We address the question: How does capillarity propel microspheres along curvature gradients? For a particle on a fluid interface, there are two conditions that can apply at the three phase contact line: Either the contact line adopts an equilibrium contact angle, or it can be pinned by kinetic trapping, e.g. at chemical heterogeneities, asperities or other pinning sites on the particle surface. We formulate the curvature capillary energy for both scenarios for particles smaller than the capillary length and far from any pinning boundaries. The scale and range of the distortion made by the particle are set by the particle radius; we use singular perturbation methods to find the distortions and to rigorously evaluate the associated capillary energies. For particles with equilibrium contact angles, contrary to the literature, we find that the capillary energy is negligible, with the first contribution bounded to fourth order in the product of the particle radius and the deviatoric curvature. For pinned contact lines, we find curvature capillary energies that are finite, with a functional form investigated previously by us for disks and microcylinders on curved interfaces. In experiments, we show microsphere migrate along deterministic trajectories toward regions of maximum deviatoric curvature with curvature capillary energies ranging from 6 × 10 3 − 5 × 10 4 kBT . These data agree with the curvature capillary energy for the case of pinned contact lines. The underlying physics of this migration is a coupling of the interface deviatoric curvature with the quadrupolar mode of nanometric disturbances in the interface owing to the particle's contact line undulations. This work is an example of the major implications of nanometric roughness and contact line pinning for colloidal dynamics.

In directed assembly, small building blocks are assembled into an organized structures under the influence of guiding fields. Capillary interactions provide a versatile route for structure formation. Colloids adsorbed on fluid interfaces distort the interface, which creates an associated energy field. When neighboring distortions overlap, colloids interact to minimize interfacial area. Contact line pinning, particle shape and surface chemistry play important roles in structure formation. Interface curvature acts like an external field; particles migrate and assemble in patterns dictated by curvature gradients. We review basic analysis and recent findings in this rapidly evolving literature. Understanding the roles of assembly is essential for tuning the mechanical, physical, and optical properties of the structure.

The capillary energy landscape for particles on curved fluid interfaces is strongly influenced by the particle wetting conditions. Contact line pinning has now been widely reported for colloidal particles, but its implications in capillary interactions have not been addressed. Here, we present experiment and analysis for disks with pinned contact lines on curved fluid interfaces. In experiment, we study microdisk migration on a host interface with zero mean curvature; the microdisks have contact lines pinned at their sharp edges and are sufficiently small that gravitational effects are negligible. The disks migrate away from planar regions toward regions of steep curvature with capillary energies inferred from the dissipation along particle trajectories which are linear in the deviatoric curvature. We derive the curvature capillary energy for an interface with arbitrary curvature, and discuss each contribution to the expression. By adsorbing to a curved interface, a particle eliminates a patch of fluid interface and perturbs the surrounding interface shape. Analysis predicts that perfectly smooth, circular disks do not migrate, and that nanometric deviations from a planar circular, contact line, like those around a weakly roughened planar disk, will drive migration with linear dependence on deviatoric curvature, in agreement with experiment.

Confined smectic A liquid crystals (SmA LCs) form topological defects called focal conic domains (FCDs) [1] that focus light as gradient-index lenses [2][3][4][5]. Here, we exploit surface curvature to self-assemble FCDs in a single step into a hierarchical structure [6,7] (coined "flower pattern") molded by the fluid interface that is pinned at the top of a micropillar. The structure resembles the compound eyes of some invertebrates, which consist of hundreds of microlenses on a curved interface, able to focus and construct images in three dimensions (3D) [8]. Here we demonstrate that these flowers are indeed "compound eyes" with important features which have not been demonstrated previously in the literature. The eccentric FCDs gradually change in size with radial distance from the edge of the micropillar, resulting in a variable microlens focal length that ranges from a few microns to a few tens of microns within a single "flower". We show that the microlenses can construct a composite 3D image from different depth of field (DOF). Moreover, the smectic "'compound eye" can be reconfigured by heating and cooling at the LC phase transition temperature; its field of view (FOV) can be manipulated by tuning the curvature of the LC interface, and the lenses are sensitive to light polarization.Insects' eyes are comprised of hundreds of microlenses (ommatidia) arranged on a curved surface [8]. Despite having modest resolution in comparison to single aperture lenses (like the human eyes), the compound eye offers attractive optical properties, including exceptionally wide FOV, fast motion detection, and polarization sensitivity. Artifical compound eyes have been created with angular sensitivity [9] and with a hemispheric FOV and near infinite DOF [10]. Typically, multiple top-down fabrication steps are required, including photolithography, replica molding, or complex micromachining processes [9][10][11]. For practical applications with wide FOV, lenses which self-align are highly desirable. * These authors contributed equally to the work † kstebe@seas.

The assembly of colloids in nematic liquid crystals via topological defects has been extensively studied for spherical particles, and investigations of other colloid shapes have revealed a wide array of new assembly behaviors. We show, using Landau-de Gennes numerical modeling, that nematic defect configurations and colloidal assembly can be strongly influenced by fine details of colloid shape, in particular the presence of sharp edges. For cylinder, microbullet, and cube colloid geometries, we obtain the particles' equilibrium alignment directions and effective pair interaction potentials as a function of simple shape parameters. We find that defects pin at sharp edges, and that the colloid consequently orients at an oblique angle relative to the far-field nematic director that depends on the colloid's shape. This shape-dependent alignment, which we confirm in experimental measurements, raises the possibility of selecting self-assembly outcomes for colloids in liquid crystals by tuning particle geometry.

Rod-like colloids distort fluid interfaces and interact by capillarity. We explore this interaction at the free surface of aligned nematic liquid crystal films. Naive comparison of capillary and elastic energies suggests that particle assembly would be determined solely by surface tension. Here, we demonstrate that, under certain circumstances, the capillary and elastic effects are complementary and each plays an important role. Particles assemble end-to-end, as dictated by capillarity, and align along the easy axis of the director field, as dictated by elasticity. On curved fluid interfaces, however, curvature capillary energies can overcome the elastic orientations and drive particle migration along curvature gradients. Domains of dominant interaction and their transition are investigated.curvature | anisotropic particles | interfacial assembly | topology | 2D superstructures

Focal conic domains (FCDs) form in smectic-A liquid crystal films with hybrid anchoring conditions with eccentricity and size distribution that depend strongly on interface curvature. Assemblies of FCDs can be exploited in settings ranging from optics to material assembly. Here, using micropost arrays with different shapes and arrangement, we assemble arrays of smectic flower patterns, revealing their internal structure as well as defect size, location, and distribution as a function of interface curvature, by imposing positive, negative, or zero Gaussian curvature at the free surface. We characterize these structures, relating free surface topography, substrate anchoring strength, and FCD distribution. Whereas the largest FCDs are located in the thickest regions of the films, the distribution of sizes is not trivially related to height, due to Apollonian tiling. Finally, we mold FCDs around microposts of complex shape and find that FCD arrangements are perturbed near the posts, but are qualitatively similar far from the posts where the details of the confining walls and associated curvature fields decay. This ability to mold FCD defects into a variety of hierarchical assemblies by manipulating the interface curvature paves the way to create new optical devices, such as compound eyes, via a directed assembly scheme.

We have studied microparticle migration on curved fluid interfaces in experiment and derived an expression for the associated capillary energy E for two cases, i.e., pinned contact lines [1] and equilibrium contact lines [2], which differ from expressions derived by others in the literature. In this problem, a particle of radius a makes a disturbance in a large domain characterized by principal radii of curvature R1 and R2. Since a is smaller than all associated geometric and physico-chemical length scales, analysis calls for a singular perturbation approach. We recapitulate these concepts, identify conceptual errors in the Comments about our work, and provide evidence from experiment and simulation that supports our view.