Abstract:
This mini-review discusses the recent contribution of theoretical and computational physics as well as experimental efforts to the understanding of the behavior of colloidal particles in confined geometries and at liquid crystalline interfaces. Theoretical approaches used to study trapping, long- and short-range interactions, and assembly of solid particles and liquid inclusions are outlined. As an example, an interaction of a spherical colloidal particle with a nematic-isotropic interface and a pair interaction potential between two colloids at this interface are obtained by minimizing the Landau-de Gennes free energy functional using the finite-element method with adaptive meshes.

Abstract:
For one-component volatile fluids governed by dispersion forces an effective interface Hamiltonian, derived from a microscopic density functional theory, is used to study complete wetting of geometrically structured substrates. Also the long range of substrate potentials is explicitly taken into account. Four types of geometrical patterns are considered: (i) one-dimensional periodic arrays of rectangular or parabolic grooves and (ii) two-dimensional lattices of cylindrical or parabolic pits. We present numerical evidence that at the centers of the cavity regions the thicknesses of the adsorbed films obey precisely the same geometrical covariance relation, which has been recently reported for complete cone and wedge filling. However, this covariance does not hold for the laterally averaged wetting film thicknesses. For sufficiently deep cavities with vertical walls and close to liquid-gas phase coexistence in the bulk, the film thicknesses exhibit an effective planar scaling regime, which as function of undersaturation is characterized by a power law with the common critical exponent -1/3 as for a flat substrate, but with the amplitude depending on the geometrical features.

Abstract:
Effect of hydrophilicity of the confining walls on lamellar phases in oil-water-surfactant mixtures is studied in a slit geometry. In contrast to strongly hydrophilic or hydrophobic walls, which induce parallel orientation of lamellae, the lamellae can be oriented perpendicularly to the neutral walls when the material properties and the termodynamical state of the sample are suitably chosen. When the elastic energy associated with compression or decompression of the lamellae parallel to very weakly hydrophilic walls is sufficiently large, then changes of the film thickness lead to a switch from the parallel to the perpendicular orientation of the lamellae. Our general arguments are confirmed by explicit mean-field calculations in a lattice vector model.

Abstract:
Complete wetting of geometrically structured substrates by one-component fluids with long-ranged interactions is studied. We consider periodic arrays of rectangular or parabolic grooves and lattices of cylindrical or parabolic pits. We show that the midpoint interfacial heights within grooves and pits are related in the same way as for complete wedge and cone filling. For sufficiently deep cavities with vertical walls and small undersaturation, an effective planar scaling regime emerges. The scaling exponent is -1/3 in all cases studied, and only the amplitudes depend on the geometrical features. We find quantitative agreement with recent experimental data for such systems.

Abstract:
Catalytically active particles suspended in a liquid can move due to self-phoresis by generating solute gradients via chemical reactions of the solvent occurring at parts of their surface. Such particles can be used as carriers at the micro-scale. As a simple model for a carrier-cargo system we consider a catalytically active particle connected by a thin rigid rod to a catalytically inert cargo particle. We show that the velocity of the composite strongly depends on the relative orientation of the carrier-cargo link. Accordingly, there is an optimal configuration for the linkage. The subtlety of such carriers is underscored by the observation that a spherical particle completely covered by catalyst, which is motionless when isolated, acts as a carrier once attached to a cargo.

Abstract:
Systems in which particles can self-assemble into mono- or bilayers can form variety of stable and metastable structures on a nanometer length scale. For this reason confinement has a particularly strong effect on such systems. We discuss in some detail effects of confinement on lamellar and cubic phases with double-diamond structure. Structural deformations in slit geometry are described for large and small unit cells of the structure (in units of the thickness of the monolayer) and for various strengths of interactions with the confining surfaces. We show how the structural changes of the confined fluid are reflected in the measurable solvation force between the confining walls.

Abstract:
The influence of finite system size on the free energy of a spherical particle floating at the surface of a sessile droplet is studied both analytically and numerically. In the special case that the contact angle at the substrate equals $\pi/2$ a capillary analogue of the method of images is applied in order to calculate small deformations of the droplet shape if an external force is applied to the particle. The type of boundary conditions for the droplet shape at the substrate determines the sign of the capillary monopole associated with the image particle. Therefore, the free energy of the particle, which is proportional to the interaction energy of the original particle with its image, can be of either sign, too. The analytic solutions, given by the Green's function of the capillary equation, are constructed such that the condition of the forces acting on the droplet being balanced and of the volume constraint are fulfilled. Besides the known phenomena of attraction of a particle to a free contact line and repulsion from a pinned one, we observe a local free energy minimum for the particle being located at the drop apex or at an intermediate angle, respectively. This peculiarity can be traced back to a non-monotonic behavior of the Green's function, which reflects the interplay between the deformations of the droplet shape and the volume constraint.

Abstract:
The effective capillary interaction potentials for small colloidal particles trapped at the surface of liquid droplets are calculated analytically. Pair potentials between capillary monopoles and dipoles, corresponding to particles floating on a droplet with a fixed center of mass and subjected to external forces and torques, respectively, exhibit a repulsion at large angular separations and an attraction at smaller separations, with the latter resembling the typical behavior for flat interfaces. This change of character is not observed for quadrupoles, corresponding to free particles on a mechanically isolated droplet. The analytical results for quadrupoles are compared with the numerical minimization of the surface free energy of the droplet in the presence of ellipsoidal particles.

Abstract:
We study the free energy landscapes of a pair of submicron spherical particles floating at the surface of a sessile droplet. The particles are subjected to radial external forces resulting in a deformation of the droplet shape relative to the reference shape of a spherical cap. This deformation leads to tangential forces on the particles. For small deformations and for the contact angle $\theta_0$ at the substrate being equal to $\pi/2$, the corresponding linearized Young-Laplace equation is solved analytically. The solution is constructed by employing the method of images from electrostatics, where each of the particles plays the role of a capillary monopole and the substrate is replaced by a virtual drop with image charges and by imposing the conditions of fixed droplet volume and vanishing total force on the droplet. The substrate boundary conditions determine the signs of the image capillary charges and therefore also the strength of the tangential forces on the particles. In the cases of an arbitrary contact angle $\theta_0$ these forces are calculated numerically by employing a finite element method to find the equilibrium shape of the droplet for those configurations in which the particles are close to the local free energy minima.

Abstract:
The Landau-de Gennes free energy is used to study theoretically the interaction of parallel cylindrical colloidal particles trapped at a nematic-isotropic interface. We find that the effective interaction potential is non-monotonic. The corresponding force-distance curves exhibit jumps and hysteresis upon approach/separation due to the creation/annihilation of topological defects. Minimization results suggest a simple empirical pair potential for the effective colloid-colloid interaction at the interface. We propose that the interface-mediated interaction can play an important role in self-organization and clustering of colloidal particles at such interfaces.