Formation of dodecagonal quasicrystals in two-dimensional systems ofpatchy particlesMarjolein N. van der Linden, Jonathan P. K. Doye, and Ard A. Louis Citation: J. Chem. Phys. 136, 054904 (2012); doi: 10.1063/1.3679653 View online: http://dx.doi.org/10.1063/1.3679653 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v136/i5 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, 054904 (2012)

Formation of dodecagonal quasicrystals in two-dimensionalsystems of patchy particles

Marjolein N. van der Linden,1,a) Jonathan P. K. Doye,1,b) and Ard A. Louis2

1Physical and Theoretical Chemistry Laboratory, Department of Chemistry, University of Oxford,South Parks Road, Oxford, OX1 3QZ, United Kingdom2Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford,OX1 3NP, United Kingdom

(Received 26 November 2011; accepted 6 January 2012; published online 6 February 2012)

The behaviour of two-dimensional patchy particles with five and seven regularly arranged patches isinvestigated by computer simulation. For higher pressures and wider patch widths, hexagonal crystalshave the lowest enthalpy, whereas at lower pressures and for narrower patches, lower density crystalswith five nearest neighbours that are based on the (32,4,3,4) tiling of squares and triangles becomelower in enthalpy. Interestingly, in regions of parameter space near to that where the hexagonalcrystals become stable, quasicrystalline structures with dodecagonal symmetry form on cooling fromhigh temperature. These quasicrystals can be considered as tilings of squares and triangles and areprobably stabilized by the large configurational entropy associated with all the different possible suchtilings. The potential for experimentally realizing such structures using DNA multi-arm motifs is alsodiscussed. © 2012 American Institute of Physics. [doi:10.1063/1.3679653]

I. INTRODUCTION

Since the formation of quasicrystals was first reported in1984 by Shechtman et al. for an Al–Mn alloy,1 many sys-tems have been found to exhibit a quasicrystalline phase.Most of these are binary or ternary metallic alloys (but nevera pure metal). The quest to find quasicrystals beyond al-loys has led to an increasing number of examples, mostlyin the field of soft condensed matter.2, 3 Examples includedendrimers,4, 5 star polymers,6 micelles,7 and binary mixturesof nanoparticles.8 All of these examples, except for one of themicellar systems,7 have dodecagonal symmetry and are of-ten found in regions of parameter space close to where crys-talline approximants, such as the Frank-Kasper σ phase, areobserved.9–14

A remaining target is to find a colloidal system that canself-assemble into a quasicrystalline structure in the absenceof an external field (two-dimensional colloidal quasicrystalscan be induced to form using quasiperiodic light fields15–17).One approach might be to use a binary or ternary colloidalmixture, but although complex crystal structures have beenreported for binary mixtures,18–22 as yet no quasicrystals havebeen observed. Another approach might be to use colloidswith anisotropic “patchy” interactions,23 where the positionsof the patches could be used to control the preferred local ge-ometry and hence influence the global structure formed. In-deed, much progress has been made in developing methodsto synthesize such types of colloidal particles.24–33 Further-more, the first experiments on the novel structures that suchpatchy interactions can enable the systems to adopt are be-

a)Present address: Soft Condensed Matter, Debye Institute for Nanomateri-als Science, Utrecht University, Princetonplein 1, 3584 CC Utrecht, TheNetherlands.

b)Author to whom correspondence should be addressed. Electronic mail:jonathan.doye@chem.ox.ac.uk.

ginning to appear,34 as well as being systematically exploredthrough computer simulations.35–44

Quasicrystals have also been found to form for a vari-ety of model potentials in computer simulations. Interestingly,these are not restricted to mixtures,45, 46 but can also occurfor one-component systems.47–58 Examples include isotropicpair potentials with both a maximum and a minimum in thepotential,49–53 micellar models,54 hard tetrahedra55 and trian-gular bipyramids,56 and water57 and silicon58 bilayers.

Here, we wish to examine in detail the behaviour of two-dimensional patchy particles with five and seven patches aspossible quasicrystal-forming systems. The reason we choosethis system is that in a recent study of two-dimensional diskswith regularly arranged patches, intriguing behaviour wasseen for the five-patch system.36, 41 Hard disks naturally forma hexagonal crystal, and six patches reinforce this tendency.For four patches, there is a competition between a square crys-tal, which is energetically stabilized by the patchy interac-tions, and a hexagonal crystal, which is stabilized by its higherdensity. However, the situation for particles with five regularlyarranged patches is more complex, since the five-fold sym-metry of the particles is incompatible with crystalline order—there is no crystal in which all the patches can point directlyat those on neighbouring particles.

In Ref. 36, we found that on cooling (at low pressure),crystallization was not observed, because of the frustrationintroduced by the geometry of the patches. The configura-tions generated did though show certain common local mo-tifs (those shown in Fig. 1) and could be considered as tilingsof squares and triangles. Although there was no overall crys-talline order, there was some evidence suggestive of longer-range orientational order; however, because of the relativelysmall systems sizes considered this was not pursued furtherat the time. Given the known tendency of random square-triangle tilings to form dodecagonal quasicrystals,59 we now

0021-9606/2012/136(5)/054904/11/$30.00 © 2012 American Institute of Physics136, 054904-1

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054904-2 van der Linden, Doye, and Louis J. Chem. Phys. 136, 054904 (2012)

ZHσ

FIG. 1. The three main local environments that are present in the simulatedstructures.

investigate in detail the suggestion made in Ref. 36 that thissystem may form quasicrystals.

II. METHODS

A. Potential

The model consists of spherical particles patterned withattractive patches. They are described by a modified Lennard-Jones potential, in which the repulsive part of the potential isisotropic, but the attractive part is anisotropic and depends onthe alignment of patches on interacting particles. Specifically,the potential is described by

Vij (rij ,�i ,�j ) ={

VLJ(rij ) r < σLJ

VLJ(rij )Vang(r̂ij ,�i ,�j ) r ≥ σLJ,

(1)where VLJ, the Lennard-Jones potential, is given by

VLJ(r) = 4ε

[(σLJ

r

)12−

(σLJ

r

)6]

. (2)

The minimum of this potential is at req = 21/6σ LJ. Vang is anangular modulation factor that depends on the orientations ofthe patches on the two interacting particles with respect to theinterparticle vector. Specifically,

Vang(r̂ij ,�i ,�j ) = Gij (r̂ij ,�i)Gji(r̂ji ,�j ), (3)

where

Gij (r̂ij ,�i) = exp

(−θ2

kminij

2σ 2pw

), (4)

σ pw is a measure of the angular width of the patches, θ kij isthe angle between patch vector k on particle i and the inter-particle vector rij , and kmin is the patch that minimizes themagnitude of this angle. Hence, only the patches on each par-ticle that are closest to the interparticle vector interact witheach other, and Vang = 1 if the patches point directly at eachother. One feature of this potential is that as σ pw → ∞, theisotropic Lennard-Jones potential is recovered. For compu-tational efficiency, the potential is truncated and shifted atr = 3 σ LJ, and the crossover distance in Eq. (1) is adjustedso that it still occurs when the potential is zero.

As well as studying crystallization in twodimensions,36, 41, 44 this patchy particle model has alsobeen previously used to study crystallization in threedimensions,36–38 and the self-assembly of monodisperseshells.60–62

B. Simulation

Systems of particles were simulated using standardMetropolis Monte Carlo (MC) in the NPT ensemble. Peri-odic boundary conditions were applied, and the simulationbox was constrained to be square. The MC moves were single-particle rotational and translational moves, as well as “vol-ume” moves in which the area of the box was scaled, the latterallowing a constant pressure ensemble to be simulated. Thenumber of particles N was always 2500.

C. Structural analysis

We employ a number of approaches to analyse the struc-tures that the systems of patchy particles adopt. In the con-densed state, we find that the five- and seven-patch particlesvirtually always adopt one of three local environments. Theseare illustrated in Fig. 1. There are two possible five-coordinateenvironments, which are based upon two different ways of lo-cally packing squares and triangles. We refer to them as the σ

and H environments by analogy to the Frank-Kasper phasesof these names.63, 64 These σ and H Frank-Kasper phasesare crystalline approximants to dodecagonal quasicrystals andcan be considered as square-triangle tilings in two of theirthree dimensions.65 The tilings containing only these local en-vironments are more formally denoted by (32, 4, 3, 4) (σ ) and(33, 42) (H), where this nomenclature refers to the sequenceof polygons around a vertex.66 The third local environment isthe six-coordinate hexagonal environment, which representsthe densest local packing, and will be denoted Z, again byanalogy to a Frank-Kasper phase.

To identify these environments, we employ a commonneighbour analysis. Neighbours are defined as all particleswithin a certain cutoff radius rc from a given particle. Thenfor each pair of neighbours, the number of neighbours com-mon to both is determined. Each local environment is charac-terized by a unique signature in terms of the number of neigh-bours with which the central particle shares a certain numberof neighbours. For example, a particle in a σ local environ-ment has a total of five neighbours, with one of which it hastwo neighbours in common and with four of which it has oneneighbour in common. Hence, this environment is denoted bythe common neighbour signature {21111}. In a similar way,the signatures for the H and Z local environments are {22110}and {222222}, respectively. Particles that are not in any ofthese three local environments are labelled “undefined” (U).The cutoff radius (rc ≈ 1.38 σ LJ) was chosen such that thefraction of U-particles was minimized.

To probe the global order of the configurations generated,the associated diffraction patterns were calculated. Quasicrys-talline configurations are characterized by diffraction patternswith non-crystallographic rotational symmetries. We calcu-lated the diffraction patterns by evaluating the real part of theinterference function

S(q) = 1

N

N−1∑i=0

N−1∑j=0

exp[2π iq · (ri − rj )], (5)

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054904-3 Dodecagonal quasicrystals of patchy particles J. Chem. Phys. 136, 054904 (2012)

where q is the wavevector and ri is the position of particle i.Unless stated otherwise, the resolution we used for the plotsof the diffraction patterns is �qx = �qy = 0.15 σ−1

LJ .When mapping out the behaviour of our system as a

function of the parameter space, it will be useful to measurethe degree of 12-fold symmetry of the diffraction pattern. Toachieve this, we take a Fourier transform around a ring ofthe interference pattern at a q value corresponding to the firstdiffraction peaks. Specifically, we evaluate

F (ν) =n−1∑i=0

S(q1st, φi) exp (−2π iνφi) , (6)

where the sum is over the n sampled points of S(q1st, φ). A12-fold symmetric diffraction pattern will have high values atν = 12 and multiples thereof. Similarly, a six-fold symmet-ric diffraction pattern, as would be expected for a hexagonalcrystal, will have high values at ν = 6 and multiples thereof(including ν = 12). Therefore, we used |F(ν = 12)| − |F(ν= 6)| as a measure of the 12-fold character of the diffrac-tion pattern. We note that the peaks at multiples of the lowestfrequency will have a lower amplitude than the main lowestfrequency peak, where this decay in amplitude is stronger ifthe diffraction peaks are more diffuse.

III. RESULTS

We will first consider the five-patch particles in detail.The behaviour of the seven-patch particles is fairly similarand so we will consider this system more briefly in Sec. III B.

A. Five-patch particles

1. Low-enthalpy structures

To begin to understand how different conditions and po-tential parameters affect the competition between differentcrystal forms for the five-patch particles, we show how thelowest enthalpy crystal structure depends on pressure andpatch width in Fig. 2. As the most stable phase at zero temper-ature is simply that with the lowest enthalpy, this is equivalentto a zero temperature phase diagram. The figure was obtainedby minimizing the enthalpy for a variety of candidate crystalstructures for different p and σ pw until the phase boundariesbetween different crystal forms were precisely located. Alsonote that the figure is slightly different from that which ap-peared in Ref. 36, because a slightly lower energy crystal atlow σ pw was subsequently identified in Ref. 41.

At sufficiently high pressure the lowest enthalpy phasewill be the crystal with highest density, which in this system isthe hexagonal crystal. At sufficiently low pressure, the crystalstructure with the lowest energy will be lowest in enthalpy andwill be the one which maximizes the patch–patch interactions.

At intermediate values of the patch width, the low-energycrystals are those based on the two five-coordinate local envi-ronments depicted in Fig. 1. Although the patches are not ableto point directly at those on neighbouring particles in thesemotifs, the loss in energy is not prohibitive because the an-gular deviations are relatively small (Fig. 3). As the average

0.01

0.1

1

0 0.2 0.4 0.6 0.8 1(radians)σ

hexagonal

pres

sure

distortedσ−phase

σ−phase

FIG. 2. Zero-temperature phase diagram showing the dependence of the low-est enthalpy structure on pressure and patch width.

angular deviation is smaller in the σ local environment, the σ

crystal is slightly lower in energy than the H crystal.As the patches become narrower, the energetic penalty

associated with the non-perfect alignment of the patches withthe interparticle vectors increases, until a point is reachedwhere it becomes favourable for the local environments todistort so that the patches point directly at three of the fiveneighbours. The resulting packings can be considered to bemade of the irregular hexagon shown in Fig. 3(c). For the Hcrystal, this distortion does not lead to any change in its spacegroup, namely, it remains as cmm. However, for the σ crys-tal there is a symmetry breaking—there are two equivalentways of dividing up the structure into these hexagons—andthe space group changes from p4g to p2. The resulting crys-tals are virtually degenerate, with the distorted σ structure justlower in energy.41

As the patch width is increased away from the intermedi-ate values at which the σ phase is favoured, the orientationaldependence of the potential becomes weaker, and there comesa point at which the hexagonal crystal becomes lowest in en-ergy because of its greater number of neighbours.

In the σ -phase, the ratio of triangles to squares is 2. Thereare also many larger unit cell crystal structures with a mix-ture of σ and Z environments that have an increasing ratio oftriangles to squares. We thought that these more complex

12

(a) (b) (c)Hσ

6o

o 18o

6o

72o

FIG. 3. Deviations of the patch vectors from the interparticle vectors in the(a) σ - and (b) H-type environments. (c) At low values of the patch width, itis favourable for these structures to distort so that three of the patches pointdirectly at the neighbouring particles. The resulting structures can be viewedas being made up of the depicted hexagon.

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054904-4 van der Linden, Doye, and Louis J. Chem. Phys. 136, 054904 (2012)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.3 0.4 0.5

Fra

ctio

n of

par

ticle

s

Temperature / εLJkB-1

σHZU

0.0

0.2

0.4

0.6

0.8

1.0

0 200000 400000 600000 800000 1×106

Fra

ctio

n of

par

ticle

s

Number of MC cycles

σHZU

(a)

(b)

U

H

σ

Z

FIG. 4. Evolution of local structural environments during (a) cooling and(b) annealing at T = 0.15 εk−1. p = 0.7 εσ−2

LJ , σ pw = 0.49.

crystals might be most stable near to the σ /hexagonal bound-ary in Fig. 2. However, we never found an instance where oneof these structures had the lowest enthalpy.

Of course, the positions of the phase boundaries betweendifferent crystal forms will depend somewhat on temperature.In particular, the hexagonal crystal is likely to have a greaterentropy than the σ crystal because it is orientationally disor-dered. Hence, we expect the σ /hexagonal phase boundary tomove to lower patch width with increasing temperature.

2. Annealing

We do not expect quasicrystalline configurations to belowest in enthalpy, both because the closest quasicrystallineapproximants (the crystals involving both σ and Z local en-vironments) are never lowest in enthalpy (Sec. III A 1) andbecause the disorder associated with quasicrystals is likely tolead them to have a somewhat higher enthalpy than the rel-evant approximant. However, it may be that a quasicrystal ismore kinetically accessible than possible crystals on coolingor is even thermodynamically stable for a particular tempera-ture range due to its greater entropy. Therefore, to search forquasicrystalline behaviour we performed a series of coolingruns for a grid of pressure and patch width values. In theseruns, the temperature was decreased linearly from 0.5 εk−1 to0 over 50 000 MC cycles. Some of the resulting configura-tions did show 12-fold diffraction patterns characteristic of adodecagonal quasicrystal. However, the peaks were very dif-fuse. Therefore, we subsequently annealed all the final con-figurations at a temperature of 0.15 εk−1 for 106 MC cycles.This temperature was chosen so that the mobility of the atoms

0

10

20

30

40

50

qx / σLJ-1

q y /

σ LJ-1

0

10

20

30

40

50

qx / σLJ-1

q y /

σ LJ-1

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4

5

0

10

20

30

40

50

qx / σLJ-1

q y /

σ LJ-1

0

10

20

30

40

50

qx / σLJ-1

q y /

σ LJ-1

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5(a) (b)

(c)

FIG. 5. (a) Configuration after annealing and (b) and (c) associated diffraction patterns (for two different ranges of q) at p = 0.7 εσ−2LJ , σ pw = 0.49.

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054904-5 Dodecagonal quasicrystals of patchy particles J. Chem. Phys. 136, 054904 (2012)

was large enough to allow significant ordering during anneal-ing. For the most part, this temperature was also below thatat which the quasicrystals formed, the exceptions occurringnear to the hexagonal/σ boundary in Fig. 2, where a hexago-nal phase formed first on cooling before transforming into thequasicrystalline phase at lower temperature.

Figure 4 shows the evolution of the number of parti-cles with different local environments for the cooling andannealing runs for a state point that led to a dodecagonalquasicrystal. On cooling, the number of σ environments in-creases rapidly between 0.2 and 0.1 εk−1 and this is due tothe formation of the quasicrystal. Prior to this, there is a tran-sient increase in the number of hexagonal particles. Closerto the hexagonal/σ boundary in Fig. 2, this transient increaseis more pronounced because of the increased stability of thehexagonal phase. Even though the hexagonal crystal is notthe lowest enthalpy structure under these conditions, it is ther-mally stabilized by its larger orientational entropy.

It is noteworthy that the number of hexagonal particlesdoes not decrease to zero on quasicrystal formation. This isnot due to incomplete ordering but is one of the features ofthe quasicrystalline structures. At the end of the cooling run,there is also still a significant fraction of particles whose localstructure cannot be assigned because of the disorder withinthe configurations.

On annealing, the number of σ environments graduallyincreases at the expense of unidentified and H environments(but not hexagonal) (Fig. 4(b)). The resulting configuration isshown in Fig. 5 along with the associated diffraction pattern.The diffraction pattern shows clear 12-fold symmetry and isvery similar to that for a “perfect” dodecagonal quasicrystalproduced by the “extended Schlottmann” inflation rules,67–69

albeit with less sharp peaks.70 The 12-fold pattern also impliesthat the orientational order is coherent across the whole box;it could be said to be a “single quasicrystal.”

The configuration is a square-triangle tiling in whichthe 12 possible orientations of the bond vectors are equallylikely—hence, the 12-fold symmetry in the diffraction pat-tern. The radial distribution function shows clear peaks outto quite long range (Fig. 6(a)) because of the square–triangleorder. Close inspection of the configuration shows that thehexagonal atoms for the most part do not cluster togetherbut are instead isolated from each other and are usually atthe centre of the dodecagonal motifs depicted in Figs. 7(a)and 7(b). These dodecagonal motifs can join together in twoways. They can share an edge as in Fig. 7(c) (their centresare separated by (2 + √

3)req = 4.189 σLJ) or interpenetrateas in Fig. 7(e) (separation (1 + √

3)req = 3.067 σLJ). Thesetwo distances are very apparent from the radial distributionfunction for particles in hexagonal environments (Fig. 6(b))with a clear preference for edge-sharing dodecagons.

The four basic ways of locally packing the dodecagonsthat are possible using interpenetrating and edge-sharingdodecagons are illustrated in Figs. 7(c)–7(f), two of whichare triangular, one of which is square and one of which isrectangular. These motifs can be used to construct a wholevariety of crystal structures with large unit cells (and varyingratios of squares to triangles), although as mentioned inSec. III A 1 we did not find an instance where these crystals

0

1

2

3

4

5

6

7

0 2 4 6 8 10

0

1

2

3

4

5

6

7

0 2 4 6 8 10

g (r)Z

g(r)

r

r

(a)

(b)

FIG. 6. Radial distribution functions for (a) all the particles and (b) the Zparticles in the quasicrystalline configuration obtained at p = 0.7 εσ−2

LJ , σ pw= 0.49. In (b) the peaks associated with interpenetrating (Fig. 7(e)) and edge-sharing (Fig. 7(c)) dodecagons are marked with arrows.

had the lowest enthalpy. The model quasicrystal that wecreated using the extended Schlottmann inflation rules70 canalso be analysed in terms of the edge-sharing dodecagonmotifs of Figs. 7(c) and 7(d).

When the configuration in Fig. 5(a) is examined, all fourof these motifs can be found, with the triangle of edge-sharingdodecagons (Fig. 7(c)) most common. When examining howthese triangular, square, and rectangular elements begin totile the plane, we find configurations like those in Fig. 1,but now, of course, on a longer length scale. Furthermore,this tiling seems to be random and does not completely tilethe plane, because of a significant fraction of defects, bothin terms of U particles with an “unidentified” coordinationshell and strings of H particles. It is noteworthy that σ andH crystals can form a coherent interface along the {11} di-rections of the σ crystals, and the latter defects seem to oftenoccur when the σ -like order in two adjacent regions is notfully in registry and needs a line of H atoms to bridge theregions.

Another source of disorder is the two possible orienta-tions of the central hexagon in the dodecagonal motif, as il-lustrated in Figs. 7(a) and 7(b). The two forms can be trans-formed into each other by a rotation of the hexagon by π /6.For the edge-sharing dodecagons in Fig. 7, the orientationsof the hexagons have been chosen so that all the particleson the edge of the dodecagons have the more favourable σ

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054904-6 van der Linden, Doye, and Louis J. Chem. Phys. 136, 054904 (2012)

(a)

(b)

(c) (d)

(e) (f)

FIG. 7. (a) and (b) Dodecagonal motifs and (c)–(f) four ways of locally packing these motifs.

environments. However, some relative orientations of the in-ternal hexagons lead to H environments. For example, thereare two H environments at the shared edge if both hexagonshave bonds parallel to the shared edge, and a few examples ofsuch H “dimers” can be found in Fig. 5(a). By contrast, forthe interpenetrating dodecagons (Figs. 7(d) and 7(e)), if the

hexagon at the centre of one of the dodecagons is rotated, thedodecagonal character of the other centres is lost, althoughthe packing is still a square–triangle tiling.

Another way to characterize the structures that we ob-serve is in terms of the ratio of triangles to squares in thetiling, and the related ratio of the number of five- and six-

0

0.2

0.4

0.6

0.8

1

Patch width / radians

Pre

ssur

e / ε

σ LJ-2

0.1 0.2 0.3 0.4 0.5 0.6

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0

0.2

0.4

0.6

0.8

1

Patch width / radians

Pre

ssur

e / ε

σ LJ-2

0.1 0.2 0.3 0.4 0.5 0.6

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

-0.2

0

0.2

0.4

0.6

Patch width / radians

Pre

ssur

e / ε

σ LJ-2

0.1 0.2 0.3 0.4 0.5 0.6

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0

0.2

0.4

0.6

0.8

1

Patch width / radians

Pre

ssur

e / ε

σ LJ-2

0.1 0.2 0.3 0.4 0.5 0.6

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

(d)

(a)

(c)

(b)

FIG. 8. Dependence on pressure and patch width of the fraction of particles in (a) σ , (b) Z and (c) H environments, and (d) |F(ν = 12)| − |F(ν = 6)|, a measureof the 12-fold character of the diffraction pattern, for the final configuration after annealing.

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054904-7 Dodecagonal quasicrystals of patchy particles J. Chem. Phys. 136, 054904 (2012)

(e)

(c)

(a) (b)

(d)

(f)

FIG. 9. Dependence of the final configuration after annealing on patch width (a) σ pw = 0.14, (b) σ pw = 0.21, (c) σ pw = 0.28, (d) σ pw = 0.35, (e) σ pw = 0.42,and (f) σ pw = 0.56. All are at p = 0.7 εσ−2

LJ .

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054904-8 van der Linden, Doye, and Louis J. Chem. Phys. 136, 054904 (2012)

σ H(b)(a)

8.6

4.3

4.3

12.9o

o

o

o

FIG. 10. Deviations of the patch vectors from the interparticle vectors in the (a) σ - and (b) H-type environments for seven-patch particles.

coordinate atoms. For example, we find the ratio of trian-gles to squares for the configuration in Fig. 5(a) to be 2.303,which compares with a value of 2 for the σ crystal and4/

√3 = 2.309 for an “extended Schlottmann” quasicrystal.71

Similarly, the ratio of five- to six-coordinate environmentsis 13.194 for the configuration in Fig. 5(a), which is againsimilar to the value for an “extended Schlottmann” quasicrys-tal, namely, (24 + 14

√3)/(2 + √

3) = 12.928. These resultsclearly indicate the close similarity of the structures we ob-tain to ideal dodecagonal quasicrystals.

So far we have only looked at the structure that results forone particular set of conditions. Figure 8 provides an overviewof the structural behaviour as a function of pressure and patchwidth, and Fig. 9 provides example final configurations fordifferent patch widths at a representative pressure.

In the top right corner of the (σ pw,P) plane, i.e., largerpatch widths and higher pressures, the hexagonal phase ismost stable (Fig. 2). As one moves away from this cornerof the parameter space to lower pressures and patch widths,the onset of quasicrystal formation is signalled by a sharpincrease in the number of σ environments at the expense ofhexagonal environments (Figs. 8(a) and 8(b)). Close to thisboundary, our measure of the twelvefoldness of the diffrac-tion pattern generally has high values, although this mea-sure should be interpreted with caution as false positives canarise. For example, the superposition of diffraction patternsfrom two different hexagonal domains leads to an anoma-lously high value of the twelvefoldness at P = 1.7 εσ−2

LJand σ pw = 0.63. The position of the hexagonal to qua-sicrystal boundary in Fig. 8 is noticeably to the left of thezero-temperature σ -hexagonal boundary in Fig. 2 because ofthe entropic stabilization of the hexagonal phase mentionedearlier.

As one moves away from this boundary, the number ofhexagonal atoms slowly decreases towards zero and the de-gree of twelvefoldness generally drops. Examining the con-figurations in Fig. 9, one sees a gradual crossover from thequasicrystalline configurations with hexagonal atoms at thecentre of the characteristic dodecagons to a pure σ phase inFig. 9(b). For example, about a half of the domains in Fig. 9(d)are pure σ and the rest are quasicrystal-like. This crossover

is driven by the hexagonal environments becoming ener-getically increasingly unfavourable as the patches becomenarrower.

At the narrowest patch width we considered, the low-energy crystals are the distorted versions of the σ andH crystals in which three of the patches point directly atneighbouring particles (Sec. III A 1). These two crystal formsare nearly degenerate, and the structure that results from an-nealing is a mixture of the two. Even in their distorted forms,these two crystals can form coherent boundaries betweenthem, and alternating series of layers of the two crystal formscan be seen in Fig. 9(a).

It is also noticeable from Fig. 8 that for higher pres-sures at the narrowest patch width hexagonal crystals againform. This change reflects the energetic destabilization of theσ phase at these patch widths because the particles can nolonger form five strong interactions.

B. Seven-patch particles

Like the five-patch particles, the seven-patch particleshave a local symmetry that is incompatible with crystallineorder. However, in addition it is also physically unfeasible fora particle to have seven neighbouring atoms at the equilib-rium pair separation without particles overlapping. The ques-tion is then how do the particles manage to maximize theirpatch–patch interactions? When the patches are reasonablynarrow, the solution the system finds is to use only five ofthe seven patches, and to adopt the same types of square–triangle tilings as for the five-patch system. Figure 10 showsthe arrangement of the particles in the σ and H environments,and it can be again seen that the σ environment has a smalleraverage deviation of the patch vectors from the interparticlevectors.

Because of this tendency to form square–triangle tilings,the behaviour of the seven-patch system is very similar to thefive-patch system, and so we will much more briefly reviewthe behaviour of this system. The zero temperature phase dia-gram shows a very similar form with a hexagonal crystal hav-ing lowest enthalpy at high pressure and patch width, and aσ crystal at lower pressures and patch widths. The crossover

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054904-9 Dodecagonal quasicrystals of patchy particles J. Chem. Phys. 136, 054904 (2012)

0

10

20

30

40

50

qx / σLJ-1

q y /

σ LJ-1

0

10

20

30

40

50

qx / σLJ-1

q y/ σ

LJ-1

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0

10

20

30

40

50

qx / σLJ-1

q y /

σ LJ-1

0

10

20

30

40

50

qx / σLJ-1

q y /

σ LJ-1

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4

5(c)

(a) (b)

FIG. 11. (a) Configuration after annealing and (b) and (c) associated diffraction patterns (for two different ranges of q) at p = 0.7 εσ−2LJ , σ pw = 0.42 for a

system of seven-patch particles.

between these forms occurs at slightly narrower patch widthsthan for the five-patch system, because the seven patchesmake the potential somewhat closer to the isotropic limit. Asfor the five-patch system at very narrow patch widths, thisstructure distorts so that three of the seven patches point di-rectly at their neighbours, but this time the acute angle in thehexagonal units is 51.43◦ not 72◦.

Again close to the hexagonal/σ boundary our anneal-ing simulations result in quasicrystalline configurations withclear 12-fold diffraction patterns (Fig. 11). Inspection of theconfigurations also again shows the same dodecagonal motifs(Fig. 7) as found for the five-patch system.

IV. CONCLUSIONS

Here, we have presented results for a simple two-dimensional patchy particle system that exhibits a richordering behaviour, in particular, showing the formation ofdodecagonal quasicrystals based on square–triangle tilingsfor certain parameter ranges for particles with five and sevenregularly arranged patches. These examples can be added tothe increasing list of model systems that have been shownto form quasicrystals in simulations.45–58 The current systemalso provides a nice model system that illustrates how thelocal structural propensities can lead to complex globallyordered structures. In particular, quasicrystals were observedin the region of parameter space near to where the enthalpi-cally preferred structure changes from five-fold to six-fold

coordination, because of the presence of both coordinationenvironments in the quasicrystal. Interestingly, the structureof the quasicrystals is quite similar to model square–triangledodecagonal quasicrystals produced by inflation rules.59, 67–69

One interesting question concerning the quasicrystalsthat we observe is whether they are thermodynamically stableor just a kinetic product. As the enthalpy difference betweenthe quasicrystal and the crystal is not that large, it is not un-reasonable that the quasicrystal could become thermodynam-ically stable due to the entropy associated with the many pos-sible configurations for the quasicrystal. However, to confirmthis hypothesis would require the free energy of the quasicrys-tal (or the free energy difference between it and the crystal) tobe computed and achieving this task is not straightforward.For example, for standard methods such as thermodynamicintegration, it is not clear what reference state to use that isneither separated from the quasicrystal by a phase transitionnor that would lead to some of the entropy associated with allthe possible quasicrystalline configurations being missed (asis likely to be the case if a single quasicrystalline configura-tion is taken as a reference). An alternative approach might beto use the direct coexistence method72 to determine the melt-ing point of the quasicrystal.

Some hints concerning the relative stability of the qua-sicrystals can be obtained from our simulations. For example,during some of the cooling runs, the system passes from aliquid to a hexagonal crystal and then to a quasicrystal as thetemperature is decreased, showing that there is a parameterrange where the quasicrystal is more stable than the hexago-

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054904-10 van der Linden, Doye, and Louis J. Chem. Phys. 136, 054904 (2012)

nal crystal. However, on heating the σ crystal, we have neverseen it transform into a quasicrystal, but instead it directlymelts.

An interesting contrast to the behaviour we see here isprovided by particles that have five-fold symmetry in theirrepulsive interactions,73, 74 rather than their attractions. Al-though hard pentagons show an interesting set of solid phasesstructures,73 none of them are quasicrystalline.

Another important question is whether a system thatshows behaviour similar to our model could be experimen-tally realized. Methods to synthesize patchy colloids andnanoparticles are developing rapidly, but as far as we areaware, there are none yet available that could produce, say,particles with a five-fold symmetric distribution of patches.By contrast, effectively restricting such colloidal systems totwo-dimensions would be relatively straightforward. For ex-ample, if there is a density mismatch between the colloids andthe solvent, sedimentation can lead to monolayer formation atthe base of the sample, as has been done in recent experimentswhere a two-dimensional Kagome lattice was assembled from“triblock Janus” particles.34

Another possible system in which these dodecagonalquasicrystals could be potentially realized is the multi-armDNA motifs produced by the group of Chengde Mao.75–82

Each arm is made of two-parallel double helices and bothhave dangling single-stranded ends that allow them to bindto other such motifs with a well-defined relative orienta-tion. These effective torsional constraints on the interactionslead to quasi-two-dimensional growth, be it into sheets75–80

or closed shells.80, 81 Interestingly, similar to the equiva-lent patchy particles, the three-, four- and six-arm mo-tifs form two-dimensional honeycomb,75, 76 square,77, 78 andhexagonal79 lattices. Furthermore, the five-arm motifs canform a σ -phase-like lattice.80

However, there are also a number of significant differ-ences between the DNA multi-arm motifs and patchy particlesystems. First, the central loop to which the stiff double he-lices are connected has a certain degree of flexibility allowingthe relative angles of the arms to vary. By contrast, the posi-tions of the patches in our model are fixed.

Second, the “valence” of the DNA motifs is fixed; i.e., thefive-arm motifs can only ever bond to five other such motifs.By contrast, for our patchy particles at intermediate values ofthe patch width, the particles can adopt five-fold or six-foldcoordination environments; it is this flexibility that allows thesystem to form the quasicrystals. Therefore, if the DNA mo-tifs are to be able to form a quasicrystal, a mixture of five-and six-arm motifs with the right stoichiometry would be re-quired. For an ideal dodecagonal quasicrystal, the requiredratio of five- and six-arm tiles would be 12.928: 1. However,even with such a mixture, it might be that the system prefers tophase separate into σ and hexagonal crystals, similar to whathas been seen for mixtures of three- and four-arm tiles.82

In future work, we intend to explore this possibility fur-ther, first by using a coarse-grained model of DNA that wehave recently developed83, 84 to characterize these DNA multi-arm motifs, including their flexibility and the angular speci-ficity of their interactions. Second, this information will thenbe used to create a patchy-particle representation where the

patch positions are not rigidly fixed but are constrained by aninternal potential.

ACKNOWLEDGMENTS

The authors are grateful for financial support from theEngineering and Physical Sciences Research Council (EP-SRC) and the Royal Society.

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