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

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 $N_b$ rings bound sideways into cylindrical bands (Fig. 1K); and type 3 multilayered circular rings made up of an $N_b$ outer ring and n inner rings resembling washers (Fig. 1L).

Fig. 1.

Particle models and stable microstructures observed in colloidal nanorings of various geometries. A–G represent the structures exhibited by rings of different symmetry but with a similar planar area: (A) smectic-A (SmA) phase for circular rings ($N_b=28$); (B) SmA phase for ellipsoidal rings ($N_b=28$) with an aspect ratio of $e=0.85$; (C) disordered phase for ellipsoidal rings ($N_b=28$) with $e=0.5$ exhibiting clustered domains of tubular morphology (only clusters of more than three particles are shown); (D) and (E) SmA phases for hexagonal and pentagonal rings ($N_b=30$), respectively; (F) SmA phase for square rings $\left(N_b=32\right)$; and (G) nematic (N) phase for triangular rings ($N_b=36$). 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 type 1 (doughnuts) of increasing thickness. Systems H and I both form SmA phases, while system J, having no ring–ring interpenetration, is isotropic (Iso). K corresponds to the SmA phase formed by multistacked cylindrical rings of type 2 (bands) comprising three layers. L corresponds to the N phase formed by circular rings of type 3 (washers) comprising one layer. For each system in H–L two particles are depicted to help visualize the maximum possible interpenetration. The static structure factor $S\left(\mathbf{k}\right)$ projected in a plane containing the director of the phase is shown for all systems. The low wave vector $S\left(\mathbf{k}\right)$ for systems A, B, D–F, H, I, and K is characterized by strong unidimensional signals due to the formation of SmA layers.

Microsctructures and Phase Boundaries

An example of the class of NC particles classified here as circular rings of type 1 is depicted in Fig. 1A. 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 $N_b=28$ bead ring system (Fig. 1A) using isothermal–isobaric ($NPT$) molecular dynamics (MD), the system exhibits a first-order phase transition from an isotropic (Iso) liquid state at packing fraction of $\varphi \sim 0.10$ to a smectic-A (SmA) phase at $\varphi \sim 0.11$ (Fig. 2). The SmA phase is characterized by the formation of layers along the director $\widehat{\mathbf{n}}$ of the system, as confirmed by the large values of the positional (layering) order parameter τ in the direction of the layers. The ring particles are oriented perpendicular to the direction of the layers, and as a consequence, the phase is characterized by negative values of the orientational (nematic) order parameter ${\lambda }_{+}$. At higher packing fractions, the order parameters take values of ${\lambda }_{+}\sim -0.5$ and $\tau \sim 1.0$ revealing a high degree of perpendicular orientational order of the rings with respect to the director. Moreover, the low packing fractions reveals that the SmA phase is highly open due to the combined effect of the layered packing and large internal cavities of the particles as is apparent from Fig. 1A. The biaxial order parameter ${\mathrm{\Delta }}_{\lambda }$ does not suggest long-range orientational order of the rings within the smectic layers, ruling out a biaxial phase. However, a close inspection of the clustering in the layers reveals that about 50% of the rings form small stacklike clusters consisting of two to four particles, with dimer clusters the more abundant. The exceptional self-assembly into this layered microstructure is driven by an interlocking process in which the ring particles, or small clusters, fill the cavities of neighboring particles to pack more efficiently. The mean-square displacement (MSD) and its projections in the directions along and perpendicular to the director (Fig. S1) for the lowest-density SmA state can be used to analyze the mobility of the particles. As is apparent from the analysis provided in 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).

Fig. 2.

Pressure-packing fraction ($P^{*}$-ϕ) phase diagram for circular colloidal nanorings with $N_b=28$ beads. The dependence of the nematic ${\lambda }_{+}$, biaxial ${\mathrm{\Delta }}_{\lambda }$, and layering τ order parameters (defined in Materials and Methods) on the packing fraction ϕ is indicated.

Fig. S1.

Mean-square displacement $\mathrm{\Delta }{\mathbf{r}}^2\left(t^{\text{*}}\right)$ of $N=2\text{,}000$ colloidal rings with circular ($N_b=28$), hexagonal ($N_b=30$), pentagonal ($N_b=30$), and square ($N_b=32$) symmetries. The states correspond to the lowest-density SmA phases formed by the systems, corresponding to $\left(P^{\text{*}}=0.084,\hspace{0.17em}\varphi =0.112\right)$ for circular rings, $\left(P^{\text{*}}=0.093,\hspace{0.17em}\varphi =0.117\right)$ for hexagonal rings, $\left(P^{\text{*}}=0.109,\hspace{0.17em}\varphi =0.123\right)$ for pentagonal rings, and $\left(P^{\text{*}}=0.128,\hspace{0.17em}\varphi =0.138\right)$ for square rings. In A the overall mean-square displacement is shown, and in B the projections in directions along and perpendicular to the director are shown.

Fig. S2.

Representative configurations at two different times for (A) circular rings with $N_b=28$ and (B) squares rings with $N_b=32$ showing particles with interlayer diffusion. The configurations shown are for the lowest-density SmA phases corresponding to the states $\left(P^{\text{*}}=0.0837,\varphi =0.112\right)$ for circular rings, and $\left(P^{\text{*}}=0.128\text{,}\hspace{0.17em}\varphi =0.138\right)$ for square rings. Left and Right correspond to ${10}^6$ time steps between the configurations.

Fig. S3.

Single-particle trajectories for circular rings with $N_b=28$. The trajectories are tracked for the lowest-density SmA phase corresponding to the states $\left(P^{\text{*}}=0.0837\text{,}\hspace{0.17em}\varphi =0.112\right)$. The red trajectory corresponds to a particle in one of the SmA layers, and the green trajectory corresponds to a particle initially trapped between layers (defects), which after some time reorientates and diffuse into an adjacent SmA layer. Left and Right correspond to the same trajectories observing the layer planes from above or the side, respectively.

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 $e=b/a$, and the number of beads forming the ring (which is kept fixed as $N_b=28$). We analyze particles with different degrees of ellipticity: Fig. 1A, $e=1.00$; Fig. 1B, $e=0.85$; Fig. 1C, $e=0.50$. The results are shown in 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 circular rings (Fig. 1A). 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 $e=0.50$, the formation of mesophases is completely inhibited over the range of conditions explored, although the slow kinetics does not allow one to completely rule out a transition; disordered structures with large stacklike clusters are formed instead (Fig. 1C), 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.

Fig. S4.

The phase behavior obtained from $NPT$–MD simulations for the pressure $P^{\text{*}}$, and order parameters ${\lambda }_{+}$, ${\mathrm{\Delta }}_{\lambda }$, and τ, as a function of the packing fraction ϕ for systems of $N=2\text{,}000$ rings: (A) $N_b=28$-membered ellipsoidal ring with an ellipticity $e=0.85$; (B) $N_b=28$-membered with $e=0.75$.

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 ${\lambda }_{+}$ and positional τ order parameters tend toward smaller values when the particle symmetry is decreased, which is in line with the decreased stability of the SmA phases. An analysis of the MSD of the polygonal rings for the low-density smectic-A states (in the vicinity of the ordering transition) reveals that polygons, particularly squares, have a much lower mobility than circular rings; a comparison of the mobility at the same packing fraction (Fig. S6) reveals that the MSD is however comparable, indicating that the particle dynamics is dominated by packing effects and is largely independent of the particle symmetry. It is also important to highlight that the concentration of trapped interlayer particles, which can be considered as defects, increases when the symmetry of the particles is decreased. Although these defects are almost absent in the SmA phase formed by circular rings (cf. Fig. S7), they are sufficiently abundant in the SmA phase of square rings to cover the entire area of the interlayer planes. As a consequence of the higher translational mobility of the interlayer particles compared with those in the smectic layers (Fig. S3), the high interlayer concentration appears to be a factor in the stabilization of the SmA layers, compensating for the low rotational symmetry of the particles. A SmA phase is not formed in the case of triangular rings, the polygons with the lowest symmetry, and instead the system exhibits an N phase via a weak first-order transition from the Iso phase.

Fig. S5.

The phase behavior obtained from $NPT$–MD simulations for the pressure $P^{\text{*}}$, and order parameters ${\lambda }_{+}$, ${\mathrm{\Delta }}_{\lambda }$, and τ, as a function of the packing fraction ϕ for systems of $N=2\text{,}000$ rings: (A) hexagonal rings ($N_b=30$; cf. Fig. 1F); (B) pentagonal rings ($N_b=30$; cf. Fig. 1G); (C) square rings ($N_b=32$; cf. Fig. 1D); and (D) triangular rings ($N_b=36$; cf. Fig. 1E).

Fig. S6.

Mean-square displacement $\mathrm{\Delta }{\mathbf{r}}^2\left(t^{\text{*}}\right)$ of $N=2\text{,}000$ colloidal rings with circular ($N_b=28$), hexagonal ($N_b=30$), pentagonal ($N_b=30$), and square ($N_b=32$) symmetries. The states correspond to SmA states of similar packing fraction, corresponding to $\left(P^{\text{*}}=0.202,\hspace{0.17em}\varphi =0.138\right)$ for circular rings, $\left(P^{\text{*}}=0.147,\hspace{0.17em}\varphi =0.138\right)$ for hexagonal rings, $\left(P^{\text{*}}=0.143,\hspace{0.17em}\varphi =0.123\right)$ for pentagonal rings, and $\left(P^{\text{*}}=0.128,\hspace{0.17em}\varphi =0.138\right)$ for square rings. In A the overall mean-square displacement is shown, and in B the projections in directions along and perpendicular to the director are shown.

Fig. S7.

Accumulation of rings trapped between two SmA layers for (A) circular rings with $N_b=28$, (B) hexagons with $N_b=30$, (C) pentagons with $N_b=30$, and (D) squares with $N_b=32$.

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 $N_{\text{cross}}$ of a given particle that can penetrate (cross) the cavity of another particle to the total number of beads $N_b$ averaged over all accessible mutual orientations $\mathrm{\Omega }$: $\chi ={\langle N_{\text{cross}}/N_b\rangle }_{\mathrm{\Omega }}$. Some examples of rings of different type have already been depicted in Fig. 1. In the case of type 1 rings, the degree of interpenetration χ increases and the particle thickness δ (relative to the ring diameter) decreases as the number of beads in the ring $N_b$ is increased (Fig. S8). For type 2 cylindrical bands (Fig. S9A), 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. S9B).

Fig. S8.

Models for circular rings of type 1 comprising different number of beads $N_s$ of diameter σ. Only systems with $N_s\ge 12$ exhibit the formation of SmA phases.

Fig. S9.

Modified models based on the original circular ring of $N_b=28$ beads: (A) rings of type 2 with thickness of one, two, and three layers have large degrees of interpenetration thus exhibiting the formation of stable SmA phases; (B) rings of type 3 with one, two, and three inner layers, where the degree of interpenetration is reduced considerably so that the systems form stable N phases rather than SmA phases.

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 $Q_{iso}$, defined as the ratio of the area of the ring to the area of a circle with the same perimeter as the ring. Particles formed from $N_b\sim 12$ beads represent the lower bound of the cavity size for circular rings at which the SmA phase is observed, and is the smallest system with a nonzero value of χ where at least one bead can penetrate the cavity of a neighboring particle. The microstructures formed by circular rings of varying cavity size are shown in Fig. 1 H–J. One would expect the formation of an N phase for circular rings with $N_b<12$; however, the small internal cavity of the rings appears to frustrate the alignment of the particles leading to the formation of dynamically arrested structures that lack long-range order and are essentially incompressible, and hypostatic, akin to the jammed phases 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 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.

Fig. 3.

Phase behavior of colloidal polygonal nanorings of different symmetry (triangles, squares, pentagons, hexagons, and circular rings). The points represent stable nematic, smectic-A, or disorder arrested states. The regions of stability are represented in terms of the degree of interpenetration χ and isoperimetric quotient $Q_{iso}$ of the particles which characterize their symmetry. Dashed lines are tentative boundaries for the different phases.

Table S1.

Summary of ordered states formed by colloidal rings

Group type	Shape	$N_b$	$Q_{iso}$	χ	α	Ordered phase
Ellipsoidal rings	Ellipse $e=0.85$	28	0.990	0.142	1.177	Smectic-A
	Ellipse $e=0.75$	28	0.970	0.156	1.333	Smectic-A
	Ellipse $e=0.50$	28	0.841	0.098	2.000	—
Regular rings of type 1 with different cavity sizes	Circles	56	1.000	0.289	1.000	Smectic-A
	Circles	42	1.000	0.255	1.000	Smectic-A
	Circles	28	1.000	0.188	1.000	Smectic-A
	Circles	21	1.000	0.129	1.000	Smectic-A
	Circles	14	1.000	0.015	1.000	Smectic-A
	Circles	12	1.000	0.004	1.000	Smectic-A
	Circles	10	1.000	0.000	1.000	—
	Circles	7	1.000	0.000	1.000	—
	Hexagons	30	0.907	0.147	1.155	Smectic-A
	Hexagons	24	0.907	0.104	1.155	Smectic-A
	Hexagons	18	0.907	0.047	1.155	Smectic-A
	Hexagons	12	0.907	0.002	1.155	—
	Hexagons	6	0.907	0.000	1.155	—
	Pentagons	30	0.865	0.111	1.236	Smectic-A
	Pentagons	25	0.865	0.079	1.236	Smectic-A
	Pentagons	20	0.865	0.046	1.236	Smectic-A
	Pentagons	15	0.865	0.019	1.236	—
	Pentagons	10	0.865	0.000	1.236	—
	Pentagons	5	0.865	0.000	1.236	—
	Squares	56	0.785	0.161	1.414	Nematic
	Squares	32	0.785	0.079	1.414	Smectic-A
	Squares	24	0.785	0.047	1.414	Smectic-A
	Squares	16	0.785	0.014	1.414	—
	Squares	12	0.785	0.007	1.414	—
	Squares	8	0.785	0.000	1.414	—
	Triangles	66	0.605	0.097	2.000	Nematic
	Triangles	36	0.605	0.059	2.000	Nematic
	Triangles	24	0.605	0.031	2.000	—
	Triangles	15	0.605	0.008	2.000	—
Rings of type 2	Circles with two layers	56	1.000	0.134	1.000	Smectic-A
	Circles with three layers	84	1.000	0.069	1.000	Smectic-A
	Circles with two inner layers	50	1.000	0.035	1.000	Nematic
Rings of type 3	Circles with three inner layers	66	1.000	0.000	1.000	Nematic
	Circles with four inner layers	76	1.000	0.000	1.000	Nematic
	Circles with five inner layers	80	1.000	0.000	1.000	Nematic

For each system we report the number of beads $N_s$ used, along with the isoperimetric quotient $Q_{iso}$, the degree of interpenetration χ, and the aspect ratio α, defined as the ratio of the circumscribed to the inscribed circles.

Fig. S10.

Relation between packing fraction ϕ, translational order τ, and average coordination number per particle Z for circular rings with different cavity sizes: The three systems correspond to circular rings with $N_b=56$ (squares); circular rings with $N_b=21$ (circles); and circular rings with $N_b=7$ (triangles). Symbols representing each state point are colored according to their average coordination number Z. For each system, a representative snapshot is shown of a particle with its Z neighbors.

Fig. S11.

(A) The center-of-mass radial distribution function $g\left(r\right)$, (B) the spherical averaged structure factor $S\left(k\right)$, and (C) mean-square displacement $\mathrm{\Delta }{\mathbf{r}}^2\left(t\right)$ of spherical rings with $N_b=7$ beads. The three states shown correspond to ($\varphi =0.353$, $\tau =0.050$, ${\lambda }_{+}=0.095$, $Z=7.4$), ($\varphi =0.374$, $\tau =0.046$, ${\lambda }_{+}=0.113$, $Z=8.1$), and ($\varphi =0.392$, $\tau =0.0418$, ${\lambda }_{+}=0.110$, $Z=8.7$).

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. 1K) 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. 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 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 $N_b$, thinner δ, and larger χ). This is equivalent to the approach of the Onsager limit for infinitely long rods and infinitely thin platelets. In the case of the more traditional rodlike (calamitic) or disklike (discotic) materials, the transition from the N phase to the SmA or C phase occurs at packing fractions of 45–50%, and is essentially insensitive to the aspect ratio (27, 32–34). The second virial coefficient of a pair of freely rotating disks diverges in the limit of infinitely 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 of our NC nanoring particles, however, the packing fraction at the transition to the SmA phase decreases with increasing aspect ratio (increasing N_b). For nanorings with the largest cavity studied, corresponding to $N_b=56$, the Iso–SmA phase transition is exhibited at $\varphi \sim 0.04$, and the transition density will decrease further for particles with larger cavities. These low values of the packing fraction suggest that colloidal nanoring particles offer promise in the formation of layered microstructures with very high porosity after the drying of any solvent, corresponding to a free volume of ∼96% observed at high (osmotic) pressures.

Fig. 4.

Global phase diagram of circular colloidal nanorings of different bead thickness. The packing fraction ϕ of the Iso–SmA transition for models comprising different numbers of beads $N_b$ and degrees of interpenetration χ, indicated in parentheses as $\left(N_b,\chi \right)$. The simulated states are represented as black dots, and the gray hatched area represents thermodynamic states not studied. The transition pressure $P^{*}$ for each system is indicated in the inset.

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 $\varphi =0.112$ and dimensionless pressure of $P^{\text{*}}=0.0837$. The trajectories of the centers of mass of a single particle located in one of the SmA layers and of another particle trapped between the SmA layers (which can be considered as a defect) are shown in Fig. S3. The trajectories span a period of $8\hspace{0.17em}\times \hspace{0.17em}{10}^6$ time steps. Two important features can be inferred from these trajectories. First, the particles in the SmA layers are more constrained and move more slowly than particles trapped in between layers. Second, the particles trapped between layers (defects) can reorient and be incorporated within an adjacent SmA layer, demonstrating a high mobility of the particles in the SmA phase and unveiling a mechanism for particle transfers between different layers. The concentration of defects increases when the symmetry decreases as shown in Fig. S7. These defects are almost absent in the SmA phase formed by circular rings, but abundant enough in the SmA phase of square rings to cover the entire area of the interlayer planes. Because interlayer particles have a higher translational mobility than those in the SmA layers, the high concentration of interlayer particles appears to be a factor in the stabilization of the SmA layers, compensating for the low rotational degrees of freedom.

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 $N_b=7$, 10, 12, 14, 21, 28, 42, and 56 beads are examined (Fig. S8). The simulations reveal that circular rings with $N_b\ge 12$ form stable SmA phases. Systems with $N_b=10$ and $N_b=7$ do not form SmA phases, forming instead disordered/jammed phases. A similar analysis is performed for regular polygonal rings, and the corresponding structures observed are presented in Table S1.

Two different modifications of the original circular ring with $N_b=28$ are also considered. These modifications correspond to type 2 rings where the thickness of the particles is increased while keeping the diameter of the cavity constant (resembling cylindrical bands), and type 3 rings where the cavity of the ring is gradually filled by adding additional inner layers (resembling washers) as depicted in Fig. S9. Type 3 rings have small cavities suppressing the formation of SmA phases, and leading instead to the formation of stable N phases. By contrast, type 2 rings with a thickness of two and three layers and an equal cavity size form stable SmA phases.

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 $r_{ij,c}$ between their two nearest spherical beads is within 5% ($\chi =1.05$) of the WCA potential cutoff, i.e., $r_{ij,c}<2^{1/6}\sigma \chi$. The results for the average Z for circular rings with $N_b=56$, 21, and 7 beads are shown in Fig. S10 (the analysis for other circular rings is not included for clarity, but they exhibit similar characteristics). For rings with $N_b=56$, corresponding to the rings with the largest internal cavity studied, the average coordination number is $Z<2$ for all values of the packing fraction explored in both isotropic (Iso) and SmA states; and for rings with $N_b=21$, the average coordination number is $Z<5$. Even at relatively high packing fractions the systems are found to exhibit low-coordination numbers with respect to what one would expect for common liquids. By contrast, for rings with $N_b=7$, which have a small cavity size, the average value of Z increases monotonically upon compression reaching a saturation value of $Z\sim 10$. The large increase in the 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 $f=6$ degrees of freedom per particle. [In general, for perfect smooth rings described as toroidal particles with azimuthal symmetry, the number of degrees of freedom per particle is $f=5$: three translational $\left(z,y,z\right)$ and two rotational $\left(\theta ,\varphi \right)$ (30). In our case, as we do not have perfect tori, the rigid particles comprising a collection of $N_b$ beads have $f=6$ degrees of freedom per particle: three translational of freedom $\left(x,y,z\right)$, and three rotational described by a set of Euler angles $\left(\varphi ,\theta ,\psi \right)$.] A system is said to be isostatic if $Z=2f$, and hypostatic if $Z<2f$ (31). For the particular case of frictionless spheres with $f=3$, it has been shown that arrested jammed phases are both isostatic, i.e., $Z=2f=6$, and hyperuniform (i.e., their density fluctuations are suppressed at low wavenumbers). However, several jammed phases of nonspherical particles are known to be hypostatic (30, 31). We observe that the arrested states found in small colloidal rings with $N_b=7$ are always hypostatic, exhibiting a similar coordination number at high packing fractions as in jammed phases of ellipsoids with $Z\sim 10$ contacts per particle (30, 46). To further understand the structure of the arrested states for rings with $N_b=7$, we calculate the radial distribution function $g\left(r\right)$ between the centers of mass of the rings, the spherically averaged structure factor $S\left(k\right)$, and the mean-square displacement $\mathrm{\Delta }{\mathbf{r}}^2\left(t\right)$ for three states of different packing fractions: (i) $\varphi =0.353$, $\tau =0.050$, ${\lambda }_{+}=0.095$, $Z=7.4$; (ii) $\varphi =0.374$, $\tau =0.046$, ${\lambda }_{+}=0.113$, $Z=8.1$; and (iii) $\varphi =0.392$, $\tau =0.0418$, ${\lambda }_{+}=0.110$, $Z=8.7$. The spherically averaged structure factor $S\left(q\right)$ is obtained from the Fourier transform of $g\left(r\right)$,

$$S\left(k\right)=1+\frac{4\pi \rho }{k}{\displaystyle {\int }_0^{\infty }r\left[g\left(r\right)-1\right]\sin \left(kr\right)\text{d}r},$$ [S1]

where k is the wave vector, r is the distance between the centers of mass of the rings, $\rho =N/V$, 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 ${\lambda }_{+}$, and from the features of $g\left(r\right)$ that the three states exhibit some degree of local structure, but lack long-range order. The first peak in $g\left(r\right)$ at $r\sim 1.2\sigma$ corresponds to the configuration of two rings parallel to each other (stacks), the second peak at $r\sim 1.8\sigma$ corresponds to a perpendicular arrangement of the rings, and $g\left(r\right)\to 1$ at long interseparations. Upon increasing the packing fraction, the features in $g\left(r\right)$ remain the same, with only a slight increment in the amplitude of the peaks confirming the system retains long-range disorder. Furthermore, the behavior of $S\left(q\right)$ at low k is related to the compressibility of the system. For the three states shown in Fig. S11, $S\left(k\right)\to 0$ as $k\to 0$ demonstrating the structures are almost incompressible. However, a more detail analysis using large system sizes is needed to completely characterize the low-k behavior of $S\left(k\right)$, and determine if the system is hyperuniform [a feature observed in various jammed states (31)], as we are restricted to wave vectors that are commensurate with the system size due to periodic boundary conditions, i.e., $k_{min}=2\pi /L$, where L is the simulation box length. Finally, analyzing the mean-square displacement $\mathrm{\Delta }{\mathbf{r}}^2\left(t\right)$, we can observe that state 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 $N_b$ tangent spherical beads of diameter σ forming a planar structure. The beads interact via a soft repulsive Weeks–Chandler–Andersen (36) potential corresponding to the Lennard–Jones (LJ) potential truncated and shifted to zero at the cutoff distance $r_c=2^{1/6}\sigma$ (36, 37). The principal orientation of a particle j is described with a unit vector ${\widehat{\mathbf{u}}}_j$, which is perpendicular to the plane of the ring. The phase behavior of the systems is obtained via compression, starting from low-density Iso states using MD simulation in the isobaric–isothermal ($NPT$) ensemble. Properties are reported in dimensionless units using the LJ parameters: pressure $P^{*}=P{\sigma }^3/\left(k_{B}T\right)$, temperature $T^{*}=k_{B}T/\mathit{ϵ}$, time $t^{*}=t{\left[\mathit{ϵ}/\left(m{\sigma }^2\right)\right]}^{1/2}$, and packing fraction $\varphi =\pi N_{b}N{\sigma }^3/\left(6V\right)$, where $k_B$ is the Boltzmann constant, ε is the well-depth of the untruncated LJ potential, 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 $\delta t^{\text{*}}=0.004$. To approximate the behavior of hard particles, a temperature of $T^{*}=1.25$ is set in all simulations (37, 38). The temperature and pressure are controlled using a Nosé–Hoover thermostat and a Hoover barostat, respectively, for cubic simulation boxes with periodic boundary conditions in the three Cartesian coordinates (39, 40). The simulations are carried out using the DL_POLY software (41).

The orientational order of the system is characterized by computing the eigenvalues ($\left|{\lambda }_{+}\right|>\left|{\lambda }_0\right|>\left|{\lambda }_{-}\right|$) of the orientational tensor $\mathbf{Q}$ from the particle orientation vectors ${\widehat{\mathbf{u}}}_j$; the director of the system $\widehat{\mathbf{n}}$ corresponds to the eigenvector with the largest absolute eigenvalue ${\lambda }_{+}$, the nematic order parameter (12). The biaxial order, indicating particle orientations perpendicular to the main director, is analyzed using the difference of the eigenvalues associated with the eigenvectors that are perpendicular to the main director, i.e., ${\mathrm{\Delta }}_{\lambda }=\left|{\lambda }_0-{\lambda }_{-}\right|$. The formation of smectic layers in the system is monitored using a one-dimensional translational order parameter (42) defined as $\tau \left({\mathbf{k}}_{\mathbf{d}}\right)=\left(1/N\right)\langle \left|{\displaystyle {\sum }_{j=1}^N\exp \left(i{\mathbf{k}}_{\mathbf{d}}\cdot {\mathbf{r}}_j\right)}\right|\rangle$, where ${\mathbf{k}}_{\mathbf{d}}$ is the wave vector in the plane of the layers that maximizes τ, and ${\mathbf{r}}_j$ is the center of mass of particle 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 $S\left(\mathbf{k}\right)=\left(1/N\right){\left|{\displaystyle {\sum }_{l,m=1}^N\exp \left(i\mathbf{k}\cdot {\mathbf{r}}_{lm}\right)}\right|}^2$, where $\mathbf{k}$ is a wave vector in the plane containing the director, and ${\mathbf{r}}_{lm}$ is the vector connecting the centers of mass of particles 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 $N_b=28$ beads) and distorting the geometry to form ellipsoidal rings with ratios of the minor to major axes of $e=b/a=0.85$, 0.75, and 0.5. The results of $NPT$ MD simulations for $e=0.85$ and 0.75 are shown in Fig. S4, where the formation of smectic-A (SmA) phases is observed. The SmA phase is characterized by large values of the translational order parameter τ and large negative values of the orientational order parameter ${\lambda }_{+}$. The transition appears to be first order based on the discontinuity of the order parameters. The system with $e=0.5$ does not form an ordered fluid structure, forming instead a jammed phase of columnar clusters.

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 ${\lambda }_{+}$ and zero value of the translational order parameter τ.