Abstract:
The phase diagram of a fluid confined between a planar and a conical walls modelling the atomic force microscope geometry displays transition between two phases, one with a liquid bridge connecting the two walls of the microscope, and the other without bridge. The structure of the corresponding coexistence line is determined and its dependence on the value of the line tension coefficient is discussed.

Abstract:
We discuss the phase diagram of fluid confined in AFM-like geometry. It combines the properties of capillary condensation and complete filling of a wedge.

Abstract:
In the reduced one-dimensional description of the adsorption on the wedge-shaped substrate the mid-point interface height serves as the order parameter. We point at the ambiguity which appears in the transfer-matrix approach to this problem. We also propose how to avoid this problem by introducing the appropriate order parameter.

Abstract:
We propose a method to describe the short-distance behavior of an interface fluctuating in the presence of the wedge-shaped substrate near the critical filling transition. Two different length scales determined by the average height of the interface at the wedge center can be identified. On one length scale the one-dimensional approximation of Parry et al. \cite{Parry} which allows to find the interfacial critical exponents is extracted from the full description. On the other scale the short-distance fluctuations are analyzed by the mean-field theory.

Abstract:
Mean field analysis of the effective interfacial Hamiltonian shows that with increasing temperature the adsorption on a periodically corrugated substrate can proceed in two steps: first, there is the filling transition in which the depressions of the substrate become partially or completely filled; then there is the wetting transition at which the substrate as a whole becomes covered with a macroscopically thick wetting layer. The actual order and location of both transitions are related to the wetting properties of the corresponding planar substrate and to the form of corrugation. Certain morphological properties of the liquid-vapor interface in the case of a saw-like corrugated substrate are discussed analytically

Abstract:
We investigate liquid layers adsorbed at spherical and corrugated cylindrical substrates. The effective Hamiltonians for the liquid-gas interfaces fluctuating in the presence of such curved substrates are derived via the mean-field density functional theory. Their structure is compared with the Helfrich Hamiltonian which is parametrized by the bending and Gaussian rigidity coefficients. For long-ranged interparticle interactions of van der Waals type these coefficients turn out to be non-universal functions of interfacial curvatures; their form varies from one interface to another. We discuss implications of the structure of these functions on the effective Hamiltonian.

Abstract:
Within the Landau-Ginzburg-Wilson approach we derive the effective Hamiltonian governing fluctuations of an interface between two phases, one of them adsorbed on a disclike substrate. For large disc radii and for temperatures close to the wetting temperature of the corresponding planar substrate the expressions for the hight-hight correlation function and the accompanying correlation angle are derived. Their dependence on the disc radius and the reduced temperature is discussed and the relevant scaling regimes are identified.

Abstract:
We present a study of the Casimir effect in an imperfect (mean-field) Bose gas contained between two infinite parallel plane walls. The derivation of the Casimir force follows from the calculation of the excess grand canonical free energy density under periodic, Dirichlet, and Neumann boundary conditions with the use of the steepest descent method. In the one-phase region the force decays exponentially fast when distance $D$ between the walls tends to infinity. When Bose-Einstein condensation point is approached the decay length in the exponential law diverges with critical exponent $\nu_{IMP}=1$, which differs from the perfect gas case where $\nu_{P}=1/2$. In the two-phase region the Casimir force is long-range, and decays following the power law $D^{-3}$, with the same amplitude as in the perfect gas.

Abstract:
We propose a one-dimensional model of a string decorated with adhesion molecules (stickers) to mimic multicomponent membranes in restricted geometries. The string is bounded by two parallel walls and it interacts with one of them by short range attractive forces while the stickers are attracted by the other wall. The exact solution of the model in the case of infinite wall separation predicts both continuous and discontinuous transitions between phases characterised by low and high concentration of stickers on the string. Our model exhibits also coexistence of these two phases, similarly to models of multicomponent membranes.

Abstract:
Within the effective interfacial Hamiltonian approach we evaluate the excess line free energy associated with cylinder-shaped droplets sessile on a stripe-like chemical inhomogeneity of a planar substrate. In the case of short-range intermolecular forces the droplet morphology and the corresponding expression for the line tension - which includes the inhomogeneity finite width effects - are derived and discussed as functions of temperature and increasing width. The width-dependent contributions to the line tension change their structure at the stripe wetting temperature T_W1: for TT_W1 the decay is algebraic. In addition, a geometric construction of the corresponding contact angle is carried out and its implications are discussed.