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# Assembly of porous smectic structures formed from interlocking high-symmetry planar nanorings

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

## Significance

The formation of low-density porous structures is currently a topic of significant interest due to the advantageous electrical, optical, and chemical properties of the materials. We have observed the formation of unique highly open liquid-crystalline smectic phases formed by the interlocking of highly symmetrical planar nanorings. In particular, we demonstrate the relationship between the size of the internal cavity of the rings and their symmetry on the formation of stable liquid-crystalline phases with free volumes of up to 95%.

## Abstract

Materials comprising porous structures, often in the form of interconnected concave cavities, are typically assembled from convex molecular building blocks. The use of nanoparticles with a characteristic nonconvex shape provides a promising strategy to create new porous materials, an approach that has been recently used with cagelike molecules to form remarkable liquids with “scrabbled” porous cavities. Nonconvex mesogenic building blocks can be engineered to form unique self-assembled open structures with tunable porosity and long-range order that is intermediate between that of isotropic liquids and of crystalline solids. Here we propose the design of highly open liquid-crystalline structures from rigid nanorings with ellipsoidal and polygonal geometry. By exploiting the entropic ordering characteristics of athermal colloidal particles, we demonstrate that high-symmetry nonconvex rings with large internal cavities interlock within a 2D layered structure leading to the formation of distinctive liquid-crystalline smectic phases. We show that these smectic phases possess uniquely high free volumes of up to ∼95%, a value significantly larger than the 50% that is typically achievable with smectic phases formed by more conventional convex rod- or disklike mesogenic particles.

Self-assembly of particles ranging in size from the nanometer to the micrometer scales can be used to fabricate structures in the mesoscale regime (1⇓⇓⇓–5), otherwise difficult to achieve with traditional methods of chemical synthesis. Strategies to produce functional materials from the self-assembly of relatively simple nonspherical (anisotropic) building blocks have undergone unprecedented growth as a result of recent advances in experimental techniques to fashion colloidal and nanoparticles of arbitrary shape and well-defined sizes (6⇓⇓–9). An appealing feature of colloidal particles is that the repulsive and attractive contributions of the interaction between the particles can be modulated by controlling the properties of both the particle surface and the solvent medium to induce different types of forces, including short-range repulsions approaching hard-core (athermal) interactions (10).

Colloidal particles are commonly represented using simplified coarse-grained convex geometries including ellipsoids, spherocylinders, polyhedra, and cut spheres (11, 12). The phase behavior of these convex models has been studied extensively by theory and simulation (13, 14). Nonconvex (NC) models have received significantly less attention. NC particles offer new possibilities for the fabrication of functional materials as a result of the self-assembly of unique structures driven by interlocking and entanglement (15⇓⇓⇓⇓–20). As featured in our current paper, NC rings can be packed into exotic highly open (low density) structures. Self-assembled NC framelike arrays are of particular interest due to their unusual optical, electrical, and mechanical properties, as a consequence of the large surface-to-volume ratios that can be attained. These emerging materials offer promise in a broad range of applications including drug delivery and therapeutics (18, 21⇓–23), novel materials for catalysis (24), optics (18, 21), photonics (22, 23), and as nanopatterned scaffolds (25, 26).

Notwithstanding the potential of such materials, the experimental assessment of the different variables affecting their synthesis and characterization is a thankless empirical task, particularly if one considers the limited range of stability of some of the more promising liquid-crystalline phases. Molecular simulation plays an invaluable role in reducing the experimental effort providing a direct link between the geometry of the constituent particles and the final microstructures adopted by the material.

We present a comprehensive investigation of the structures formed by a particular class of framelike particles, namely colloidal rings, by direct molecular simulation. The nanorings are taken to be planar and perfectly rigid. Two geometries are investigated starting from a circular shape as the basis: ellipsoidal and polygonal rings. Our model particles are represented as a number of tangent spherical segments of diameter σ forming a planar ring interacting via repulsive interactions. The diameter of each bead is much smaller than the characteristic dimensions of the particles, allowing the beads to be distributed in space to form circular rings of different diameters, rings of different ellipticity, polygons of various symmetries, and thick-walled open cylindrical structures resembling doughnuts, bands, and washers.

More specifically, three types of circular and equilateral polygonal rings are modeled by systematically varying the symmetry, the cavity size, and the width of the rings: type 1 single rings (doughnuts) with different numbers *N _{b}* of tangent beads and symmetries (Fig. 1

*A–J*); type 2 multistacked circular rings made up of

*n*identical

*K*); and type 3 multilayered circular rings made up of an

*n*inner rings resembling washers (Fig. 1

*L*).

## Microsctructures and Phase Boundaries

An example of the class of NC particles classified here as circular rings of type 1 is depicted in Fig. 1*A*. In contrast to the phases commonly associated with colloidal platelets and molecular discotics (27, 28), circular rings do not form (discotic) nematic (N) or columnar (C) phases as a consequence of the large particle cavity. On sequential compression of an *A*) using isothermal–isobaric (*A*. The biaxial order parameter *Dynamics of the Smectic Phases* and Figs. S2 and S3, the particles exhibit considerable mobility in their own layers with occasional hopping of particles to adjacent layers (not dissimilar to the dynamics of colloidal rods in smectic layered structures) ruling out dynamically arrested states (29).

To better understand the effect of the particle symmetry on the formation of the SmA phases exhibited by circular rings, the shape of the rings can be deformed uniaxially into an ellipsoidal geometry. Our model ellipsoidal ring particles are characterized by the ratio of the minor and major axis *A*, *B*, *C*, *B*) form a SmA phase that is similar to that found for the circular rings (Fig. 1*A*). The Iso–SmA phase transition is shifted toward higher pressures and packing fractions relative to that of the circular rings; very long simulations are required to stabilize the SmA phase, indicating that is more difficult to form the layered structure when the particle cavity is less symmetric. Indeed, for the particles with an ellipticity of *C*), so that the phase does not possess long-range orientational nor translational order. This suggests that the interlocking of particles is not sufficient to facilitate the layering into SmA phases; particles with high degrees of rotational symmetry are required to stabilize SmA layers of uniform thickness regardless of the orientation of the interlocking particles.

To further understand the role of the symmetry of the particle cavity in the stabilization of the liquid-crystalline phases, equilateral polygonal rings of type 1 with decreasing symmetry are also investigated: hexagons, pentagons, squares, and triangles (Fig. 1 *D–G* and Fig. S5); the 2D in-plane order of discrete rotational symmetry for these polygons is 6, 5, 4, and 3, respectively. Ring particles with hexagonal, pentagonal, and square symmetries are still found to exhibit a first-order transition from an Iso to a SmA phase. However, a reduction in the symmetry of the rings results in a diminished stability of the SmA phase with respect to the N phase, which is consistent with the behavior observed for ellipsoidal rings. This reduced proclivity for self-assembly can be inferred from the considerable increase in the transition pressure and the very long simulation runs required for the formation of the SmA phase, particularly in the case of square rings. Furthermore, the discontinuity in the packing fraction at the transition becomes smaller, suggesting that the first-order nature of the phase transition becomes weaker as the particle symmetry is decreased. Both the nematic

Having established that high-symmetry particles are ideal for the formation of the SmA phase with interlocking neighboring particles, we proceed to quantify how the extent of interlocking affects the viability of forming the SmA phase. For this purpose, it is convenient to define the degree of interpenetration χ as the ratio of the maximum number of beads *A*), χ decreases as *n* is increased because of the concomitant increase in δ (the cylinder height), and the cavity diameter remains constant. By contrast in the case of type 3 rings (washers), χ decreases as the number of inner layers *n* is increased keeping δ constant (Fig. S9*B*).

The generic phase behavior expected for polygonal rings of type 1 with different cavity sizes and particles symmetries, including those already described, is presented in Fig. 3 and Table S1, indicating the regions of stability of the ordered phases (for further details, see *Effect of the Cavity Size on the Phase Behavior of Circular Rings*). The cavity size and particle symmetry are characterized here in terms of the interpenetration parameter χ and the isoperimetric quotient *H–J*. One would expect the formation of an N phase for circular rings with *Characterization of Disorder Arrested States* and Figs. S10 and S11). A similar behavior is observed for hexagonal and pentagonal rings for which SmA phases are observed only when the cavity is large enough. Interestingly, although square rings with small internal cavities exhibit SmA phases, systems of square rings with large cavities tend to form N phases instead. This further demonstrates that square rings represent a lower-symmetry bound for the formation of the SmA phases. Therefore, particles that possess both a high symmetry and a large internal cavity are likely candidates to form 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 the particles, is analyzed using circular rings of types 1, 2, and 3. In the case of type 2 cylindrical bands (Fig. 1*K*) with a thickness of two and three layers (corresponding to an interpenetrability of χ = 0.134 and 0.069, respectively), the system still forms stable SmA phases, demonstrating the role of the degree of interpenetration for a fixed circular symmetry. Conversely, for type 3 rings (washers), the incorporation of just a single inner layer reduces the size of the cavity enough to make the degree of interpenetration very small, promoting the formation of an N phase instead of a SmA phase (Fig. 1*L*).

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 it is apparent that the Iso–SmA phase transition shifts progressively to lower packing fractions for rings with increasingly open cavities (corresponding to a larger *N _{b}*). For nanorings with the largest cavity studied, corresponding to

## Dynamics of the Smectic Phases

The mean-square displacement (MSD) of the particles in SmA phases formed by colloidal rings with circular, hexagonal, pentagonal, and square geometries is used to analyze the translational dynamics of the particles. A comparison of the MSD, and its projections in directions along and perpendicular to the director is carried out for (*i*) the lowest-density SmA states, and (*ii*) SmA states of similar density. The results are shown in Figs. S1 and S6, respectively. It is apparent that circular rings in the lowest-density SmA phase exhibit higher mobility than the particles with other ring geometries. Although the dynamics of circular rings is similar to the dynamics exhibited by rings with hexagonal and pentagonal symmetries, it is considerable different from that observed for square rings. Rings with higher symmetry are also found to move faster. The analysis of the MSD perpendicular and parallel to the director reveals a high mobility of the rings in the SmA layers and a low mobility along the director. The projection of the MSD along the director first exhibits a ballistic regime, followed by a plateau, with only a small increment of mobility at longer times, implying a low probability of particles moving into other layers. This behavior is expected as the mechanism for the formation of SmA phases in colloidal rings is driven by the interlocking of particles in the SmA layers. An analysis of the MSD in configurations of similar packing fraction (Fig. S6) reveals that at these conditions the mobility of the various particle types is very similar, in particular within the smectic layers. Interlayer diffusion of particles can also be detected in Fig. S2 where some particles in the same lowest-density SmA states of circular rings have been colored at two different times. A similar analysis of individual particles is carried out for square rings revealing that particles in the SmA phases are also able to move to adjacent layers.

To further understand the dynamics of circular rings in the SmA phase, some individual particles are tracked in the lowest-density SmA state formed by the system, corresponding to a packing fraction of

## Effect of the Cavity Size on the Phase Behavior of Circular Rings

The type 1 models for circular and polygonal rings are used to study the effect of the cavity size on the formation of the SmA phase. Type 1 models for circular rings with

Two different modifications of the original circular ring with

## Characterization of Disorder Arrested States

As described in the main text and in the previous section, we do not observe the formation of SmA phases in rings with small cavity sizes, leading instead to arrested disordered states. We cannot, however, completely rule out the possibility that the suppression of ordered phases, in particularly columnar phases, is due to slow kinetics or the roughness of our polybead particle surfaces [the latter has been observed to have a considerable effect in the suppression of liquid crystalline phases of rodlike particles formed by tangent spheres (44, 45)]. To understand the suppression of ordered phases, we have calculated the average coordination number per particle *Z* for circular rings with different cavity sizes. The instantaneous coordination number of a ring *i* is calculated by determining the total number of neighboring rings *Z* making direct contact. Two particles *i* and *j* are said to be in contact if the distance *Z* for circular rings with *Z* increases monotonically upon compression reaching a saturation value of *Z* observed in rings with small cavities correlates with the dynamic slowing down of the system and the formation of arrested disordered states.

It is instructive to analyze the relationship between *Z* and the number of degrees of freedom per particle *f* for the different ring systems (30). This relationship has proved to be useful in describing the structural stability of various colloidal systems in jammed states (31, 46). The models of colloidal rings used in our current work have *i*) *ii*) *iii*) *k* is the wave vector, *r* is the distance between the centers of mass of the rings, *N* is the total number of rings, and *V* is the total volume of the system. The results for the three states are shown in Fig. S11. It is evident from the magnitudes of the ordered parameters τ and *k* is related to the compressibility of the system. For the three states shown in Fig. S11, *k* behavior of *L* is the simulation box length. Finally, analyzing the mean-square displacement *i* exhibits some degree of fluidity (this was also confirmed by visual inspection of the trajectories), but for systems *ii* and *iii* the mean-square displacement exhibits minimal particle mobility confirming the suppression of fluid dynamics. All of these features, i.e., lack of long-range structural order, high coordination numbers, and arrested dynamics, suggest that systems with small cavities form disordered jammed states.

## Conclusions

We reported a detailed mapping for the formation of a distinctive class of self-assembled layered liquid-crystalline phases from colloidal nanorings in terms of both the symmetry of the particles and the degree of interpenetration. These highly porous structures are unique candidates as adsorption and storage materials due to their large void volumes and fluidity (35). One would expect a significant enhancement in the diffusivity of small particles within the smectic layers compared with isotropic liquid phases of equivalent porosity. Inclusion compounds comprising these low-density layered structures could provide a route to additional control of the mechanical, optical, and electronic properties of the functional material.

## Materials and Methods

The nonconvex colloidal nanoring particles are modeled as rigid necklaces of *j* is described with a unit vector *m* is the mass of the bead, *N* is the total number of ring particles, and *V* is the volume of the system. The nanorings are treated as perfectly rigid bodies, and the equations of motion are integrated using the leap-frog algorithm with a time step of

The orientational order of the system is characterized by computing the eigenvalues (*j*. This parameter is the discrete one-dimensional analog of the translational order metric reported by Torquato et al. (43) evaluated using the wave vector with the same phase offset as the layering of the smectic states. The smectic layers can also be characterized using the structure factor *l* and *m*.

## Effect of the Particle Symmetry and Cavity Size on the Phase Behavior of Colloidal Rings

The effect of the distortion of the cavity of the rings on phase behavior is analyzed by taking the base case (circular ring with

An analysis of the in-plane (2D) discrete rotational symmetry is undertaken by examining rings with hexagonal, pentagonal, square, and triangular symmetries. The corresponding simulation data are presented in Fig. S5. All of the polygons except the triangular rings exhibit the formation of stable SmA phases. The system of triangular rings exhibits a nematic (N) phase characterized by large positive values of the nematic order parameter

## Acknowledgments

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

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: carlos.avendano{at}manchester.ac.uk.

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.

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

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- Effect of the Cavity Size on the Phase Behavior of Circular Rings
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