Assembly of porous smectic structures formed frominterlocking high-symmetry planar nanoringsCarlos Avendañoa,1, George Jacksonb, Erich A. Müllerb, and Fernando A. Escobedoc

aSchool of Chemical Engineering and Analytical Science, The University of Manchester, Manchester M13 9PL, United Kingdom; bDepartment of ChemicalEngineering, Centre for Process Systems Engineering, Imperial College London, London SW7 2AZ, United Kingdom; and cSchool of Chemical andBiomolecular Engineering, Cornell University, Ithaca, NY 14853

Edited by Michael L. Klein, Temple University, Philadelphia, PA, and approved July 5, 2016 (received for review March 22, 2016)

Materials comprising porous structures, often in the form of inter-connected concave cavities, are typically assembled from convexmolecular building blocks. The use of nanoparticles with a charac-teristic nonconvex shape provides a promising strategy to createnew porous materials, an approach that has been recently usedwith cagelikemolecules to form remarkable liquids with “scrabbled”porous cavities. Nonconvex mesogenic building blocks can be engi-neered to form unique self-assembled open structures with tunableporosity and long-range order that is intermediate between that ofisotropic liquids and of crystalline solids. Here we propose the de-sign of highly open liquid-crystalline structures from rigid nanoringswith ellipsoidal and polygonal geometry. By exploiting the entropicordering characteristics of athermal colloidal particles, we demon-strate that high-symmetry nonconvex rings with large internal cav-ities interlock within a 2D layered structure leading to the formationof distinctive liquid-crystalline smectic phases. We show that thesesmectic phases possess uniquely high free volumes of up to ∼95%,a value significantly larger than the 50% that is typically achievablewith smectic phases formed by more conventional convex rod- ordisklike mesogenic particles.

nanorings | porous liquid crystals | self-assembly

Self-assembly of particles ranging in size from the nanometerto the micrometer scales can be used to fabricate structures

in the mesoscale regime (1–5), otherwise difficult to achieve withtraditional methods of chemical synthesis. Strategies to producefunctional materials from the self-assembly of relatively simplenonspherical (anisotropic) building blocks have undergone un-precedented growth as a result of recent advances in experimentaltechniques to fashion colloidal and nanoparticles of arbitrary shapeand well-defined sizes (6–9). An appealing feature of colloidalparticles is that the repulsive and attractive contributions of theinteraction between the particles can be modulated by controllingthe properties of both the particle surface and the solvent mediumto induce different types of forces, including short-range repulsionsapproaching hard-core (athermal) interactions (10).Colloidal particles are commonly represented using simplified

coarse-grained convex geometries including ellipsoids, spher-ocylinders, polyhedra, and cut spheres (11, 12). The phase be-havior of these convex models has been studied extensively bytheory and simulation (13, 14). Nonconvex (NC) models havereceived significantly less attention. NC particles offer newpossibilities for the fabrication of functional materials as a resultof the self-assembly of unique structures driven by interlockingand entanglement (15–20). As featured in our current paper, NCrings can be packed into exotic highly open (low density) struc-tures. Self-assembled NC framelike arrays are of particularinterest due to their unusual optical, electrical, and mechani-cal properties, as a consequence of the large surface-to-volumeratios that can be attained. These emerging materials offerpromise in a broad range of applications including drug deliveryand therapeutics (18, 21–23), novel materials for catalysis (24),optics (18, 21), photonics (22, 23), and as nanopatterned scaf-folds (25, 26).

Notwithstanding the potential of such materials, the experimentalassessment of the different variables affecting their synthesis andcharacterization is a thankless empirical task, particularly if oneconsiders the limited range of stability of some of the more prom-ising liquid-crystalline phases. Molecular simulation plays an in-valuable role in reducing the experimental effort providing a directlink between the geometry of the constituent particles and the finalmicrostructures adopted by the material.We present a comprehensive investigation of the structures

formed by a particular class of framelike particles, namely colloidalrings, by direct molecular simulation. The nanorings are taken to beplanar and perfectly rigid. Two geometries are investigated startingfrom a circular shape as the basis: ellipsoidal and polygonal rings.Our model particles are represented as a number of tangentspherical segments of diameter σ forming a planar ring interactingvia repulsive interactions. The diameter of each bead is muchsmaller than the characteristic dimensions of the particles, allowingthe beads to be distributed in space to form circular rings of dif-ferent diameters, rings of different ellipticity, polygons of varioussymmetries, and thick-walled open cylindrical structures resemblingdoughnuts, bands, and washers.More specifically, three types of circular and equilateral po-

lygonal rings are modeled by systematically varying the sym-metry, the cavity size, and the width of the rings: type 1 singlerings (doughnuts) with different numbers Nb of tangent beadsand symmetries (Fig. 1 A–J); type 2 multistacked circular ringsmade up of n identical Nb rings bound sideways into cylindricalbands (Fig. 1K); and type 3 multilayered circular rings madeup of an Nb outer ring and n inner rings resembling washers(Fig. 1L).

Microsctructures and Phase BoundariesAn example of the class of NC particles classified here as circularrings of type 1 is depicted in Fig. 1A. In contrast to the phasescommonly associated with colloidal platelets and molecular

Significance

The formation of low-density porous structures is currently atopic of significant interest due to the advantageous electrical,optical, and chemical properties of the materials. We haveobserved the formation of unique highly open liquid-crystal-line smectic phases formed by the interlocking of highly sym-metrical planar nanorings. In particular, we demonstrate therelationship between the size of the internal cavity of the ringsand their symmetry on the formation of stable liquid-crystal-line phases with free volumes of up to 95%.

Author contributions: C.A., E.A.M., and F.A.E. designed research; C.A. performed research;C.A., G.J., E.A.M., and F.A.E. analyzed data; and C.A., G.J., E.A.M., and F.A.E. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. Email: carlos.avendano@manchester.ac.uk.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1604717113/-/DCSupplemental.

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discotics (27, 28), circular rings do not form (discotic) nematic(N) or columnar (C) phases as a consequence of the large par-ticle cavity. On sequential compression of an Nb = 28 bead ringsystem (Fig. 1A) using isothermal–isobaric (NPT) moleculardynamics (MD), the system exhibits a first-order phase transitionfrom an isotropic (Iso) liquid state at packing fraction of ϕ∼ 0.10to a smectic-A (SmA) phase at ϕ∼ 0.11 (Fig. 2). The SmA phaseis characterized by the formation of layers along the director n ofthe system, as confirmed by the large values of the positional(layering) order parameter τ in the direction of the layers. Thering particles are oriented perpendicular to the direction of thelayers, and as a consequence, the phase is characterized by negativevalues of the orientational (nematic) order parameter λ+. At higher

packing fractions, the order parameters take values of λ+ ∼ − 0.5and τ∼ 1.0 revealing a high degree of perpendicular orientationalorder of the rings with respect to the director. Moreover, the lowpacking fractions reveals that the SmA phase is highly open due tothe combined effect of the layered packing and large internal cav-ities of the particles as is apparent from Fig. 1A. The biaxial orderparameter Δλ does not suggest long-range orientational order of therings within the smectic layers, ruling out a biaxial phase. However,a close inspection of the clustering in the layers reveals that about50% of the rings form small stacklike clusters consisting of two to fourparticles, with dimer clusters the more abundant. The exceptionalself-assembly into this layered microstructure is driven by an inter-locking process in which the ring particles, or small clusters, fill the

Fig. 1. Particle models and stable microstructures observed in colloidal nanorings of various geometries. A–G represent the structures exhibited by rings ofdifferent symmetry but with a similar planar area: (A) smectic-A (SmA) phase for circular rings (Nb = 28); (B) SmA phase for ellipsoidal rings (Nb = 28) with anaspect ratio of e= 0.85; (C) disordered phase for ellipsoidal rings (Nb = 28) with e= 0.5 exhibiting clustered domains of tubular morphology (only clusters of morethan three particles are shown); (D) and (E) SmA phases for hexagonal and pentagonal rings (Nb = 30), respectively; (F) SmA phase for square rings ðNb = 32Þ; and(G) nematic (N) phase for triangular rings (Nb = 36). In each case we present a single particle with the corresponding number of beads, a snapshot of the liquid-crystalline structure, and a single layer of the SmA phase or N phase observed along the director. H–J correspond to the phases observed in circular rings of type1 (doughnuts) of increasing thickness. Systems H and I both form SmA phases, while system J, having no ring–ring interpenetration, is isotropic (Iso). K cor-responds to the SmA phase formed by multistacked cylindrical rings of type 2 (bands) comprising three layers. L corresponds to the N phase formed by circularrings of type 3 (washers) comprising one layer. For each system in H–L two particles are depicted to help visualize the maximum possible interpenetration. Thestatic structure factor SðkÞ projected in a plane containing the director of the phase is shown for all systems. The low wave vector SðkÞ for systems A, B, D–F, H, I,and K is characterized by strong unidimensional signals due to the formation of SmA layers.

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cavities of neighboring particles to pack more efficiently. The mean-square displacement (MSD) and its projections in the directionsalong and perpendicular to the director (Fig. S1) for the lowest-density SmA state can be used to analyze the mobility of theparticles. As is apparent from the analysis provided in Dynamics ofthe Smectic Phases and Figs. S2 and S3, the particles exhibit con-siderable mobility in their own layers with occasional hopping ofparticles to adjacent layers (not dissimilar to the dynamics of col-loidal rods in smectic layered structures) ruling out dynamicallyarrested states (29).To better understand the effect of the particle symmetry on

the formation of the SmA phases exhibited by circular rings, theshape of the rings can be deformed uniaxially into an ellipsoidalgeometry. Our model ellipsoidal ring particles are characterizedby the ratio of the minor and major axis e= b=a, and the numberof beads forming the ring (which is kept fixed as Nb = 28). Weanalyze particles with different degrees of ellipticity: Fig. 1A,e= 1.00; Fig. 1B, e= 0.85; Fig. 1C, e= 0.50. The results are shownin Fig. S4. The ring particles with an ellipticity of e= 0.85 (Fig.1B) form a SmA phase that is similar to that found for the cir-cular rings (Fig. 1A). The Iso–SmA phase transition is shiftedtoward higher pressures and packing fractions relative to that ofthe circular rings; very long simulations are required to stabilizethe SmA phase, indicating that is more difficult to form thelayered structure when the particle cavity is less symmetric. In-deed, for the particles with an ellipticity of e= 0.50, the forma-tion of mesophases is completely inhibited over the range ofconditions explored, although the slow kinetics does not allowone to completely rule out a transition; disordered structureswith large stacklike clusters are formed instead (Fig. 1C), so thatthe phase does not possess long-range orientational nor trans-lational order. This suggests that the interlocking of particles isnot sufficient to facilitate the layering into SmA phases; particleswith high degrees of rotational symmetry are required to stabilizeSmA layers of uniform thickness regardless of the orientation ofthe interlocking particles.To further understand the role of the symmetry of the particle

cavity in the stabilization of the liquid-crystalline phases, equi-lateral polygonal rings of type 1 with decreasing symmetry are alsoinvestigated: hexagons, pentagons, squares, and triangles (Fig. 1D–G and Fig. S5); the 2D in-plane order of discrete rotationalsymmetry for these polygons is 6, 5, 4, and 3, respectively. Ringparticles with hexagonal, pentagonal, and square symmetries arestill found to exhibit a first-order transition from an Iso to a SmA

phase. However, a reduction in the symmetry of the rings resultsin a diminished stability of the SmA phase with respect to the Nphase, which is consistent with the behavior observed for ellip-soidal rings. This reduced proclivity for self-assembly can be in-ferred from the considerable increase in the transition pressure andthe very long simulation runs required for the formation of the SmAphase, particularly in the case of square rings. Furthermore, thediscontinuity in the packing fraction at the transition becomessmaller, suggesting that the first-order nature of the phase transitionbecomes weaker as the particle symmetry is decreased. Both thenematic λ+ and positional τ order parameters tend toward smallervalues when the particle symmetry is decreased, which is in line withthe decreased stability of the SmA phases. An analysis of the MSDof the polygonal rings for the low-density smectic-A states (in thevicinity of the ordering transition) reveals that polygons, particularlysquares, have a much lower mobility than circular rings; a com-parison of the mobility at the same packing fraction (Fig. S6) revealsthat the MSD is however comparable, indicating that the particledynamics is dominated by packing effects and is largely independentof the particle symmetry. It is also important to highlight that theconcentration of trapped interlayer particles, which can be consid-ered as defects, increases when the symmetry of the particles isdecreased. Although these defects are almost absent in the SmAphase formed by circular rings (cf. Fig. S7), they are sufficientlyabundant in the SmA phase of square rings to cover the entire areaof the interlayer planes. As a consequence of the higher transla-tional mobility of the interlayer particles compared with those in thesmectic layers (Fig. S3), the high interlayer concentration appears tobe a factor in the stabilization of the SmA layers, compensating forthe low rotational symmetry of the particles. A SmA phase is notformed in the case of triangular rings, the polygons with the lowestsymmetry, and instead the system exhibits an N phase via a weakfirst-order transition from the Iso phase.Having established that high-symmetry particles are ideal for

the formation of the SmA phase with interlocking neighboringparticles, we proceed to quantify how the extent of interlockingaffects the viability of forming the SmA phase. For this purpose, it isconvenient to define the degree of interpenetration χ as the ratio ofthe maximum number of beads Ncross of a given particle that canpenetrate (cross) the cavity of another particle to the total numberof beads Nb averaged over all accessible mutual orientationsΩ: χ = hNcross=NbiΩ. Some examples of rings of different typehave already been depicted in Fig. 1. In the case of type 1 rings,the degree of interpenetration χ increases and the particle thick-ness δ (relative to the ring diameter) decreases as the number ofbeads in the ring Nb is increased (Fig. S8). For type 2 cylindricalbands (Fig. S9A), χ decreases as n is increased because of theconcomitant increase in δ (the cylinder height), and the cavitydiameter remains constant. By contrast in the case of type 3 rings(washers), χ decreases as the number of inner layers n is increasedkeeping δ constant (Fig. S9B).The generic phase behavior expected for polygonal rings of type 1

with different cavity sizes and particles symmetries, including thosealready described, is presented in Fig. 3 and Table S1, indicating theregions of stability of the ordered phases (for further details, seeEffect of the Cavity Size on the Phase Behavior of Circular Rings). Thecavity size and particle symmetry are characterized here in terms ofthe interpenetration parameter χ and the isoperimetric quotientQiso, defined as the ratio of the area of the ring to the area of a circlewith the same perimeter as the ring. Particles formed from Nb ∼ 12beads represent the lower bound of the cavity size for circular ringsat which the SmA phase is observed, and is the smallest system witha nonzero value of χ where at least one bead can penetrate thecavity of a neighboring particle. The microstructures formed bycircular rings of varying cavity size are shown in Fig. 1 H–J. Onewould expect the formation of an N phase for circular rings withNb < 12; however, the small internal cavity of the rings appears tofrustrate the alignment of the particles leading to the formation of

Fig. 2. Pressure-packing fraction (P*-ϕ) phase diagram for circular colloidalnanorings with Nb = 28 beads. The dependence of the nematic λ+, biaxial Δλ,and layering τ order parameters (defined in Materials and Methods) on thepacking fraction ϕ is indicated.

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dynamically arrested structures that lack long-range order and areessentially incompressible, and hypostatic, akin to the jammedphases observed in various anisotropic colloidal systems (30, 31) (cf.Characterization of Disorder Arrested States and Figs. S10 and S11).A similar behavior is observed for hexagonal and pentagonal ringsfor which SmA phases are observed only when the cavity is largeenough. Interestingly, although square rings with small internalcavities exhibit SmA phases, systems of square rings with largecavities tend to form N phases instead. This further demonstratesthat square rings represent a lower-symmetry bound for the for-mation of the SmA phases. Therefore, particles that possess both ahigh symmetry and a large internal cavity are likely candidates toform stable layered phases with a low packing fraction.The role of the degree of interpenetration on the formation of

liquid-crystalline phases, independently of the symmetry of theparticles, is analyzed using circular rings of types 1, 2, and 3. In thecase of type 2 cylindrical bands (Fig. 1K) with a thickness of twoand three layers (corresponding to an interpenetrability of χ =0.134 and 0.069, respectively), the system still forms stable SmAphases, demonstrating the role of the degree of interpenetrationfor a fixed circular symmetry. Conversely, for type 3 rings (wash-ers), the incorporation of just a single inner layer reduces the sizeof the cavity enough to make the degree of interpenetration verysmall, promoting the formation of an N phase instead of a SmAphase (Fig. 1L).Finally, the global phase diagram of type 1 ring particles is

shown in Fig. 4 (see sample structures in Fig. 1 A, H–J), where itis apparent that the Iso–SmA phase transition shifts progres-sively to lower packing fractions for rings with increasingly opencavities (corresponding to a larger Nb, thinner δ, and larger χ).This is equivalent to the approach of the Onsager limit for in-finitely long rods and infinitely thin platelets. In the case of themore traditional rodlike (calamitic) or disklike (discotic) mate-rials, the transition from the N phase to the SmA or C phaseoccurs at packing fractions of 45–50%, and is essentially in-sensitive to the aspect ratio (27, 32–34). The second virial co-efficient of a pair of freely rotating disks diverges in the limit ofinfinitely thin particles as the disks cannot interpenetrate (34);though there is a degree of interpenetration the same is true for the

second virial coefficient of the infinitely thin rings. In the case ofour NC nanoring particles, however, the packing fraction at thetransition to the SmA phase decreases with increasing aspect ratio(increasing Nb). For nanorings with the largest cavity studied, cor-responding to Nb = 56, the Iso–SmA phase transition is exhibited atϕ∼ 0.04, and the transition density will decrease further for particleswith larger cavities. These low values of the packing fraction suggestthat colloidal nanoring particles offer promise in the formationof layered microstructures with very high porosity after the dryingof any solvent, corresponding to a free volume of ∼96% observedat high (osmotic) pressures.

ConclusionsWe reported a detailed mapping for the formation of a distinc-tive class of self-assembled layered liquid-crystalline phases fromcolloidal nanorings in terms of both the symmetry of the particlesand the degree of interpenetration. These highly porous struc-tures are unique candidates as adsorption and storage materialsdue to their large void volumes and fluidity (35). One wouldexpect a significant enhancement in the diffusivity of small par-ticles within the smectic layers compared with isotropic liquidphases of equivalent porosity. Inclusion compounds comprisingthese low-density layered structures could provide a route to ad-ditional control of the mechanical, optical, and electronic prop-erties of the functional material.

Materials and MethodsThe nonconvex colloidal nanoring particles are modeled as rigid necklaces ofNb tangent spherical beads of diameter σ forming a planar structure. Thebeads interact via a soft repulsive Weeks–Chandler–Andersen (36) potentialcorresponding to the Lennard–Jones (LJ) potential truncated and shifted tozero at the cutoff distance rc = 21=6σ (36, 37). The principal orientation ofa particle j is described with a unit vector uj, which is perpendicular tothe plane of the ring. The phase behavior of the systems is obtained viacompression, starting from low-density Iso states using MD simulationin the isobaric–isothermal (NPT) ensemble. Properties are reported in

Fig. 3. Phase behavior of colloidal polygonal nanorings of different symmetry(triangles, squares, pentagons, hexagons, and circular rings). The points representstable nematic, smectic-A, or disorder arrested states. The regions of stability arerepresented in terms of the degree of interpenetration χ and isoperimetricquotientQiso of the particles which characterize their symmetry. Dashed lines aretentative boundaries for the different phases.

Fig. 4. Global phase diagram of circular colloidal nanorings of differentbead thickness. The packing fraction ϕ of the Iso–SmA transition for modelscomprising different numbers of beads Nb and degrees of interpenetrationχ, indicated in parentheses as ðNb, χÞ. The simulated states are representedas black dots, and the gray hatched area represents thermodynamicstates not studied. The transition pressure P* for each system is indicatedin the inset.

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dimensionless units using the LJ parameters: pressure P* = Pσ3=ðkBTÞ,temperature T* = kBT=e, time t* = t½e=ðmσ2Þ�1=2, and packing fractionϕ= πNbNσ3=ð6VÞ, where kB is the Boltzmann constant, e is the well-depth of theuntruncated LJ potential,m is the mass of the bead,N is the total number of ringparticles, and V is the volume of the system. The nanorings are treated asperfectly rigid bodies, and the equations of motion are integrated usingthe leap-frog algorithm with a time step of δt* = 0.004. To approximatethe behavior of hard particles, a temperature of T* = 1.25 is set in allsimulations (37, 38). The temperature and pressure are controlled usinga Nosé–Hoover thermostat and a Hoover barostat, respectively, forcubic simulation boxes with periodic boundary conditions in the threeCartesian coordinates (39, 40). The simulations are carried out using theDL_POLY software (41).

The orientational order of the system is characterized by computing theeigenvalues (jλ+j> jλ0j> jλ−j) of the orientational tensor Q from the particleorientation vectors uj; the director of the system n corresponds to the eigen-vector with the largest absolute eigenvalue λ+, the nematic order parameter(12). The biaxial order, indicating particle orientations perpendicular to themain director, is analyzed using the difference of the eigenvalues associated

with the eigenvectors that are perpendicular to the main director, i.e.,Δλ = jλ0 − λ−j. The formation of smectic layers in the system is monitoredusing a one-dimensional translational order parameter (42) defined asτðkdÞ= ð1=NÞÆ��PN

j=1expðikd · rjÞ��æ, where kd is the wave vector in the plane of

the layers that maximizes τ, and rj is the center of mass of particle j. This pa-rameter is the discrete one-dimensional analog of the translational order metricreported by Torquato et al. (43) evaluated using the wave vector with the samephase offset as the layering of the smectic states. The smectic layers can alsobe characterized using the structure factor SðkÞ= ð1=NÞ��PN

l,m=1expðik · rlmÞ��2,

where k is a wave vector in the plane containing the director, and rlm is thevector connecting the centers of mass of particles l and m.

ACKNOWLEDGMENTS. C.A. thanks A. Patti (University of Manchester) forhelpful discussions. G.J. and E.A.M. acknowledge funding from the Engi-neering and Physical Sciences Research Council (Grants EP/E016340, EP/L020564, EP/J010502, and EP/J014958), and F.A.E. acknowledges fundingfrom the US Department of Energy, Office of Basic Energy Sciences, Divisionof Materials Sciences and Engineering (Grant ER46517) and the NationalScience Foundation (Award CBET-1402117).

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