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In directed assembly, small building clocks 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., Comment: 14 pages, 9 figures

Directed assembly of colloids is an exciting field in materials science to form structures with new symmetries and responses. Fluid interfaces have been widely exploited to make densely packed ordered structures. We have been studying how interface curvature can be used in new ways to guide structure formation. On a fluid interface, the area of the deformation field around adsorbed microparticles depends on interface curvature; particles move to minimize the excess area of the distortions that they make in the interface. For particles that are sufficiently small, this area decreases as particles move along principle axes to sites of high deviatoric curvature. We have studied this migration for microparticles on a curved host interface with zero mean curvature created by pinning an oil-water interface around a micropost. Here, on a similar interface, we demonstrate capillary curvature repulsion, that is, we identify conditions in which microparticles migrate away from high curvature sites. Using theory and experiment, we discuss the origin of these interactions and their relationship to the particle's undulated contact line. We discuss the implications of this new type of interaction in various contexts from materials science to microrobotics., Comment: 9 pages, 7 figures

A particle placed in soft matter distorts its host and creates an energy landscape. This can occur, for example, for particles in liquid crystals, for particles on lipid bilayers or for particles trapped at fluid interfaces. Such energies can be used to direct particles to assemble with remarkable degrees of control over orientation and structure. These notes explore that concept for capillary interactions, beginning with particle trapping at fluid interfaces, addressing pair interactions on planar interfaces and culminating with curvature capillary migration. Particular care is given to the solution of the associated boundary value problems to determine the energies of interaction. Experimental exploration of these interactions on planar and curved interfaces is described. Theory and experiment are compared. These interactions provide a rich toolkit for directed assembly of materials, and, owing to their close analogy to related systems, pave the way to new explorations in materials science., Comment: 37 pages, 28 figures

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 and equilibrium contact lines, 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 $R_1$ and $R_2$. 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., Comment: 6 pages, 2 figures

Confined smectic A liquid crystals (SmA LCs) form topological defects called focal conic domains (FCDs) that focus light as gradient-index lenses. Here, we exploit surface curvature to self-assemble FCDs in a single step into a hierarchical structure (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. 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. Moreover, the smectic "compound eye" can be reconfigured by heating and cooling at the LC phase transition temperature; its field of view can be manipulated by tuning the curvature of the LC interface, and the lenses are sensitive to light polarization., Comment: 17 pages, 9 figures (4 supplementary), 3 supplementary movies. This is the pre-peer reviewed version of the article which has been published in its final form in Advanced Optical Materials and is available at http://onlinelibrary.wiley.com/doi/10.1002/adom.201500153/abstract

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 $6 \times10^3 - 5 \times 10^4~k_BT$. 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., Comment: 9 pages, 8 figures

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.

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., Comment: 10 pages, 3 figures