archimedean-like colloidal tilings on substrates with ... · archimedean-like colloidal tilings on...

DOI 10.1140/epje/i2010-10587-1

Regular Article

Eur. Phys. J. E 32, 25–34 (2010) THE EUROPEANPHYSICAL JOURNAL E

Archimedean-like colloidal tilings on substrates with decagonaland tetradecagonal symmetry

M. Schmiedeberg1,a, J. Mikhael2, S. Rausch2, J. Roth3, L. Helden2, C. Bechinger2,4, and H. Stark5

1 Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104, USA2 2. Physikalisches Institut, Universitat Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany3 Institut fur Theoretische und Angewandte Physik, Universitat Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany4 Max-Planck-Institut fur Metallforschung, Heisenbergstraße 3, D-70569 Stuttgart, Germany5 Institut fur Theoretische Physik, Technische Universitat Berlin, Hardenbergstr. 36, D-10623 Berlin, Germany

Received 16 December 2009 and Received in final form 1 April 2010Published online: 22 May 2010 – c© EDP Sciences / Societa Italiana di Fisica / Springer-Verlag 2010

Abstract. Two-dimensional colloidal suspensions subjected to laser interference patterns with decagonalsymmetry can form an Archimedean-like tiling phase where rows of squares and triangles order aperiodicallyalong one direction (J. Mikhael et al., Nature 454, 501 (2008)). In experiments as well as in MonteCarlo and Brownian dynamics simulations, we identify a similar phase when the laser field possessestetradecagonal symmetry. We characterize the structure of both Archimedean-like tilings in detail and pointout how the tilings differ from each other. Furthermore, we also estimate specific particle densities wherethe Archimedean-like tiling phases occur. Finally, using Brownian dynamics simulations we demonstratehow phasonic distortions of the decagonal laser field influence the Archimedean-like tiling. In particular,the domain size of the tiling can be enlarged by phasonic drifts and constant gradients in the phasonicdisplacement. We demonstrate that the latter occurs when the interfering laser beams are not ideallyadjusted.

1 Introduction

Unlike ordinary crystals, quasicrystals possess point-sym-metries, e.g., with five- or ten-fold rotational axes, whichare not allowed in periodic crystals [1,2]. Nevertheless,quasicrystals exhibit Bragg reflections due to their long-range positional and orientational order. Since they showexceptional material properties [3], there is much interestin understanding and controlling the growth of quasicrys-tals. Therefore, a lot of research has been devoted to studythe ordering of atoms on quasicrystalline surfaces [4–15].However, in these experiments it is very difficult to de-termine the exact positions of the atoms. Therefore, amodel system consisting of micron-sized colloidal parti-cles subject to a laser interference pattern has been usedto investigate the dynamics and ordering of particles in atwo-dimensional quasicrystalline potential [16–20]. Inter-esting structures have been observed, such as phases with20 bond directions [19] or colloidal orderings consisting ofrows of squares and rows of triangles termed Archimedean-like tilings [18,19] (see also fig. 1). The latter will be themain topic of this work.

Interference patterns of five or seven laser beams havedecagonal or tetradecagonal symmetry, respectively (seefig. 2). In such laser fields, the colloids are forced towards

a e-mail: [email protected]

Fig. 1. Archimedean-like tiling phases induced by (a,b)decagonal and (c,d) tetradecagonal substrates. (a,c) show ex-perimental results, (b,d) are obtained by Monte Carlo simula-tions. The figures show Delaunay triangulations of colloidalpoint patterns. Bonds longer than 1.1aS (1.2aS for experi-ments) are omitted, where aS is the spacing of the parti-cles when placed on an ideal triangular lattice. The potentialstrength is V0/(kBT ) = 13 in (a), V0/(kBT ) = 10 in (c), andV0/(kBT ) = 20 in the simulations (b,d). The density is givenby aS/aV = 0.57 in (a), aS/aV = 0.58 in (b), and aS/aV = 0.7in (c,d).

26 The European Physical Journal E

Fig. 2. The external potentials are created by interfering laser beams with a symmetric arrangement. An interference patternwith 10-fold rotational symmetry is obtained with five laser beams (a-d), 14-fold symmetry is realized with seven beams (e-h).(a,e) show how the laser beams are arranged and (b,f) the wave vectors projected onto the sample plane. (c,g) demonstrate theinterference patterns observed in experiments, and in (d,h) they are calculated from eq. (1). The bar in the upper right cornersof (d,h) denotes the length scale aV = 2π/|Gj |, where Gj are the projected wave vectors as shown in (b,f).

the regions with highest laser intensity, i.e., for the col-loidal particles the interference patterns act like a sub-strate with quasicrystalline symmetry. If the influence ofthe laser field is weak, the colloidal particles form a trian-gular lattice in case of sufficiently strong repulsion forcesor they are in a fluid phase when colloidal interactions areweak. For large laser intensities, the ordering of the col-loids is determined by the interference pattern, i.e., a qua-sicrystalline phase is observed (see also the phase diagramsin fig. 3 and in [19]). In this work we are especially inter-ested in phases that occur at intermediate laser strengthswhere there is a competition between the repulsive col-loidal interactions (that alone would lead to periodic tri-angular ordering) and the external forces due to the laserfield which favor aperiodic quasicrystalline structures. Forcertain particle densities that we will determine later inthis article, the competition leads to a colloidal orderingthat consists of rows of triangles and rows of squares thatestablish an aperiodic order in one spatial direction (seefig. 1). The structure observed is close to an Archimedeantiling, where the rows of squares and triangles form atwo-dimensional periodic order. It was therefore namedArchimedean-like tiling when it was first realized in ex-periments using laser fields with decagonal symmetry [18].Later Archimedean-like tilings also occured in simulationsthat studied the complete phase behavior on a decagonalsubstrate (ref. [19], see also fig. 3).

Fig. 3. (Colour on-line) Phase diagram obtained with MonteCarlo simulations for colloids on the decagonal substrate (thecomplete phase diagrams are presented and explained in [19]).Unlike quasicrystalline phases that we observed at all densi-ties, Archimedean-like tiling structures are only found for in-termediate potential strengths within a small range of densities(marked green).

The purpose of this article is to characterize the struc-ture of the Archimedean-like tiling phases in detail, todiscuss the differences of the ordering in laser fields withdecagonal and tetradecagonal symmetry, and finally todetermine the densities where these phases occur. Inaddition, we study the influence of phasonic distortions oncolloidal ordering in laser fields with decagonal symmetry.Phasons are excitations that only exist in quasicrystals.

M. Schmiedeberg et al.: Archimedean-like colloidal tilings on substrates with decagonal and tetradecagonal symmetry 27

Like phonons, they are hydrodynamic modes, i.e., they donot cost free energy in the long-wavelength limit [21,22].Phasons are an ongoing main topic in current research onquasicrystals and intensively discussed in the field [23–25].

In sect. 2 we introduce our system. The details of theexperimental realization as well as the numerical simu-lations are explained in sect. 3. In sect. 4 we analyze thestructure of the Archimedean-like tilings and calculate thedensities where these phases occur. We study the conse-quences of phasonic distortions on the Archimedean-liketilings in sect. 5 and finally conclude in sect. 6.

2 Model

A charged stabilized suspension is confined between twoglass plates and subjected to laser interference pat-terns with quasicrystalline symmetries. Five or sevenlaser beams with identical linear polarizations are em-ployed to obtain interference patterns with decagonalor tetradecagonal rotational symmetry, respectively (seefig. 2). For vanishing tilt angle θ → 0 of the beams, thepotential in the xy plane is [17,26]

V (r) = − V0

N2

N−1∑

j=0

N−1∑

k=0

cos [(Gj − Gk) · r + φj − φk] , (1)

where N is the number of beams, φj are the phases of thelaser light waves, and Gj their wave vectors projected ontothe xy plane (cf. fig. 2(b) and (f)). The prefactor is chosensuch that −V0 gives the minimum value of the potential.We usually set φj = 0 for all j corresponding to a laserfield without any phononic or phasonic displacements anda center of exact 10- or 14-fold rotational symmetry atr = 0. In addition, for the decagonal potential, we ex-plicitly analyze the effect of phasonic distortions on theArchimedean-like tiling in the last section. In this case,the phases φj are used to specify the phononic displace-ment field u(r, t) = [ux(r, t), uy(r, t)] and the phasonicfield w(r, t) = [wx(r, t), wy(r, t)] following the conventionof refs. [21,22],

φj (r, t) = u (r, t) · Gj + w (r, t) · G3j mod 5. (2)

In sect. 5 we will show how a uniform phasonic displace-ment w(r, t), which increases linearly in time, or how astatic w(r, t) with a constant gradient can be used to alignthe rows of the Archimedean-like tiling phases along spe-cific directions of the underlying substrate potential.

3 Methods

In the following, we shortly introduce our methods, whichwe employed to analyze the results from experiments andsimulations.

3.1 Experiment

The colloidal system we used is an aqueous suspension ofhighly charged sulphate-terminated polystyrene particleswith a radius of R = 1.45μm from Interfacial Dynam-ics Corporation with an average polydispersity below 4%.The pair interaction of particles can be described by ascreened Coulomb potential as introduced below in eq. (3).As sample cell we used a silica glass cuvette with 200 μmspacing. First, the sample cell was connected to a deion-ization circuit to reduce the salt concentration of the sus-pension. This circuit also included an electrical conductiv-ity probe (typical ionic conductivities below 0.08μS/cm),a vessel of ion exchange resin, and a peristaltic pump.After deionization, the suspension was inserted into thecell which was sealed afterwards. Before applying thequasiperiodic potential, the particle density was increasedin the field of view. This is achieved by thermophoreticand optical forces using vertically incident fiber coupledinfrared laser with a wavelength of λ = 1070 nm andmaximum output power Pmax = 5W and an argon ionlaser (λ = 514 nm, Pmax = 7W). A circular monolayerwas formed on the lower surface of the sample cell witha diameter of about 500μm. The vertical fluctuations ofthe particles are also suppressed by the argon ion beam,therefore the system can be considered as two dimen-sional. The monolayer spontaneously adapts the hexag-onal packing. After reaching the desired particle density,the infrared laser was turned off. The high particle densitywas kept constant using an optical tweezer (λ = 488 nm,Pmax = 7W) scanned around the central region of thesystem, i.e., forming a circular corral. The quasiperiodicpotential was created by interference of five or seven lin-early polarized beams of a Nd:YVO4 laser (λ = 532 nm,Pmax = 18W). Due to optical gradient forces, this inter-ference pattern acts as an external potential on the parti-cles. The depth V0 of the singular deepest potential well(see sect. 2) scales linearly with the laser power. V0 wasdetermined by calibrating the depth of potential wells ina triangular pattern created by three laser beams. Thelength scale aV was tuned by changing the angle of inci-dence of the laser beams with respect to the sample plane.Different colloidal phases were identified with the help of aDelaunay triangulation. It creates nearest-neighbor bondsbetween vertices that were defined by the maxima of thecolloidal density distribution. For Archimedean-like tilingstructures the bond length is a bimodal distribution withthe ratio of the peak positions close to

√2. Bonds longer

than 1.2aS were removed from the triangulation whichresulted in the characteristic square-triangle tiling.

3.2 Simulations

The interaction between the colloids is modeled by thepair potential

φ(d) =[Z∗e exp(κR)]2

4πε0εr(1 + κR)2exp(−κd)

d(3)


of the Derjaguin-Landau-Verwey-Overbeek theory [27,28],where d is the distance between two colloids and κ the in-verse Debye screening length. The prefactor also dependson the radius R of a colloid, its effective surface chargeZ∗, and the dielectric constant of water εr.

We quantify the particle density by the spacing aS

of the particles when placed on an ideal triangular lat-tice. The occurrence of Archimedean-like tilings mainlydepends on the density of the system. All other parame-ters can be varied over huge ranges. The main limitationto the choice of parameters is that there has to be a com-petition between the colloidal interactions and the inter-action with the substrate. Therefore, we usually choosethe parameters such that the density used in the simula-tions is slightly above the density where the triangular-to-liquid phase transition occurs. For example, the pa-rameter set for the decagonal substrate employed alreadyin [19] is suitable to observe Archimedean-like tilings, i.e.,R = 1.2μm, Z∗ = 1000, εr = 78, T = 300K, aV = 5.0μm,and κ−1 = 0.25μm. Whereas a parameter set closer to theexperimental values R = 1.45μm, Z∗ = 400000, εr = 78,T = 300K, aV = 6.5μm, and κ−1 = 0.2μm (cf. [20]) wasemployed to simulate colloidal ordering in the tetradecago-nal laser field. We usually performed Monte Carlo simula-tions using the Metropolis algorithm [30]. Periodic bound-ary conditions were implemented for simulation boxes thatcontained about 500 to 1400 colloids. For comparison or tostudy dynamical properties, we also employed Browniandynamics simulations as specified in [17].

When domains with Archimedean-like tiling struc-tures occur, their rows are aligned along the respectivefive or seven equivalent directions of the laser potential.Usually within one simulation box, one finds many do-mains with different orientations. There is a tendency thatsmall domains anneal out in time, i.e., adjacent domainsgrow slowly within the simulation process. The structuresshown in fig. 1 are about the largest domains we achievedby pure Monte Carlo simulations. However, we discovereddifferent methods to support the formation of larger do-mains. While in Monte Carlo simulations the directionof a domain may change quite easily, such domain flipsare rare in Brownian dynamics simulations. Therefore,Brownian dynamics simulations can help to grow largeuniform domains. It is also possible to predefine a pre-ferred direction. For example, by using small simulationboxes where the potential is discontinuous at the bound-aries, the rows of an Archimedean-like tiling start to formalong the boundaries. Furthermore, phasonic distortionssupport the growth of Archimedean-like tilings along spe-cific directions as discussed in sect. 5.

4 Archimedean-like tilings

The structures we observe in the two-dimensional qua-sicrystalline potentials at certain particle densities consistof rows of squares and rows of triangles and therefore aresimilar to the Archimedean tiling shown in fig. 4. In gen-eral, Archimedean tilings are formed by regular polygonsand are characterized by the number of edges of the poly-

Fig. 4. (Colour on-line) Archimedean tiling of type (33 · 42).

gons that meet in each vertex [31]. The tiling shown infig. 4 is called (33 · 42)-Archimedean tiling, because goingaround a vertex, one first finds three regular triangles andthen two squares (see red arrow in fig. 4). Very recently,similar Archimedean tilings were also observed for binarymixtures of nanoparticles situated between a cubic anda dodecagonal phase [32]. Furthermore, an Archimedeantiling of type (32 · 4 · 3 · 4) occurred in simulations ofmonoatomic systems on periodic substrates [33–36].

4.1 On decagonal substrates

The Archimedean-like tiling on a decagonal substratediffers from the perfect (33 · 42)-Archimedean tiling bythe frequent appearance of double rows of triangles (seefigs. 1(a,b)). From the simulation data we were able toidentify the sequence over a range of −6aV ≤ y ≤ 100aV

that corresponds to 77 single or double rows of triangles.Within this range the observed sequence is exactly thesequence of the Fibonacci chain, if one identifies a dou-ble row with L and a single row with S (cf. fig. 1(b)). AFibonacci chain is obtained by starting with element Land repeatedly applying the iteration rules L → LS andS → L. For example, the first iterations are L, LS, LSL,LSLLS, LSLLSLSL, etc.

Note that the squares and triangles of the Archime-dean-like tiling are not perfect regular polygons. A row oftriangles has almost the same height as a row of squares. Inthe simulations, we find the ratio of the mean row heightly divided by the mean particle spacing lx in the directionalong the rows to be ly/lx ≈ 0.95, i.e., the triangles arestretched in the direction perpendicular to the rows whilethe squares are stretched in the direction along the rows.We will calculate the ratio ly/lx in subsect. 4.3.

At a first glance, the Archimedean-like tiling structureseems to be periodic along the rows of triangles. However,some aperiodic modulations along the rows can be ob-served. For example, in the structures shown in figs. 1(a,b)only very few bonds are perfectly horizontal. Whereasmost of the bonds between two neighboring rows of trian-gles are almost horizontal, the direction of all other bondsdeviate from the horizontal direction in an aperiodic way.Furthermore, there is a modulation of the bond lengthsalong the direction of the rows. As a result, some rowsof squares exist along a direction that is rotated by 2π/5from the main horizontal row direction.


Fig. 5. Colloidal order close to symmetry centers in simu-lations: (a) On decagonal substrates double rows of trianglesoccur at symmetry centers, (b) in tetradecagonal laser fieldsdouble rows of squares are found.

4.2 On tetradecagonal substrates

The Archimedean-like tiling phase on a tetradecagonalsubstrate consists of large periodic regions that corre-spond to a perfect Archimedean tiling of type (33 · 42).The periodicity is only interrupted by few double rowsof squares. Figure 5(b) shows the position of the colloidson the substrate. Obviously, a double row of squares oc-curs at a local symmetry center of the laser field. Thisbright spot corresponds to a deep potential minimum thatis surrounded by 14 shallower potential wells. It there-fore constitutes the center of a region with 14-fold ro-tational symmetry. Since in a tetradecagonal laser fieldlocal symmetry centers are rare [20], only few doublerows of squares exist in the Archimedean-like tiling struc-ture. Figure 5(a) demonstrates that for the decagonal sub-strate double rows of triangles occur at symmetry cen-ters. Since there are much more symmetry centers in adecagonal laser field [20], much more double rows exist inthe Archimedean-like tiling phase of a decagonal substratecompared to a tetradecagonal laser field.

4.3 Densities with Archimedean-like tilings

In the following, we want to estimate the densities whereArchimedean-like tiling structures occur. At a suitabledensity, the rows of colloids should be oriented alonglines with many minima of the potential landscape. Inthe decagonal laser field, we can determine the distancesbetween such lines by averaging the potential V (r) alongthe x-direction, i.e.,

〈V 〉x =V0

25[−5 − 2 cos (2πy/λ1) − 2 cos (2πy/λ2)] , (4)

with

λ1 =

√5 −

√5

10aV

and

λ2 =

√5 +

√5

10aV .

We now calculate the density of an Archimedean-like tilingwhose mean row height ly fits to one of the lengths λ1

or λ2.

Fig. 6. (Colour on-line) We calculate the mean bond lengthin horizontal direction lx in a way that many colloids lie alongthe 2π/5-direction (circled in red). The mean row hight ly ischosen to fit one of the two length scales of the substrate inthe y-direction.

The simulation results show that the mean bond lengthlx in the direction of the rows differs from the row heightly. Therefore, in the following, we aim at calculating theratio ly/lx. So far we have determined the row height lyof an Archimedean-like tiling that fits to the decagonalsubstrate. To estimate the bond length lx, we argue asfollows. The rows are oriented along a symmetry directionof the decagonal potential. On this direction many poten-tial minima are situated that are occupied by the particles.However, an equivalent symmetry direction is found at anangle of 2π/5 with the rows. It should also be occupied byas much particles as possible. Figure 6 demonstrates howthis direction can be fitted to the Archimedean-like tiling.Colloids close to the red line are marked by red circles. Incase of a single row of triangles, to move from one particleto another close to the red line, one steps lx/2 in the x-direction and 2ly in the y-direction. In case of a double rowof triangles, the step lenghts are lx in the x-direction and3ly in the y-direction. The sequence of single and doublerows corresponds to the sequence of the Fibonacci chain.Therefore, there are τ times as many double rows as sin-gle rows of triangles, where τ = (1 +

√5)/2 ≈ 1.618 is

the number of the golden mean. As a consequence, theorientation of the red line is determined by

tan(2π/5) =(2 + 3τ)ly(

12 + τ

)lx

, (5)

i.e., we find the ratio ly/lx ≈ 0.951 which agrees remark-ably well with the value determined from simulations insubsect. 4.1.

To calculate the mean particle spacing aS , we comparethe number density in the Archimedean-like tiling,

ρ =1

lxly, (6)

with the number density of a triangular lattice ρ =2/(

√3a2

S). By using lx from eq. (5), we find for the mean


-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

inte

ract

ion

ener

gy c

ollo

ids-

subs

trat

e

particle spacing aS/aV

Archimedean-like tilingtriangular

Fig. 7. Average potential energy of a colloid in the decagonalpotential calculated for a triangular and for an Archimedean-like tiling structure as a function of density, which is given bythe particle spacing aS/aV . To determine the average potentialenergy, we constructed a perfect triangular or Archimedean-like tiling structure with the corresponding particle spacingand calculated the average potential depth at the positions ofthe colloids of the structure.

particle spacing of the Archimedean-like tiling,

aS =

√2√3

2+3τ12 +τ

1tan(2π/5)

ly =[

815

(5−

√5)] 1

4

ly, (7)

where ly has to be one of the length scales λ1,2 intro-duced in eq. (4). Therefore, Archimedean-like tilings onthe decagonal substrate occur for

aS/aV = 2[

175

(5 − 2

√5)] 1

4

≈ 0.579,

or aS/aV =[

875

(5 +

√5)] 1

4

≈ 0.937. (8)

Indeed, in experiments and in simulations, we find theArchimedean-like tiling structures for densities close tothese theoretical values (see, e.g., phase diagram in fig. 3).

In fig. 7 we show how the average potential energy ofa colloid varies with density when it is part of either atriangular lattice or an Archimedean-like tiling structureunder the influence of the decagonal substrate potential.We assumed that the Archimedean-like tiling structure isstretched perpendicular to the direction of the rows suchthat eq. (5) holds. From fig. 7 we can clearly identify twodensities where the Archimedean-like tiling fits to the sub-strate very well. Most of the colloids are located in minimaof the potential and therefore the average potential energyexhibits two sharp minima. These densities correspond tothe values calculated in eq. (8). Note that we also deter-mined the average potential energy for other Archimedeantilings, but did not find any other structure that showspronounced minima. For example, an Archimedean tilingof type (32 · 4 · 3 · 4) has exactly the same particle inter-action energy as the tiling of type (33 · 42). However, bycomparing the average potential energies, we find that thesubstrate favors the row structure corresponding to type(33 · 42).

-0.28

-0.26

-0.24

-0.22

-0.2

-0.18

-0.16

-0.14

-0.12

-0.1

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

inte

ract

ion

ener

gy c

ollo

ids-

subs

trat

e

particle spacing aS/aV

Archimedean tilingtriangular

Fig. 8. Average potential energy of a colloid in thetetradecagonal potential calculated for a triangular and foran Archimedean-like tiling structure as a function of density,which is given by the particle spacing aS/aV .

To determine the densities where Archimedean-liketilings occur under the influence of the tetradecagonal sub-strate potential, we calculate the average potential energyof a colloid within an Archimedean tiling structure as afunction of the particle spacing (see fig. 8). Again, we findpronounced minima of the potential energy (although notas sharp as for decagonal potentials) at aS/aV = 0.557,aS/aV = 0.691, and aS/aV = 0.861. At these values wewere indeed able to observe Archimedean-like tilings inthe simulations.

We just argued that Archimedean-like tilings shouldonly occur at specific densities. Correspondingly, whenusing Monte Carlo simulations to find the equilibriumphases, we only observe them close to the calculated densi-ties, e.g., in a narrow density region in the phase diagramof fig. 3. Whereas in the simulations the density is a con-stant quantity, it can to a certain degree change its valuein experiments; the experimental system is large enoughthat the density can become non-uniform and colloids canleave the boundary box. Indeed, when in the experimentwe bring the density close to the ones predicted by eq. (8),we observe that the colloids self-adjust their density sothat the Archimedean-like tiling forms. Figure 7 explainsthis observation since the energy is lowered by forming anArchimedean-like tiling, at least relative to the triangu-lar phase. To further investigate the idea of self-adjustingthe colloidal density, we performed the following Brow-nian dynamics simulations: We confine the colloids at adensity given by aS/aV = 0.565 that is larger than theone suitable for Archimedean-like tilings (aS/aV = 0.58).We then remove the confining boundaries so that the col-loids are able to float over the substrate potential and thesystem expands. Indeed, we find that particles adjust tothe potential by forming the characteristic rows of squaresand triangles of an Archimedean-like tiling (see fig. 9). Ofcourse, this structure is not the thermodynamic groundstate in such an open system. However, it remains presentduring the duration of the simulation which is the time afree particle would need to diffuse a distance aV . Only atthe edges of an Archimedean-like tiling domain does thestructure start to dissolve. This demonstrates that, while


Fig. 9. Brownian dynamics simulation without confiningboundaries started at an initial density that corresponds toaS/aV = 0.565. (a) Delaunay construction (as in fig. 1) dis-playing the colloidal configuration when the simulation starts(t = 0). After time t = 0.05γa2

V /kBT , where γ is the frictioncoefficient of a colloid, an Archimedean-like tiling structure canbe observed which remains almost unchanged up to the end ofour simulation at time t = 0.25γa2

V /kBT shown in (b). Notethat, while the Archimedean-like tilings shown in fig. 1 areequilibrium phases that occur in systems with densities fixedat special values, the structure in (b) is only metastable be-cause it will slowly dissolve when the density decreases due tothe missing boundaries.

Archimedean-like tilings are equilibrium phases at specialdensities only, they also occur as metastable structures insystems with open boundaries.

5 Phasonic distortions

In this section we demonstrate how the ordering of colloidsin the Archimedean-like tiling that forms in the decago-nal laser field is affected by manipulating the phasonicdegrees of freedom as defined in eq. (2). The consequencesof a constant phasonic displacement on a quasicrystallinephase with decagonal symmetry were presented in [19].

5.1 Inducing phasonic flips

We first study how a constant phasonic displacementinfluences the Archimedean-like tiling. Figure 10 showspatterns of the Archimedean-like tiling in the decagonallaser field for different phasonic components wy = 0 andwy = 0.015aV . Note that a whole row of triangles and arow of squares have interchanged their position. This cor-responds to a phasonic flip in the Fibonacci chain definedby the sequence of long and short distances between thesquare rows.

Figure 2(d) shows very clearly sets of parallel lines oflow intensity in the decagonal laser field that are orientedalong its five equivalent directions. One can choose pha-sonic displacements such that one set of dark parallel linesdoes not change. So if the square and triangular rows areparallel to this set of lines, the whole Archimedean-liketiling is not affected by the phasonic displacement. Forexample, a phasonic displacement wx does not influencethe square and triangular rows when they are oriented

Fig. 10. Inducing a phasonic flip: Delaunay construction (asin fig. 1) for the Archimedean-like tiling phase (aS/aV = 0.577,V0/(kBT ) = 20) obtained by Monte Carlo simulations forwy = 0 (top) and wy = 0.015aV (bottom). The flip in theFibonacci chain defined by the single and double rows of tri-angles is marked by dotted circles.

along the x-direction. As we show in the next subsec-tion, a phasonic drift in wx may even help to stabilizethe Archimedean-like tiling structure.

5.2 Stabilization by phasonic drifts and gradients

A steadily growing wx in time orients the Archimedean-like tiling along the x-direction. This is demonstrated infig. 11 for a phasonic drift velocity Δwx = 10−4aV perunit time. Domains that are not oriented along the x-direction are reduced since a non-zero Δwx constantlyinduces phasonic flips in the rows of squares and trian-gles of the Archimedean-like tiling unless they are ori-ented along the x-direction. Another way of explainingthe consequences of the phasonic drift velocity Δwx goesas follows. One can demonstrate that a steadily growingphasonic displacement rearranges the decagonal potentiallandscape such that existing potential wells disappear andreappear at other locations. Now consider a colloid in thedecagonal laser field that experiences a sufficiently fastphasonic drift in the wx-direction, i.e., the relaxation timeof the colloid within a potential well is much larger thanthe time scale on which the potential well vanishes dueto the phasonic drift. Then one can introduce an effectivepotential 〈V 〉wx

(y) for the colloids that is the decagonal


Fig. 11. Stabilization by a phasonic drift: Archimedean-liketiling phase (same parameters as in fig. 10) obtained by MonteCarlo simulations without phasonic displacement (top) and ina potential with a phasonic drift velocity of Δwx = 104aV

per Monte Carlo step (bottom). The small graph on the right-hand side shows the decagonal potential averaged over wx asa function of y.

potential V (x, y) averaged over wx. It is illustrated on theright-hand side of fig. 11. From eq. (1) it is clear that〈V 〉wx

(y) does not depend on x or equals V (x, y) aver-aged over the spatial coordinate x already introduced ineq. (4), i.e.,

〈V 〉wx(y) = 〈V 〉x(y). (9)

Clearly, such an effective potential can only support do-mains of Archimedean-like tilings oriented along the x-direction.

Another possibility of orienting domains of Archime-dean-like tilings preferentially along the x-direction is tointroduce a phasonic displacement field with a constantgradient in the x-direction. Figure 12 shows the resultingdecagonal interference pattern. The gradient in wx de-stroys all continuous lines of low intensity in the potentiallandscape, except those in x-direction, as indicated by thered lines. Therefore, the Archimedean-like tiling is mainlyoriented along the direction of the phasonic gradient, asillustrated in fig. 13.

In the experimental interference pattern of fig. 2c someof the lines of low intensity just end in the middle ofthe pattern. This suggests that phasonic distortions arepresent, as we will now demonstrate. Suppose that the fivelaser beams in fig. 2a are not perfectly arranged aroundthe vertical. For example, if the tilt angle θ0 of beam 0 islarger than the other angles, the projected wave vector be-comes G′

0 = G0 + ΔG0, with |G′0| > |Gi| (i = 0, 1 . . . 4).

From eq. (1) one finds that the deviation ΔG0 can be in-

Fig. 12. (Colour on-line) Interference pattern of the decagonallaser field calculated from eq. (1) when the phase φ0 attachedto the wave vector G0 varies with a constant gradient, φ0 =0.2x/aV . Via eq. (2) one can show that it creates a constantgradient in the x components of the phononic and phasonicdisplacements. As a result, the lines of low intensity are infinitlyextended in the x-direction whereas in all other directions theyhave a finite length (see, e.g., the dark stripes framed by thered lines). Similar jags are studied in [24,25].

terpreted as a constant gradient in the phase φ0 attachedto G0, φ0 = ΔG0 · r. Using eq. (2), this phase φ0 can bedecomposed into x components of the phononic and pha-sonic displacement fields with constant gradients. There-fore, laser beams that are not perfectly adjusted relativeto the vertical lead to a substrate potential with phononicand phasonic distortions. Furthermore, such distortionscan also be controlled intentionally by displacing one ofthe laser beams radially from its position on an ideal pen-tagon. This way one can create a preferred orientation forthe Archimedean-like tiling, as discussed above.

In summary, phasonic drifts or distortions can be em-ployed to grow large domains of an Archimedean-like tilingstructure with a preferred orientation. These domains arestable when the phasonic drift is stopped or the phasonicdistortion is removed.

6 Outlook and conclusions

In this article we have studied the structure of Archime-dean-like tiling phases that form in decagonal andtetradecagonal laser fields at certain densities. We havealso demonstrated how the domain size of such tilings isinfluenced by phasonic distortions. In particular, if the in-terfering laser beams are not ideally adjusted, phasonicdisplacement fields with a constant gradient arise. Inter-estingly, simulations using substrate potentials with fur-


Fig. 13. Stabilization by a phasonic gradient: Archimedean-like tiling phase (same parameters as in fig. 10) obtained byMonte Carlo simulations without phasonic displacement (top)and in a potential with a phasonic gradient with wx/x = 0.15(bottom).

ther quasicrystalline symmetries also show similar phases.For example, in an interference pattern of eight beamswe find rows of squares and triple or four-fold rows oftriangles (fig. 14(a)). A laser field created by ten beamsfavors exactly the same structure as the decagonal poten-tial obtained by five laser beams (fig. 14(b)). For the caseof eleven beams, we also observe the characteristic rowsof triangles and squares (fig. 14(c)). However, we are notable to analyze this ordering in more detail with respectto the order perpendicular to the rows, since the domainsare too small. It is an interesting question for future re-search to understand, why such Archimedean-like tilingsform at specific densities on substrates with different qua-sicrystalline symmetries. Depending on the specific rota-tional symmetry, all these structures consist of single anddouble (or even triple or four-fold) rows of triangles orsquares that ultimately form a quasiperiodic order in onedirection.

While on the decagonal substrate the rows in the Ar-chimedean-like tiling form a Fibonacci chain, they showdomains with periodic order on the tetradecagonal sub-strate interrupted by double rows of squares. We find thatthese double rows, as the double rows of triangles on thedecagonal substrate, occur at symmetry centers of the sub-strate potential. As discussed in [20], the number of thesymmetry centers is by a factor of 100 larger in the decago-nal compared to the tetradecagonal substrate. Therefore,they enforce the formation of a Fibonacci chain in the firstcase, whereas in the tetradecagonal laser field larger pe-riodic domains form due to the much smaller number ofsymmetry centers.

Fig. 14. Delaunay construction (as in fig. 1) of simulationresults that consist of rows of triangles and rows of squaresin laser fields (of strength V0/(kBT ) = 50) created by (a)eight beams (aS/aV = 0.8), (b) ten beams (aS/aV = 0.7),and (c) eleven beams (aS/aV = 0.74).

Amongst other phases (see ref. [19]), Archimedean-liketilings appear as a compromise between structures favoredeither by the particle interaction or by the substrate po-tential. There is some evidence that they occur in exper-iments that study how atoms order on atomic surfaceswith decagonal symmetry [18]. For the specific exampleof the Archimedean-like tiling, we have presented hereBrownian dynamics simulations to demonstrate how thecolloidal adsorbate reacts on phasonic distortions in thesubstrate. This rises the question if phasonic excitationsare detectable in atomic substrates. The Archimedean-liketiling as an adsorbate phase might be a good candidateto visualize these excitations in the underlying substratesince resulting rearrangements in the rows of triangles andsquares should be visible.

We would like to thank R. Lifshitz and H.-R. Trebinfor helpful discussions. We acknowledge financial supportfrom the Deutsche Forschungsgemeinschaft under GrantNo. RO 924/5-1 and BE 1788/5.

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