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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:
[email protected].
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
2468 | www.pnas.org/cgi/doi/10.1073/pnas.1312670111
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http://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1312670111/-/DCSupplemental/pnas.201312670SI.pdf?targetid=nameddest=SF1http://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1312670111/-/DCSupplemental/pnas.201312670SI.pdf?targetid=nameddest=STXTwww.pnas.org/cgi/doi/10.1073/pnas.1312670111
-
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.
Dontabhaktuni et al. PNAS | February 18, 2014 | vol. 111 | no. 7
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