Quasicrystalline tilings with nematic colloidal plateletsJayasri Dontabhaktunia,b, Miha Ravnika, and Slobodan Žumera,c,1

aFaculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia; bCentre for Modelling Simulation and Design, University ofHyderabad, Hyderabad 500 046, India; and cJozef Stefan Institute, SI-1000 Ljubljana, Slovenia

Edited* by T. C. Lubensky, University of Pennsylvania, Philadelphia, PA, and approved January 8, 2014 (received for review July 4, 2013)

Complex nematic fluids have the remarkable capability for self-assembling regular colloidal structures of various symmetries anddimensionality according to their micromolecular orientationalorder. Colloidal chains, clusters, and crystals were demonstratedrecently, exhibiting soft-matter functionalities of robust binding,spontaneous chiral symmetry breaking, entanglement, shape-driven and topological driven assembly, and even memory im-printing. However, no quasicrystalline structures were found.Here, we show with numerical modeling that quasicrystallinecolloidal lattices can be achieved in the form of original Penrose P1tiling by using pentagonal colloidal platelets in layers of nematicliquid crystals. The tilings are energetically stabilized with bindingenergies up to 2500 kBT for micrometer-sized platelets and furtherallow for hierarchical substitution tiling, i.e., hierarchical pentagu-lation. Quasicrystalline structures are constructed bottom-up byassembling the boat, rhombus, and star maximum density clusters,thus avoiding other (nonquasicrystalline) stable or metastable con-figurations of platelets. Central to our design of the quasicrystal-line tilings is the symmetry breaking imposed by the platelet shapeand the surface anchoring conditions at the colloidal platelets,which are misaligning and asymmetric over two perpendicularmirror planes. Finally, the design of the quasicrystalline tilings asplatelets in nematic liquid crystals is inherently capable of a con-tinuous variety of length scales of the tiling, ranging over threeorders of magnitude in the typical length (from ∼10 nm to∼ 10 μm), which could allow for the design of quasicrystallinephotonics at multiple frequency ranges.

colloids | quasicrystals | Penrose tiling | hierarchy

Quasicrystals are aperiodic crystalline materials, distinguishedby noncrystallographic rotational symmetry of fivefold,sevenfold, eightfold, and higher rotational symmetry axes (1–3).These symmetries are typically found in atomic lattices of dis-tinct metallic alloys (1, 4). However, more recently, a uniqueclass of soft-matter quasicrystals is emerging (5–8), where thebasic building blocks are not single atoms but rather macro-molecules (9, 10), copolymers (11), molecular liquid crystallinefields (12, 13), or colloidal particles (14, 15). Two-dimensionalrealizations of materials with quasicrystalline symmetries arequasicrystalline tilings (2, 16, 17). In tilings, the structures ofpolygons or platelets—tiles—cover an area in complex pat-terns, typically following geometric rules. Tilings with fivefold(18, 19), sevenfold (20), eightfold (21), ninefold (22), tenfold(23), twelvefold (24), and other (25) quasicrystalline symme-tries were realized, demonstrating analogous ordering mech-anisms as in quasicrystals (26, 27). These ordering mechanismsand the formation dynamics were particularly explored inquasicrystalline colloidal monolayers stabilized by interferinglaser beams (28, 29).Nematic liquid crystals are fluids with molecular orientational

order, called the director field, and it is by designing the profilesof this field that colloidal structures of various functionalitiescan be self-assembled. The self-assembly is based on effectivestructural forces emerging between the particles (typically ∼1−10pN for micrometer-sized particles), caused by the inhomo-geneous and anisotropic director profiles imposed by the particlesurfaces or general shapes (30). Already simple spherical col-loidal particles were shown to self-assemble into chains (31),

clusters (32), 2D (33), and 3D colloidal crystals (34, 35). Aspecifically strong way to affect the self-assembly is by shape-controlled colloidal interactions (36–38) and faceted colloidalparticles (39), where the shape of the particles determines theinterparticle potentials and the symmetry of the structures (40)as well as their rotational dynamics (41). However, despite usingparticles with geometrically quasicrystalline symmetry, e.g.,pentagonal or heptagonal platelets, generically, structures withstrictly crystalline symmetry are found (36, 40, 42). It was shownthat such platelets effectively lose their faceted nature and be-have as dipoles and quadrupoles in the distortion field of thefluid (36), exactly as already known for spherical particles. Moregenerally, therefore, finding relations between the inherentsymmetry of the building blocks and the actual symmetry of thestructures made from these building blocks presents a far-reaching challenge in the design of advanced quasicrystalline andcrystalline materials.In this paper, we combine the energy-based concept of struc-

tural forces in complex nematic fluids and geometry of build-ing blocks to self-assemble quasicrystalline Penrose P1 tiling ofpentagonal colloidal platelets. More specifically, we considersubmicrometer-sized platelets whose top and bottom surfacesare treated to impose different alignment directions on the di-rector field (Fig. 1), which generates interparticle potentialscompatible with the quasicrystalline fivefold symmetry. Theplatelets are pre-positioned according to the symmetry of thePenrose lattice in thin nematic cells, typically ∼5 times plateletthickness, whose surfaces are taken to yield strong uniformplanar anchoring along a common direction (denoted as therubbing direction), and then relaxed to equilibrium. Such anapproach creates strongly bound equilibrium platelet structures,which, however, do not necessarily correspond to the global

Significance

Complex nematic fluids have the remarkable capability to or-ganize microparticles and nanoparticles into regular structuresof various symmetries and dimensionality, according to theirmicromolecular orientational order. Structures of particles suchas chains, clusters, and crystals are found, but no quasicrystals.In this paper, we demonstrate that quasicrystalline structurescan be achieved in the form of Penrose tiling by assemblingmicroplatelets in the shape of pentagons within a thin layerof nematic fluid. The tiling is energetically stabilized withbinding energies of typically several orders of magnitudehigher than thermal energy and further allows for hierar-chical substitution of individual pentagonal tiles with smallertiles, which is interesting for the design of photonics atmultiple frequency ranges.

Author contributions: S.�Z. designed research; J.D. and M.R. performed research; J.D., M.R.,and S.�Z. analyzed data; and J.D., M.R., and S.�Z. wrote the paper.

The authors declare no conflict of interest.

*This Direct Submission article had a prearranged editor.

Freely available online through the PNAS open access option.1To whom correspondence should be addressed. E-mail: slobodan.zumer@fmf.uni-lj.si.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1312670111/-/DCSupplemental.

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ground states of the system. The quasicrystalline assembly isrobust over multiple length scales of the tiling pattern, and allowsa hierarchy of scales, i.e., quasicrystalline tilings at one scale, withsmaller-scale (quasicrystalline) substitutions. Our main method-ological approach is phenomenological numerical modelingbased on the minimization of the Landau-de Gennes free energy(see SI Text and Theory and Method), which proves particularlyefficient exactly in strongly confined systems with multiple par-ticles, like tilings, and which can give full quantitative or quali-tative agreement with experiments (36, 40). Finally, experimentalstrategies for the self-assembly of Penrose nematic colloidaltilings are proposed, suggesting optical fields or quasicrystallineseed colloidal particles as possible approaches.

Prototiles: Pentagonal Colloidal PlateletsThe assembly of quasicrystalline tilings depends centrally on thesymmetry properties of the elementary building blocks—theprototiles—which determine the prototile-to-prototile interac-tion potential. Our prototiles are regular pentagonal platelets ofuniform size (edge length d= 300 nm), with designed surfaces.The top and bottom large faces of the platelets are taken toimpose in-plane uniform alignment of the nematic molecules—strong uniform planar anchoring—in mutually perpendiculardirections (Fig. 1 A−C). All other (side) surfaces induce de-generate planar anchoring. Experimentally realizing such surfaceanchoring on all platelet surfaces could possibly be achieved byusing photoimprinting of the surface anchoring profile. Sucha choice of anchoring surface preparation aligns the platelets atan angle of 0° or 72° relative to the far-field director—see potential

minima at 0° and 72° in Fig. 1D—and, more importantly, breaksthe top−bottom symmetry about the midplane passing throughthe platelet, which effectively removes the repulsive directions inthe interaction potential of platelets, common for elastic dipolesor quadrupoles (30, 36). This additional symmetry breaking inturn gives rise to complex interactions between the platelets,leading to various possibilities of assembling them into qua-siperiodic arrangements.The frustration of preferred molecular orientations at the

edges and corners of the platelets gives rise to topological defectsalong the edges of the platelets, as shown in Fig. 1C. The dis-clination lines surrounding individual platelets are surface defects,i.e., effective generalizations of surface boojum defects, that windaround the edges of the platelets, alternating between the top andbottom surfaces (for more, see SI Text and Fig. S1).The pair interaction potential between two pentagonal pro-

totiles in a selected region of separations—calculated as changesin the free energy upon shifting the platelets (see SI Text)—isshown in Fig. 1E. Interestingly, the potential exhibits multipleminima, corresponding to different alignments of bound colloi-dal pairs with respect to the undistorted nematic direction (Fig. 1F−H). At short separations, as the platelets get laterally close toone another with only a thin layer of nematic ð∼ 10 nmÞ be-tween them, the surface anchoring on the side walls of the pla-telets dictates the assembly. Because the surface anchoring onthe side walls of two platelets is compatible, the platelets attract.In all equilibrium pair configurations, the attraction is almostexactly side surface to side surface, which proves central to sta-bilize the quasicrystalline tilings, but could also stabilize other,

A B C

D E

F G H

Fig. 1. Colloidal pentagonal platelets as prototiles. Uniform planar anchoring imposed (A) at an angle 36° on the top surface and (B) at an angle 126° on thebottom surface. (C) Faceted surface generates surface topological defects (dark green) in the director field along the edges of the platelets. Molecularorientational profile—the director—is shown in black; defects are visualized as isosurfaces of nematic degree of order S= 0:5. (D) Rotation potential asfunction of the platelet angle with respect to the rubbing direction exhibits a minimum at 0° and 72°. (E) Pair interaction potential F of two pentagonalprototiles. Note free energy minimum regions 1, 2, and 3, which lead to the formation of equilibirum pair structures. Rubbing direction at cell walls is at anangle of 72° with respect to X (see inset). (F−H) The three equilibrium platelet pair structures, as corresponding to the free energy minima. Blue double-headed arrows in F−H indicate the imposed anchoring direction on top surface of both considered platelets.

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e.g., crystalline, structures. Notably, this side surface to side sur-face pair attraction is very different from the strictly steric re-pulsion in the typical geometric tilings, where the tiles can easilyslide with respect to each other. Also important is that we considerthe platelet assembly in a rather thin layer of nematic planar cellwith a thickness comparable to the platelet thickness (ratio of ∼5).Working in this regime prevents: (i) possible top−bottom flippingof the platelets and (ii) out-of-plane stacking of the platelets (Fig.S2). Having all platelets with equal top and bottom surfaces, i.e.,nonflipped, prevents possible structural defects in the tiling, asflips notably change the pair interaction potential (Fig. S3). Al-ternatively, the assembly in the regime of large cell thickness toplatelet thickness ratio could be importantly affected by thestacking of platelets, likely into columns, building 3D structuresof diverse complexity. Finally, the proposed design of plateletassembly in thin nematic layers gives equilibrium directions ofthe platelet pairs with respect to the undistorted nematic di-rection that correspond well to the nearest neighbor directions,characteristic for the Penrose tilings.

Elementary Quasi-Building Units: Star, Rhombus, Boat, andDecagonal Colloidal ClustersQuasicrystalline tilings require a distinct approach to assembly,as by definition they cannot be constructed by simply re-peating a unit cell according to the Bravais lattice vectors.Two approaches are known to construct the quasicrystallinetilings (16, 17, 43–45), and we show that actually both can be—within some limitations—used with our nematic platelets to as-semble the Penrose lattice.The first approach for the Penrose P1 lattice is based on three

basic quasi-building units (clusters), called the star, rhombus,and boat structures (named after the shapes of voids formedbetween the pentagonal platelets in these structures), and usesthem as elements of the tiling. When assembled into the tiling,the neighboring clusters partially overlap, i.e., share some tiles,effectively covering the whole tiling according to distinct overlaprules. Fig. 2 A−C shows that star, rhombus, and boat quasi-building units form from pentagonal nematic platelets, and it isthe pair potential originating from the designed nematic distor-tion field between the platelets that stabilizes the quasi-buildingblocks (for more, see SI Text and Fig. S4). Topological defectlines wind along the edges of the platelets in all of the quasi-building units, acting energetically as effective “sticky” edgesconnecting the adjacent platelets. Short defect line segments ofwinding number −1=2 are observed also in the voids between theplatelets (see inset in Fig. 2C), and typically emerge at distinctorientation angles of the building units. In equilibrium, the quasi-building units are oriented at stable configuration angles of 0°(rhombus), 0° (boat), and 72° (star) with respect to the far-fieldnematic director, as seen in Figs. 2 D−F; the boat and the starclusters have the total free energy for 470 kBT and 780 kBThigher, respectively, than the rhombus cluster (kBT is thermalenergy at room temperature T and kB is Boltzmann constant).Note, however, that the quasi-building-unit clusters also remainbound if oriented at other (nonequilibrium) angles, which isimportant for achieving the stability of the full-plane tiling.Alternatively, under the second approach (17, 43–45), the

Penrose tiling is constructed by assembling only one basic quasi-building unit—the decagonal colloidal cluster—which also par-tially overlaps with neighboring clusters when assembled into thetiling. This decagonal cluster is also frequently called the quasi-unit cell of the Penrose tiling. Indeed, we show that the deca-gonal colloidal cluster can be assembled from nematic pentagonsas presented in Fig. 2G. The decagon is stable against lateralshifts and dissociation modes of individual pentagons, and isagain bound by the effective elastic distortion of the nematic (seealso SI Text and Fig. S4). The decagon is stable at orientationangles of 70° and 260° (and metastable at 170° and 350°) with

respect to the rubbing direction, as depicted in Fig. 2H, but,similarly to the rhombus, star, and boat clusters of the first ap-proach, also remains laterally bound at all other orientationangles. The total free energy of the decagonal cluster is for 1130kBT higher than for the rhombus cluster. More generally, findingthe colloidal decagonal cluster stable indicates that in the tiling,there will be overlapping regions of attraction, effectively cor-responding to the overlapping decagonal clusters, which arecompatible with the quasicrystalline symmetry and will stabilizethe tiling. Finally, it is important to comment that the concept ofoverlapping decagonal clusters as quasi-unit cells also requiresthat the decagons overlap cleanly only in two ways (17, 43–45),which is not observed in our nematic colloidal tiling, where wesee more different overlaps, in particular caused by the locallydifferent structure of the nematic orientational field.

Penrose Tilings: Simple and HierarchicalHaving confirmed the stability of the individual tiling quasi-building units, we design a quasicrystallite with N = 55 particles(Fig. 3A) to form a robust and stable structure. Indeed, uponstretching under the “breathing mode” (i.e., equally increasingall platelet-to-platelet separations), the free energy of the struc-ture increases by several thousand kBT (Fig. 3B). Finally, we de-sign and demonstrate colloidal Penrose tiling with as many asN = 176 platelets, as shown in Fig. 3C. The calculations of thewhole structure are performed directly, within a single simula-tion box, using large-scale parallelization and edge of computerresources. The presented structure is the Penrose P1 tiling with

A B C

D E F

G H

Fig. 2. Elementary building units of the Penrose tiling. (A) Rhombus, (B)boat, and (C) star colloidal clusters assembled from the pentagonal plateletsand bound by nematic structural forces. Note the rubbing direction in-dicating the undistorted nematic director. Inset in C shows a closer view ofthe −1=2 defect line segment that can connect adjacent platelets. Reor-ientation potential of (D) rhombus, (E) boat, and (F) star cluster with respectto the rubbing direction. (G) Decagonal cluster assembled from nematicpentagonal platelets. (H) Reorientation potential of the decagonal clusterwith respect to the rubbing direction. Blue double-headed arrows in A−Cand G indicate the same imposed anchoring direction on top surface of allplatelets.

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distinguished fivefold symmetry axes and rhombus, star, and boatvoids. Importantly, this fivefold symmetry applies to the posi-tional order of the platelets, and it breaks if also considering thesymmetry of the surrounding nematic field, which is also evidentfrom the anchoring direction imposed by the top (and bottom)surfaces of the platelets that is equal for all platelets in the tiling(blue double-headed arrow in Fig. 3 A and C). The stability ofthe tiling is ensured by the complex director field surroundingthe tilings as seen in Fig. 3 D and E. More generally, the complex

director is reflected in the highly anisotropic structural forcesbetween the tiles, which are caused by a remarkable interplayof inherent symmetries in the material, i.e., quasicrystalline fromthe positioning of the tiles and the uniaxial of the nematic orderparameter field.The Penrose tilings assembled from pentagonal platelets in

nematics have an interesting property: Tile interaction potentialsallow for hierarchical substitution of individual prototiles, by ageometrically matching and energetically bound structure ofphysically smaller prototiles. Fig. 4A shows three platelet struc-tures, assembled from various-size pentagons, but of the samethickness, that are stable and that effectively behave as onelarger-scale prototile. Indeed, because of the nematic elasticorigin, the interaction forces between the platelets of varioussizes follow roughly the same profile of the interaction potentialover a range of length scales from a few hundred nanometers tomicroscopic scales, as illustrated in Fig. 4B, which is usually morecomplex in other soft-matter quasicrystalline systems (9, 11, 28).Therefore, if the structures from smaller-size pentagons geo-metrically match the size of one larger platelet, hierarchical Penrose

A

C

D E

B

Fig. 3. Penrose P1 tiling structures. (A) Quasicrystallite assembled with 55particles. (B) Stretching potential of the quasicrystallite upon the breathingmode. (C) Colloidal quasicrystalline Penrose tiling P1 assembled from 176particles in nematic liquid crystal layer with the indicated fivefold symmetryaxis. (D and E) Close-up view of the director field in the P1 tiling. Dark greenregions surrounding the platelets correspond to surface and bulk topologicaldefects, visualized as isosurfaces of the nematic degree of order S= 0:52.Rubbing direction of the undistorted nematic director is indicated by thewhite or black double-headed arrows; blue double-headed arrows indicatethe imposed anchoring direction on top surface of the platelets, which is thesame for all platelets in the tiling.

A

B C

D

Fig. 4. Hierarchical tiling. (A) A single pentagonal prototile of edge length400 nm can be replaced with six smaller-sized (160 nm) pentagonal tiles andfurther dissociated into 36 pentagonal tiles of edge length 60 nm. (B) In-teraction forces of a pair of particles as a function of their separation foredge lengths 60 nm, 250 nm, and 1 mm. (C) Scheme of Penrose P1 tiling withhierarchically organized tiles of three length scales. (D) Numerically calcu-lated hierarchical Penrose P1 tiling, with one pentagonal tile replaced withsmaller-scale pentagonal tiles. Gray streamlines show the projection of di-rector field on the visualized plane.

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tiling can be achieved, as presented in Fig. 4 C and D. Effectively,this tiling still carries the general properties of the Penrose lat-tice, yet with refined smaller-scale additions. More quantita-tively, we compared the free energies of one pentagonal prototileand one effective prototile assembled from smaller-size penta-gons. We observed that the free energies are comparable to eachother except for the interaction energy of the smaller-size pen-tagons in the voids. In the hierarchical Penrose tilings, the ne-matic-embedded pentagons are energetically bound at all scales,whereas in a strictly geometric hierarchical tiling of pentagons(e.g., with only hard-core repulsions), the smaller-scale tiles inparticular can become loose and break the tiling symmetry. Fi-nally, knowing that quasicrystalline materials can have distinctphotonic properties such as complete band gaps (46–48), thePenrose tiling from hierarchically organized nematic plateletspresented here may offer unique photonic responses, for exam-ple as multiple-frequency photonic crystals.

DiscussionExperimentally realizing the proposed nematic Penrose tilingswill critically depend on being able to minimize the possibilityof forming conformations in the tiling patterns that break thequasicrystalline symmetry. The platelet−platelet interactionpotentials presented here are typically in the range of 100−1000kBT, which suggests that thermal fluctuations will be unable tomake any changes to the structure once bound. We list twopossible experimental approaches that could avoid, or at leastreduce, the structural imperfections: (i) the optical tweezers-assisted assembly and (ii) the step-wise assembly by a pre-assembly of colloidal clusters as seeds. Optical tweezers andtweezers with complex beams have proven to be robust tools forprecise assembly of liquid crystal colloids, including dynamicmanipulation and rotation (41), and could be used to assemblethe tiling. Alternatively, colloidal protoclusters or particles ofcomplex shapes (e.g., rhombi, stars, or boats) could be used asquasicrystalline seeds in the dispersion of pentagons, in partic-ular if combined with a slow and gradual assembly of the tiling(order of magnitude of one platelet at a time). Indeed, theconcept of creating sticky edges by designing surface anchoringcould be generalized also to particles of other—more complex—shapes, possibly allowing for their use as quasicrystalline seeds inthe assembly.The demonstrated tiling reveals an interesting combination of

the fivefold quasicrystalline symmetry of the platelets and theinherent uniaxial symmetry of the anisotropic nematic host,which originates from the soft nematic response via the forma-tion of topological defects and the effective elastic deformationsof the director. The topological defects surrounding the platelets(e.g., Fig.1 and Fig. S1) allow the uniaxial nematic director tocompensate the fivefold geometrical symmetry of the platelets byusing the defect lines with core singularities, where the nematicorder is lost. Further insight into the interplay of symmetries canbe obtained by roughly generalizing the nematic field into twocharacteristic regions: (i) the voids between the platelets and (ii)the layers above and below the platelets. In the voids, the ne-matic field is primarily determined by the boundary conditionsimposed by the tiles as opposed to nematic elasticity. In thenematic layers below and above, however, the director is de-termined primarily by the competition between the uniformplanar alignment at the cell walls and the in-plane alignment atthe top and bottom surfaces of the platelets. In all structures ofplatelets, the imposed direction by the top or bottom surface ofthe platelets forms an angle in respect to the rubbing directionimposed by the cell walls, which results in a twist deformation ofthe nematic director above and below the platelets. And it is thistwist that effectively couples the detailed structure and geo-metrical symmetry of the platelets with the uniform uniaxialsymmetry of the undistorted nematic imposed by the nematic

planar cell. Finally, it is the softness of the nematic deformationthat combines these elementary diverse—quasicrystalline anduniaxial—symmetries in the same system.In summary, we have demonstrated that quasicrystalline Penrose

P1 tiling can be assembled from pentagonal colloidal platelets usingfluid-mediated interactions in nematic liquid crystals. The assemblyis approached by constructing elementary Penrose tiling quasi-building units of the star, boat, rhombus, and decagonal colloidalclusters. They are stable and bound at arbitrary orientation angles,which proves central for the stability of the tiling as a whole. Themechanism of hierarchical pentagulation at three hierarchicalscales is shown, where targeted tiles are replaced with energet-ically bound clusters of smaller-size but same-thickness tiles,which is an approach toward the assembly of fractal tilings. Thedemonstrated Penrose tiling is achieved only from one type ofprototile, i.e., pentagonal platelets, which is uncommon for col-loidal quasicrystals, which typically need two or more species ofparticles to exhibit quasicrystalline structuring. Here, it is thestructure of the nematic fluid that supports the quasicrystallineassembly. A notable difference between these nematic-embed-ded tilings and purely geometric ones is also that nematic me-diated interplatelet interactions favor side surface to side surfacealignment with, importantly, no sliding, which causes inherentenergetically favorable positioning of the platelets along symmetrydirections. From a broader perspective, our results demonstrateunique materials with quasicrystalline symmetry, hierarchicalsubstitutions, and robust assembly, which are all among cutting-edge characteristics for advanced optic and photonic materials.Finally, the demonstrated nematic Penrose tilings have revealed aunique compatibility of the general material symmetries, combin-ing the locally uniaxial symmetry of the nematic complex fluids andthe quasicrystalline fivefold symmetry of the individual platelets,which could lead to a previously unknown insight into the fun-daments of quasicrystal formation.

Theory and MethodPhenomenological modeling based on Landau-de Gennes free energy wasused as the central methodological approach formodeling and predicting thenematic Penrose tilings (see also SI Text). This has proven to be a particularlystrong approach for modeling nematic colloids, as it can give qualitative andquantitative agreement with experiments (49, 50). The modeling is based ontotal free energy F written in terms of tensorial order parameter Qij, andconstitutes contributions from elasticity of the nematic medium, tempera-ture-dependent bulk order, and the surface anchoring:

F ¼ZLC

�L2

�∂Qij=∂xk

�2 þ 12AQijQji þ 13BQijQjkQki þ

14C�QijQji

�2�dV þ

Zsurf

fS dS,

where L is the single elastic constant, A, B, and C are nematic materialparameters, fS is surface free energy density,

RLC indicates integration over

the whole nematic volume, andRsurf indicates integration over the surface

of the tiles. Surface free energy for degenerate planar anchoring on the sidesurfaces of the tiles is taken as fS =Wð~Qij − ~Q⊥ij Þ2 (for more details, see ref.51), whereas surface free energy for uniform planar anchoring on the topand bottom surfaces of the tiles is taken as fS =WU2 ðQij −Q0ijÞðQji −Q0jiÞ, whereW and WU are the anchoring strengths and Q0ij is the surface preferred orderparameter. Free energy is minimized using explicit finite difference re-laxation method on a cubic mesh (51). Surface mesh point allocation isadapted for strongly confined regions of a few mesh points size. The topand bottom substrates of the cell induce strong uniform planar anchoring,and periodic boundary conditions are assumed along lateral XY directions.The material parameters are those of a standard nematic liquid crystal; ifnot stated differently, we take: pentagonal edge lengths d = 300 nm, L =40 pN, A = -0.172 MJ/m3, B = -2.12 MJ/m3, C = 1.73 MJ/m3,W =WU = 10

-3 J/m2,and surface anchoring alignment angle of 36° (126°) at the bottom(top) surface.

ACKNOWLEDGMENTS. We acknowledge discussions with I. I. Smalyukh. J.D.acknowledges financial support from the European Commission-funded

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Marie Curie Actions Hierarchical Assembly in Controllable Matrices projectand M.R. acknowledges support from Marie Curie Actions Grant Channel-free liquid crystal microfluidics FREEFLUID. We acknowledge funding fromCentre of Excellence, Ljubljana, Slovenian Research Agency Grant Z1-5441,and Programme P1-0099. J.D. acknowledges the Centre for Modelling

Simulation and Design (University of Hyderabad), including for computa-tional time. This work was supported in part by the National Science Foun-dation under Grant NSF PHY11-25915, under the Kavli Institute for TheoreticalPhysics Program Knotted Fields and the Isaac Newton Institute for Mathemat-ical Sciences program The Mathematics of Liquid Crystals.

1. Shechtman D, Blech I, Gratias D, Cahn JW (1984) Metallic phase with long-rangeorientational order and no translational symmetry. Phys Rev Lett 53(20):1951–1954.

2. Levine D, Steinhardt PJ (1984) Quasicrystals: A new class of ordered structures. PhysRev Lett 53(26):2477–2500.

3. Dotera T (2012) Toward the discovery of new soft quasicrystals: From a numericalstudy viewpoint. J Polym Sci Pol Phys 50(3):155–167.

4. Dubois J-M, Lifshitz R (2011) Quasicrystals: Diversity and complexity. Philos Mag91(19−21):2971–2982.

5. Lifshitz R, Diamant H (2007) Soft quasicrystals: Why are they stable? Philos Mag87(18−21):3021–3030.

6. Barkan K, Diamant H, Lifshitz R (2011) Stability of quasicrystals composed of softisotropic particles. Phys Rev B 83(17):172–201.

7. Iacovella CR, Keys AS, Glotzer SC (2011) Self-assembly of soft-matter quasicrystals andtheir approximants. Proc Natl Acad Sci USA 108(52):20935–20940.

8. Ungar G, Percec V, Zeng X, Leowanawat P (2011) Liquid quasicrystals. Isr J Chem51(11−12):1206–1215.

9. Zeng X, et al. (2004) Supramolecular dendritic liquid quasicrystals. Nature 428(6979):157–160.

10. Fischer S, et al. (2011) Colloidal quasicrystals with 12-fold and 18-fold diffractionsymmetry. Proc Natl Acad Sci USA 108(5):1810–1814.

11. Hayashida K, Dotera T, Takano A, Matsushita Y (2007) Polymeric quasicrystal: Meso-scopic quasicrystalline tiling in ABC star polymers. Phys Rev Lett 98(19):195502.

12. Schwarz MH, Pelcovits RA (2009) Nematic cells with quasicrystalline-patternedalignment layers. Phys Rev E Stat Nonlin Soft Matter Phys 79(2 Pt 1):022701.

13. Ackerman PJ, Qi Z, Smalyukh II (2012) Optical generation of crystalline, quasicrys-talline, and arbitrary arrays of torons in confined cholesteric liquid crystals for pat-terning of optical vortices in laser beams. Phys Rev E Stat Nonlin Soft Matter Phys86(2 Pt 1):021703.

14. Talapin DV, et al. (2009) Quasicrystalline order in self-assembled binary nanoparticlesuperlattices. Nature 461(7266):964–967.

15. Haji-Akbari A, et al. (2009) Disordered, quasicrystalline and crystalline phases ofdensely packed tetrahedra. Nature 462(7274):773–777.

16. Penrose R (1974) The role of aesthetics in pure and applied mathematical research.Bull Inst Math Appl 10:266–271.

17. Steinhardt PJ (2000) Penrose tilings, cluster models and the quasi-unit cell picture.Mater Sci Eng A 294–296, 205–210.

18. Caspar DLD, Fontano E (1996) Five-fold symmetry in crystalline quasicrystal lattices.Proc Natl Acad Sci USA 93(25):14271–14278.

19. Ledieu J, et al. (2001) Tiling of the fivefold surface of Al70Pd21Mn9. Surf Sci 492(3):L729–L734.

20. Mikhael J, et al. (2010) Proliferation of anomalous symmetries in colloidal monolayerssubjected to quasiperiodic light fields. Proc Natl Acad Sci USA 107(16):7214–7218.

21. de Gier J, Nienhuis B (1996) Exact solution of an octagonal random tiling model. PhysRev Lett 76(16):2918–2921.

22. Gorkhali SP, Qi J, Crawford GP (2006) Switchable quasi-crystal structures with five-,seven-, and ninefold symmetries. J Opt Soc Am B 23(1):149–158.

23. Mackay A, Quinquangula DN (1981) On the pentagonal snowflake. Sov Phys Crys-tallogr 26:517–522.

24. Oxborrow M, Henley CL (1993) Random square-triangle tilings: A model for twelve-fold-symmetric quasicrystals. Phys Rev B 48(10):6966–6998.

25. Yamamoto A (1996) Crystallography of quasiperiodic crystals. Acta Crystallogr A 52:509–560.

26. Gahler F, Reichert M (2002) Cluster models of decagonal tilings and quasicrystals.J Alloy Comp 342(1-2):180–185.

27. Gummelt P (1996) Penrose tilings as coverings of congruent decagons. Geom Dedicata62(1):1–17.

28. Mikhael J, Roth J, Helden L, Bechinger C (2008) Archimedean-like tiling on decagonalquasicrystalline surfaces. Nature 454(7203):501–504.

29. Mikhael J, Gera G, Bohlein T, Bechinger C (2011) Phase behavior of colloidal mono-layers in quasiperiodic light fields. Soft Matter 7(4):1352–1357.

30. Lubensky TC, Pettey D, Currier N, Stark H (1998) Topological defects and interactionsin nematic emulsions. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Top 57(1):610–625.

31. Poulin P, Stark H, Lubensky TC, Weitz DA (1997) Novel colloidal interactions in an-isotropic fluids. Science 275(5307):1770–1773.

32. Yada M, Yamamoto J, Yokoyama H (2004) Direct observation of anisotropic in-terparticle forces in nematic colloids with optical tweezers. Phys Rev Lett 92(18):185501.

33. Mu�sevi�c I, �Skarabot M, Tkalec U, Ravnik M, �Zumer S (2006) Two-dimensional nematiccolloidal crystals self-assembled by topological defects. Science 313(5789):954–958.

34. Ravnik M, Alexander GP, Yeomans JM, �Zumer S (2011) Three-dimensional colloidalcrystals in liquid crystalline blue phases. Proc Natl Acad Sci USA 108(13):5188–5192.

35. Trivedi RP, Klevets II, Senyuk B, Lee T, Smalyukh II (2012) Reconfigurable interactionsand three-dimensional patterning of colloidal particles and defects in lamellar softmedia. Proc Natl Acad Sci USA 109(13):4744–4749.

36. Lapointe CP, Mason TG, Smalyukh II (2009) Shape-controlled colloidal interactions innematic liquid crystals. Science 326(5956):1083–1086.

37. Senyuk B, et al. (2012) Shape-dependent oriented trapping and scaffolding of plas-monic nanoparticles by topological defects for self-assembly of colloidal dimers inliquid crystals. Nano Lett 12(2):955–963.

38. Senyuk B, Smalyukh II (2012) Elastic interactions between colloidal microspheresand elongated convex and concave nanoprisms in nematic liquid crystals. Soft Matter8(33):8729–8734.

39. Hung FR, Bale S (2009) Faceted nanoparticles in a nematic liquid crystal: Defectstructures and potentials of mean force. Mol Simul 35(10−11):822–834.

40. Dontabhaktuni J, Ravnik M, �Zumer S (2012) Shape-tuning the colloidal assemblies innematic liquid crystals. Soft Matter 8(5):1657–1663.

41. Lapointe CP, Hopkins S, Mason TG, Smalyukh II (2010) Electrically driven multiaxisrotational dynamics of colloidal platelets in nematic liquid crystals. Phys Rev Lett105(17):178301.

42. Nelson DR (2002) Defects and Geometry in Condensed Matter Physics (CambridgeUniversity Press, New York).

43. Lord EA and Ranganathan S (2001) The Gummelt decagon as a quasi unit cell. ActaCryst A 57(Part 5):531–539.

44. Steinhardt PJ, et al. (1996) A simpler approach to Penrose tiling with implications forquasicrystal formation. Nature 382:433–435.

45. Steinhardt PJ, et al. (1998) Experimental verification of the quasi-unit-cell model ofquasicrystal structure. Nature 396:55–57.

46. Kaliteevski MA, et al. (2004) Directionality of light transmission and reflectionin two-dimensional Penrose tiled photonic quasicrystals. J Phys Condens Matter16(8):1269–1278.

47. Lavrentovich OD, Lazo I, Pishnyak OP (2010) Nonlinear electrophoresis of dielectricand metal spheres in a nematic liquid crystal. Nature 467(7318):947–950.

48. Sengupta A, Bahr C, Herminghaus S (2013) Topological microfluidics for flexiblemicro-cargo concepts. Soft Matter 9(30):7251–7260.

49. Ravnik M, et al. (2007) Entangled nematic colloidal dimers and wires. Phys Rev Lett99(24):247801.

50. Sengupta A, et al. (2013) Liquid crystal microfluidics for tunable flow shaping. PhysRev Lett 110(4):048303.

51. Ravnik M, �Zumer S (2009) Landau-de Gennes modelling of nematic liquid crystalcolloids. Liq Cryst 36(10−11):1201–1214.

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