the position of the deepest minimum.Figure 2 shows substrate potentials calculated with differentnumbers of laser beams. Large laser intensities generateminima of the substrate potential and are depicted in whiteor yellow while low laser intensities appear in blue or black.The patterns are classified by their rank (see the caption offigure 2).

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J. Phys.: Condens. Matter 24 (2012) 284101 M Schmiedeberg and H Stark

Figure 2. Interference patterns calculated using equation (2) for different numbers of laser beams. The patterns are classified according tothe rank D which is the minimal number of incommensurate wavevectors needed to obtain the reciprocal lattice. The first row shows allpossible crystalline interference patterns with rank D = 2. In the second row all patterns with D = 4 are shown that correspond to thequasicrystals that have been observed in nature. All laser fields with rank D = 6 are displayed in the third row. Finally, the last row showssome examples of patterns with rank D ≥ 8. Quasicrystals with D ≥ 6 are probably hard to grow in real systems. The global symmetrycenter at (x, y) = (0, 0) is marked by a red circle. The side length of the displayed areas is 25aV , where aV = 2π/|Gj| is the wavelengthassociated with the lattice vectors Gj (cf figure 1(b)).

In the following, we present results of Monte Carlosimulations according to the Metropolis algorithm [24].In all simulations the system is first equilibrated for onemillion Monte Carlo steps before the quantities of interestare determined by averaging over 100 configurations thatwe determine with 10 000 steps between two subsequentconfigurations. The simulation box contains 3844 colloids andis quadratic with a side length of 50aV . We assumed periodicboundary conditions for the colloids. Since the substrate is notperiodic, it is discontinuous at the box boundaries. Thereforewhen evaluating the particle positions we neglect all colloidsclose to the boundaries because their positions might bedistorted by the discontinuities.

3. The fluid-to-quasicrystal transition

We first consider a system that is fluid without an externalpotential. We study how quasicrystalline order is induceddepending on the number of laser beams by means of whichthe substrate is realized and the strength of the substrate.The quasicrystalline order is quantified by means of the bondorientational order parameter (cf [17])

ψm =

⟨∣∣∣∣∣ 1N

N∑j=1

1nj

nj∑k=1

eimθjk

∣∣∣∣∣⟩, (3)

where θjk is the angle of the bond connecting colloids j andk with respect to some arbitrary reference direction, nj is

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J. Phys.: Condens. Matter 24 (2012) 284101 M Schmiedeberg and H Stark

Figure 3. (a) Bond orientational order parameter ψm as a functionof potential strength V0/kBT for colloidal ordering on a substraterealized by means of different numbers n of interfering laser beams.For vanishing substrate strength V0 = 0 the colloids are in the fluidphase (for details see section 2). We choose the order parameter ψmsuch that it measures quasicrystalline order corresponding to thesymmetry of the substrate. Therefore, m = n if n is even andm = 2n if n is odd. Furthermore, ψm is divided by the value ψ (max)

mthat is reached when only the deepest minima of the potential areoccupied. For quasicrystalline substrate potentials with rank D = 4the data points are shown by blue, solid symbols, for D = 6 by red,open symbols, and for D ≥ 8 by orange stars or crosses. (b) Bondorientational order parameter ψflowers

m calculated only for bonds forwhich one colloid is in a local symmetry center, i.e., if the potentialvalue at its position is smaller than −0.7V0. The value −0.7V0 ischosen such that for all substrates there are some local symmetrycenters within the simulation box.

the number of nearest neighbors of colloid j, and N is thetotal number of particles. For n laser beams the substrate has2n-fold rotational symmetry if n is odd and n-fold symmetry ifn is even. Therefore we calculate ψm with m = 2n for an oddnumber n of beams and with m = n if n is even. Furthermore,in order to compare the bond orientational order parameterψm

for substrates with different symmetries, we dividedψm by thebond orientational order parameter ψ (max)

m of a configurationwhere the colloids occupy the deepest minima of the substrate.Therefore, ψm/ψ

(max)m → 1 for V0/(kBT)→∞.

Figure 3(a) shows how the bond orientational orderparameter increases with increasing substrate strengthV0/(kBT) for different substrate potentials. For substratepatterns with rank D = 4 (depicted with solid blue symbolsin figure 3(a)) a low substrate strength is sufficient toinduce some quasicrystalline order. However, to obtainquasicrystalline colloidal ordering for patterns with rank D >

4 (shown by red, open symbols for D = 6 and by orange

Figure 4. Bond orientational order parameter ψ6 indicating thetriangular order of colloids as a function of potential strengthV0/kBT . For vanishing substrate strength V0 = 0 the colloids are inthe triangular phase, which is destroyed by the incommensurablesubstrate if the substrate strength V0/kBT is increased.

stars or crosses for D ≥ 8), a much larger substrate strengthis needed.

To explore the influence of local symmetry centers,which are also known as flowers [19], we calculated thebond orientational order parameter only for bonds whereone colloid is close to a symmetry center (see figure 3(b)).Local symmetry centers can be recognized by their deepcentral minima. Therefore, we considered colloids to beclose to a symmetry center if the potential values V(r) attheir positions were smaller than −0.7V0. Unlike the overallbond orientational order parameter, the quasicrystalline orderclose to symmetry centers does not show a clear dependenceon the rank of the substrate potential. To achieve a localquasicrystalline structure in the vicinity of a symmetry center,a low substrate strength is sufficient for all substrates studied.

4. The triangular solid-to-quasicrystal transition

We adjust the parameters here such that the colloids forma triangular lattice for vanishing substrate strength V0 = 0.Figure 4 then shows the bond orientational order parameterψ6 that quantifies the amount of triangular ordering in acolloidal system. If subjected to a substrate, the triangularorder survives at a low substrate strength. However, ifthe substrate strength is increased, the triangular order isdestroyed and a new structure determined by the substrate isinduced. According to figure 4, triangular order is demolishedmost efficiently by quasicrystalline substrate potentials withrank D = 4. Substrate potentials with D > 4 require largerstrengths to affect the triangular structure.

5. The importance of local symmetry centers

Figure 5(c) shows the fraction of deep minima since itis a measure for the number density of local symmetrycenters [10]. The density of symmetry centers stronglydepends on the rank. To be specific, for the substrates studied,the fraction of deep minima differs by at least one orderof magnitude between patterns with different ranks. The

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J. Phys.: Condens. Matter 24 (2012) 284101 M Schmiedeberg and H Stark

#beams n

Figure 5. (a) Quasicrystalline order parameter for colloidalordering on substrates with strength V0/kBT = 200 plotted againstthe number n of interfering laser beams. At vanishing substratestrength the colloidal suspension is fluid. The order is measured byψm/ψ

(max)m as explained in the caption of figure 3(a). (b) Loss of

triangular order on substrates with strength V0/kBT = 200 plottedagainst the number n of beams. At vanishing substrate strength thecolloidal suspension is in the triangular phase. Note that we did notcarry out any simulations for n ≥ 14. (c) Number of minima that aredeeper than −0.99V0 divided by the number of local minima whosedepth was determined (up to 10 millions) versus the number n ofbeams (cf [10]). Such deep minima are the centers of flowers, i.e.,local symmetry centers that resemble the global symmetry center at(x, y) = (0, 0). For substrates with rank D ≥ 8 (i.e., n = 11, 13, 15,16, 17, and 19) only an upper limit is given.

differences in the density of symmetry centers are alreadyapparent in figure 2. On the one hand, quasicrystalline patternswith rank D = 4 (second row in figure 2) exhibit many localsymmetry centers, i.e. local structures that are similar to theglobal symmetry center (marked by a red circle). On the otherhand, patterns with rank D = 6 (third row in figure 2) onlyshow a few local structures that look like distorted symmetrycenters. For D ≥ 6 (bottom row) no symmetry centers asidefrom the global symmetry center can be found in the depictedarea.

In figure 5 we compare the density of local symmetrycenters to the induced quasicrystalline colloidal ordering onsubstrates with strength V0/kBT = 200 (figure 5(a)) for thesystem studied in section 3 and to the loss of triangular orderon a substrate of the same strength (figure 5(b)) when thesystem is triangular at zero potential strength (cf section 4).Interestingly, all quantities depicted in figure 5 can begrouped according to the rank of the substrate pattern. For aquasicrystal with many local symmetry centers (rank D = 4),the quasicrystalline order depicted in 5(a) and the loss oftriangular order shown in 5(b) are larger than those for othersubstrates.

Note that the rank D of the pattern generated by ninterfering laser beams is equal to the value of Euler’s totientfunction ϕ(n) (cf [10]; see also the second row of the labels onthe x-axis in figure 5). It is defined as the number of positiveintegers that are less than or equal to n and that are coprimewith n, i.e., the greatest common divisor of these numbers andn is 1.

6. Conclusions

Using Monte Carlo simulations we studied how the rotationalsymmetry of different quasicrystalline substrates determinesthe induced quasicrystalline ordering in a colloidal monolayer.We are able to confirm the experimental findings for colloidsin decagonal and tetradecagonal laser fields presented in [10]and to generalize them to substrates that correspond tointerference patterns of up to 13 laser beams. We find thatthe amount of quasicrystalline order mainly depends on therank D of a substrate pattern. While quasicrystalline colloidalordering occurs at low strengths in substrate potentials withrank D = 4, a larger substrate strength is needed when D > 4.The main reason for this difference seems to be the lownumber of local symmetry centers in quasicrystals with rankD > 4. In future, in order to better understand why mostof the observed quasicrystals in nature have rank D = 4, itmight be interesting to study how important local symmetrycenters are in other systems, e.g., in atomic quasicrystalsor in colloidal systems that form intrinsic quasicrystallinestructures due to a special pair interaction potential [25–27].In particular, a detailed investigation of the growth dynamicsshould reveal the role of local symmetry centers in suchintrinsic quasicrystals. For example, in the case of a layer bylayer growth process we expect the local symmetry centers toaffect the order of the particles in a new layer in a similar wayto in the colloidal system.

Acknowledgments

We would like to thank C Bechinger, R Lifshitz, J Mikhael,and J Roth for helpful discussions. We acknowledge financialsupport from the Deutsche Forschungsgemeinschaft (DFG)under Grant No. RO 924/5-1. MS was also supported by theDFG within the Emmy Noether program (Grant No. SCHM2657/2-1) and within the SFB TR6 and by the GermanAcademic Exchange Service within the postdoc program.

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