## Abstract

Robust control over the positions, orientations, and assembly of nonspherical colloids may aid in the creation of new types of structured composite materials that are important from both technological and fundamental standpoints. With the use of lithographically fabricated equilateral polygonal platelets, we demonstrate that colloidal interactions and self-assembly in anisotropic nematic fluids can be effectively tailored via control over the particles’ shapes. The particles disturb the uniform alignment of the surrounding nematic host, resulting in both a distinct equilibrium alignment and highly directional pair interactions. Interparticle forces between polygonal platelets exhibit either dipolar or quadrupolar symmetries, depending on whether their number of sides is odd or even, and drive the assembly of a number of ensuing self-assembled colloidal structures.

Self-assembly of micrometer- and nanometer-scale colloidal particles into ordered structures is of wide-ranging interest for both fundamental science and technological applications (*1*). In isotropic liquids such as water, the electrostatic and entropic forces that drive the assembly of spherical colloids are typically isotropic, limiting the overall landscape of possible structures. Concentrated suspensions of monodisperse spherical particles are an important example; these can form three-dimensional (3D) colloidal crystals that are markedly similar to their atomic counterparts. However, colloidal crystals formed in this fashion are restricted to lattices with high packing fractions, such as hexagonal close-packed or face-centered cubic (*2*). The generation of anisotropic interactions is necessary to increase the complexity and diversity of colloidal architectures formed by such interactions (*3*–*6*). Oriented assemblies of particles can be produced by means such as nonuniform patterning of their surfaces (*3*), anisotropic deposition of colloids onto solid substrates (*4*), or application of external fields (*5*). Alternatively, introducing anisotropy directly into a solvent by using a nematic liquid crystal (NLC), one can engender anisotropic interaction forces between colloids that are not present in ordinary fluids (*7*). NLCs are composed of rod-shaped molecules with long molecular axes aligned along a common direction (*8*). The local average molecular orientation is often represented by a unit vector **n** with inversion symmetry **n** ≡ –**n**, referred to as the director. The dependence of **n** as a function of spatial position **r** is described with a director field **n**(**r**). Anisotropic molecular interactions at NLC surfaces, known as surface anchoring, result in a preferential alignment and boundary conditions for **n**(**r**). Colloids immersed in NLCs deform the surrounding director field because of this surface anchoring and induce point or line defects [regions where **n**(**r**) is discontinuous] in the nematic bulk (Fig. 1, A and B) or at the nematic-particle interface (Fig. 1C), unless the surface anchoring is weak or the particles are small (supporting online material fig. S1) (*9*). The particles and accompanying defects introduce long-range gradients in **n**(**r**) that depend on particle size (*9*), type and strength of surface anchoring (*10*), confinement (*11*, *12*), and external fields (*13*). The elastic energy due to these gradients depends on the particles’ relative positions and gives rise to interactions mediated by elasticity. Even for spherical particles in NLCs (Fig. 1, A to C), elastic interactions are highly anisotropic and can lead to a host of self-assembled structures ranging from linear and branched chains to 2D crystals (*7*, *9*–*16*). Reminiscent of electrostatic interactions exhibited by charge distributions, elastic colloidal interactions bear qualitatively different symmetries that mimic the dipolar (Fig. 1A) or quadrupolar (Fig. 1, B and C) symmetries of **n**(**r**) around isolated particles.

We demonstrate that altering the shapes of particles can lead to marked changes in the symmetry of their elastic interactions and the resulting colloidal assemblies in NLCs. Optical polarizing microscopy (PM) and fluorescence confocal polarizing microscopy (FCPM) show that platelet colloids with equilateral polygonal shapes exhibit well-defined alignment and elastic deformations of **n**(**r**) that have either dipolar or quadrupolar symmetry. Colloidal polygons with an odd number of sides form elastic dipoles, whereas even-sided particles form elastic quadrupoles. Using model polygonal platelets shaped as triangles, squares, and pentagons, we demonstrate that their shape dictates the resulting **n**(**r**) symmetry as well as the symmetry of the ensuing elastic interactions. Particle tracking video microscopy (*17*), combined with optical tweezing of particle pairs, provides direct measurements of anisotropic interaction forces.

Monodisperse platelet colloids of uniform thickness and predesigned shapes are fabricated with the use of photolithography (*18*). Micron-sized polygonal colloids of triangular, square, and pentagonal shapes are produced using an ultraviolet-sensitive photoresist (SU-8) on Si wafers (*19*). After exposure and development, the particles are released from the wafers into an organic solvent and transferred into pentylcyanobiphenyl (5CB), a room temperature NLC. Sample cells consisting of parallel glass plates separated by 10- to 60-μm spacers are filled with colloidal dispersions in 5CB by capillary action and sealed with epoxy. The far-field alignment direction **n _{0}** is set by unidirectional rubbing of the polyimide coated inner surfaces of the cell. The samples are studied with an inverted optical microscope equipped with a confocal laser scanning unit and a holographic optical tweezers system (

*20*) operating at λ = 1064 nm. The 3D structure of

**n**(

**r**) around the colloids is determined with lateral and vertical resolution of ~0.5 μm with the use of FCPM (

*21*). For FCPM observations, 0.01 weight percent of anisotropic fluorescent dye was dissolved homogeneously in 5CB (

*19*); at this concentration, the rodlike dye molecules do not alter the NLC properties and orient parallel to 5CB molecules so that the contrast in the fluorescence image arises from spatial changes in

**n**(

**r**) (

*21*). Imaging and optical tweezing are performed simultaneously with a 100× oil-immersion objective.

PM images reveal the **n**(**r**) deformations surrounding isolated particles of each type suspended in aligned 5CB (Fig. 1). When **n _{0}** is oriented along the linear polarization of incident light, distorted regions where

**n**(

**r**) departs from

**n**alter the polarization state of transmitted light and appear bright when viewed through the analyzer. Polygonal platelets always orient with their larger-area top and bottom surfaces parallel to

_{0}**n**, suggesting planar degenerate anchoring at the interface of SU-8 and 5CB. Polygons that have an odd number of sides (

_{0}*N*), such as triangles and pentagons, orient with one of their sides along

**n**, and bright lobes are visible near their other sides (Fig. 1, D and F). However, colloids with even

_{0}*N*, such as squares, align with one diagonal axis along

**n**, and bright regions appear symmetrically along all outer and inner edges (Fig. 1H). PM and FCPM textures indicate the presence of three mirror symmetry planes of the

_{0}**n**(

**r**) deformations, which intersect the particle’s center of mass: one coplanar with both

**n**and the unit vector

_{0}**v**normal to the platelet’s larger-area faces, a second parallel to the faces, and a third plane orthogonal to

**n**. Thus, the

_{0}**n**(

**r**) structure is quadrupolar, as schematically shown in Fig. 1K, resembling the symmetry of elastic quadrupoles formed by spherical particles (Fig. 1, B and C). Further, because the strongest FCPM signal corresponds to regions where

**n**(

**r**) is parallel to the linear FCPM polarization, the fluorescence images in Fig. 1, I and J, demonstrate that

**n**(

**r**) is indeed quadrupolar and consistent with surface anchoring of 5CB on SU-8 photoresist being degenerate planar (

*22*).

In the case of triangles and pentagons with odd *N*, however, the mirror symmetry plane that is coplanar with both **n _{0}** and

**v**is broken so that the

**n**(

**r**) structure is dipolar (Fig. 1, E and G), unlike that of other previously studied colloids promoting planar surface anchoring (

*15*,

*23*). Moreover, the elastic dipole moment

**p**is orthogonal to

**n**(Fig. 1E), in contrast to what is seen for colloids with vertical surface anchoring and

_{0}**p**parallel to

**n**(

_{0}*7*,

*11*,

*14*,

*24*), as shown in Fig. 1A. Examples of dipoles that align orthogonally to field lines are rare but can be formed by dipolar pairs of line defects in NLCs (

*19*) and vortex spin configurations in ferromagnets (

*8*). Similar to a sphere with planar anchoring shown in Fig. 1C, the shape-dictated dipolar structures of odd-

*N*platelets do not give rise to point or line defects in the NLC bulk. The dipolar

**n**(

**r**) symmetry of odd-

*N*platelets should be stable with respect to varying particle size and the strength of surface anchoring at their interfaces (fig. S1) (

*19*). This is different from the case of dipoles formed by spherical colloids accompanied by bulk point defects (Fig. 1A) observed only for strong anchoring and for particle sizes larger than ~1 μm, but not for smaller colloids for which a quadrupolar

**n**(

**r**) (Fig. 1B) is of lower energy (

*9*). For odd-

*N*polygonal platelets, although the magnitude of

**p**decreases with decreasing particle size or weakening anchoring strength (fig. S1), the alignment and dipolar symmetry should retain down to particle sizes of ~50 to 100 nm, at which the planar boundary conditions at the platelet surfaces are expected to partially relax (

*19*). For

*N*= 5, the magnitude of

**p**is smaller than that for

*N*= 3 (Fig. 1), and we expect that it decreases further as

*N*increases and ultimately vanishes in the limit

*N*→ ∞, corresponding to a circular disc with quadrupolar

**n**(

**r**) (

*9*).

The director field configurations surrounding regular polygons (Fig. 1) can be understood within an elegant theoretical framework that is built on an analogy with electrostatics (*25*–*27*). Within the one–elastic constant approximation (*19*), minimization of the NLC elastic energy *U*_{el} = (*K*/2)∫*d*^{3}**r**(**∇n**)^{2} [where *K* is an average Frank elastic constant, and (**∇n**)^{2} = (**∇** · **n**)^{2} + (**∇** × **n**)^{2}] leads to Laplace’s equation for **n**(**r**). Far from the particle, deviations from **n _{0}** are small, and

**n**(

**r**) can be expanded in a multipole series containing elastic monopole, dipole, and quadrupole terms that decay with distance

*r*as 1/

*r*, 1/

*r*

^{3}, and 1/

*r*

^{5}, respectively. The predicted absence of an elastic monopole, when no external torque is present (

*25*), is consistent with the observed dipolar symmetry of

**n**(

**r**), as well as the equilibrium orientation of polygons with odd

*N*. For example,

**n**(

**r**) would have no planes of mirror symmetry for a triangle or pentagon oriented so that all edges are neither parallel nor perpendicular to

**n**. Consequently, an elastic torque would be present, and the system would not be in mechanical equilibrium. There are two possible orientations for a triangle or a pentagon with ensuing

_{0}**n**(

**r**) having at least two planes of mirror symmetry: (i) one with a side along

**n**giving an elastic dipole with

_{0}**p**perpendicular to

**n**and (ii) another with a side oriented perpendicular to

_{0}**n**(in this case,

_{0}**p**would be parallel to

**n**). Evidently, the former has lower elastic energy because this is the equilibrium orientation observed in the experiments. The alignment of colloidal polygons with even

_{0}*N*, such as square-shaped particles, can be understood in a similar fashion. Orientations for which neither of the diagonals are parallel to

**n**would give rise to an elastic torque and are unstable. When the sample is heated into the isotropic phase, no preferred orientation is observed (fig. S2), confirming the elastic nature of the alignment of polygons in the nematic phase. Furthermore, observations during multiple heating and cooling cycles show that different sides (odd

_{0}*N*) and diagonals (even

*N*) can align along

**n**each time the sample is quenched into the nematic phase, demonstrating that there is no preference in the selection of these sides or diagonals.

_{0}Although the orientations of the polygonal edges are constrained relative to **n _{0}**, a platelet’s surface normal

**v**is free to rotate about

**n**in the bulk of a ≈60-μm-thick NLC cell, indicating that the elastic energy is independent of such rotations (

_{0}*28*). Confinement to cells of thickness comparable to the lateral size of platelets (≈10 μm) inhibits rotations about

**n**, and the platelike colloids orient parallel to the cell substrates to minimize the elastic energy due to the planar anchoring at the top and bottom surfaces of the colloids. To explore the directionality and strength of anisotropic elastic-pair interactions, we control the initial positions and orientations of particles with the use of optical tweezers (

_{0}*11*,

*12*,

*15*) and then track their motion using video microscopy after release from the laser traps. When the center-to-center separation vector

**R**for two triangles is along

**n**, elastic repulsion occurs for parallel dipoles (Fig. 2A), whereas attraction takes place for antiparallel dipoles (Fig. 2B). The opposite is true for situations when

_{0}**R**is perpendicular to

**n**; antiparallel dipoles repel (Fig. 2C) and parallel dipoles attract (Fig. 2D). Two types of self-assembled chainlike aggregates are observed: (i) antiparallel dipole chains in which the triangles aggregate along

_{0}**n**(Fig. 2E) and (ii) chains perpendicular to

_{0}**n**consisting of parallel dipoles (Fig. 2F). Chaining of triangular colloids perpendicular to

_{0}**n**is a consequence of the dipoles’ alignment orthogonal to

_{0}**n**. The dipolar nature of the elastic interaction is further evidenced by the time dependence of particle separation

_{0}*R*(

*t*) for a pair of triangles aggregating along

**n**(Fig. 2G). Because the system is highly overdamped (Reynolds number << 1), inertial forces are negligible and the elastic force

_{0}*F*

_{el}is balanced by a viscous Stokes drag

*F*

_{drag}= –ζ

*dR*/

*dt*, where ζ is a drag coefficient, and

*dR*/

*dt*is the time-derivative of the particle separation

*R*(

*t*). For an elastic dipolar force

*F*

_{el}= –κ

_{d}/

*R*

^{4}(where κ

_{d}is a constant that depends on

*K*and the geometry and size of the particle), integration of the equation of motion

*F*

_{el}+

*F*

_{drag}≈ 0 yields

*R*(

*t*) = (

*R*

_{0}

^{5}– 5α

_{d}

*t*)

^{1/5}, where α

_{d}= κ

_{d}/ζ, and

*R*

_{0}is the initial separation at time

*t*= 0 when particles are released from the traps.

*R*(

*t*) fits the data well with one adjustable parameter α

_{d}= 63.1 ± 0.5 μm

^{5}/s (red curve in Fig. 2G). Using an estimate of the drag coefficient ζ ~ (2 to 4) × 10

^{−6}kg/s (

*29*) and the maximum relative velocity

*dR*/

*dt*≈ 1 μm/s determined from the data in Fig. 2G, one obtains a maximum attractive elastic force of 2 to 4 pN near contact at

*R*≈ 2.6 μm. This force and the corresponding binding energy ≈5 × 10

^{−18}J (≈1200

*k*

_{B}

*T*, where

*k*

_{B}is Boltzmann’s constant) for a pair of triangles are comparable to those measured for spherical colloids of similar size (

*11*,

*12*,

*15*).

Square-shaped platelets aggregate at angles intermediate between 0 and 90° relative to **n _{0}**, suggesting a nondipolar symmetry of elastic interactions. A time series of video frames in Fig. 3A shows two interacting squares after release from the optical traps used to position them initially with

**R**parallel to

**n**. The squares repel while gradually moving sideways (frames 1 and 2), then attract along ≈45° to

_{0}**n**(frames 3 and 4), and ultimately aggregate with adjacent sides touching to form a chain that equilibrates at ≈40° to

_{0}**n**. This equilibration angle decreases with the addition of more particles into the linear chain, consistent with the planar anchoring at the NLC-colloid surfaces. Kinked chains as well as more symmetric structures are also possible. For example, a square and a two-particle chain can attract (Fig. 3, C and D) and form a structure in which the individual square orientations match those of isolated ones.

_{0}To elucidate the angular dependence of elastic interactions between square particles, two are positioned at a fixed center-to-center separation of *R* = 12.3 μm and various angles between **R** and **n _{0}**: θ = 0, ±π/8, ±π/4, and 3π/8, as shown in the inset of Fig. 3E. For each θ, the particles are released from the optical traps, tracked with video microscopy for 13 s while the traps are off, and then moved back to the same initial locations. Because the elastic forces at

*R*= 12.3 μm are weak (~10

^{−2}pN), we time-average an ensemble of 10 particle trajectories for each θ to mitigate the effects of Brownian motion. The average relative trajectories at various θ are shown in Fig. 3E. Elastic repulsion occurs for pair orientations parallel (θ = 0) and perpendicular (θ = π/2) to

**n**, whereas strong attraction along

_{0}**R**takes place at θ = ±π/4. At θ = ±π/8, the elastic force drives the particles sideways toward θ = ±π/4 while gradually becoming attractive. The angular dependence of the expected force between quadrupoles at a large fixed separation (shown by black arrows in Fig. 3E) (

*9*,

*15*) exhibits marked correlation with the measured displacements, confirming the quadrupolar nature of elastic forces between colloidal squares. These results imply that the presence of the hole in a colloidal square and, more generally, other modifications to the platelet’s topology are inconsequential to the anisotropy of interactions, as long as the quadrupolar

**n**(

**r**) symmetry is preserved.

Quadrupolar forces are expected to decay with distance as ~*R*^{−6} (*9*, *15*). To test if square platelets interact in this manner, we have measured the relative positions of two colloidal squares along θ = π/4 from initial separations *R*_{0} = 14.3 and 12.9 μm (Fig. 3F). From a balance of a quadrupolar elastic force *F*_{elq} = –κ_{q}/*R*^{6} with a viscous drag, one obtains the time-dependent particle separation *R*(*t*) = (*R*_{0}^{7} – 7α_{q}*t*)^{1/7}, where α_{q} = κ_{q}/ζ . The two sets of data in Fig. 3F can be fit with *R*(*t*) using only one adjustable parameter α_{q} = (1.6 ± 0.1) × 10^{5} μm^{7}/s. Taking the average elastic constant *K* ≈ 7 pN (*30*), an effective viscosity η ≈ 0.075 Pa⋅s for 5CB, as well as the side length *L* = 4.5 μm of the platelet, dimensional analysis gives an estimate of α_{q} ~ *KL*^{5}/η = 1.7 × 10^{5} μm^{7}/s, in reasonable agreement with these experiments. Using a drag coefficient ζ ≈ 1.9 × 10^{−6} kg/s of a square platelet in 5CB [determined by probing its diffusive motion with video microscopy (fig. S3)] and α_{q} = 1.6 × 10^{5} μm^{7}/s, we calculate a maximum attractive elastic force of ≈20 pN near contact at *R* = 4.5 μm and corresponding binding energy ≈3 × 10^{−17} J (≈7000*k*_{B}*T*).

In conclusion, elastic colloidal interactions in NLCs are sensitive to the colloids’ shapes. Equilibrium director field configurations around equilateral polygonal colloids exhibit dipolar symmetry if they have odd *N* (i.e., triangles or pentagons) and quadrupolar symmetry if *N* is even, giving rise to dipolar and quadrupolar elastic colloidal interactions, respectively. Elastic dipole moments of polygonal platelets orient perpendicular to the far-field director **n _{0}**. Dipole-dipole forces drive their assembly into chains perpendicular to

**n**if their dipoles are parallel and chains along

_{0}**n**if their dipoles are antiparallel. Although the symmetry of these highly directional elastic forces should not change over a broad range of particle sizes (~50 nm to tens of microns), the strength can vary substantially. One can envision the design of such interactions for the assembly of colloidal architectures ranging from anisotropic aggregates to new types of colloidal crystals and optical metamaterials with well-defined alignment relative to the far-field director.

_{0}## Supporting Online Material

www.sciencemag.org/cgi/content/full/326/5956/1083/DC1

Materials and Methods

Figs. S1 to S3

References

## References and Notes

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Materials and methods are available as supporting material on
*Science*Online. - ↵
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Because of a density mismatch of ~0.2 g/cm
^{3}between SU-8 and 5CB, particles tend to sediment toward the lower half of the cell and come to rest at a height ≈5 μm at which the repulsive particle-substrate interaction due to the**n**(**r**) deformations balances gravity (*23*). - ↵
The drag coefficient of a triangular platelet can be estimated as that of a thin disk with the radius
*a*circumscribing the edges of the triangle: ζ ≈ 32η*a*/3. Using a representative value of shear viscosity η ≈ 0.075 Pa·s for 5CB (*9*), one finds ζ ≈ 2 × 10^{−6}kg/s. Although this analysis is only approximate, it gives reasonable estimates for platelet colloids, as verified experimentally (*19*). - ↵
- We thank K. Zhao for assistance with the fabrication of colloids, and we acknowledge support from the Institute for Complex and Adaptive Matter and from NSF grants DMR 0645461, DMR 0847782, CHE 0450022, and DMR 0820579.