The effect of gravity on a two dimensional fluid, or Ising magnet, confined between opposing walls is analyzed by density matrix renormalization. Gravity restores two phase coexistence up to the bulk critical point, in agreement with mean field calculations. A detailed finite size scaling analysis of the critical point shift is performed. Density matrix renormalization results are very accurate and the technique is promising and best suited to study equilibrium properties of two dimensional classical systems in contact with walls or with free surfaces.

The phase boundaries for corner wetting (filling) in square and diagonal lattice Ising models are exactly determined and show a universal shift relative to wetting near the bulk criticality. More generally, scaling theory predicts that the filling phase boundary shift for wedges and cones is determined by a universal scaling function $R_d\left(\psi \right)$ depending only on the opening angle $2\psi$. $R_d\left(\psi \right)$ is determined exactly in $d=2\mathrm{}$ and approximately in higher dimensions using nonclassical local functional and mean-field theory. Detailed numerical transfer matrix studies of the magnetization profile in finite-size Ising squares support the conjectured connection between filling and the strong-fluctuation regime of wetting.

We consider an Ising model confined in an $L\times \infty$ geometry with identical surface fields at the boundaries. According to the Kelvin equation the bulk coexistence field scales as $1/L$ for large $L;$ thermodynamics and scaling arguments predict higher-order corrections of the type ${1/L}^2$ and ${1/L}^{5/3}$ at partial and complete wetting, respectively. Our numerical results, obtained by density-matrix renormalization techniques for systems of widths up to $L=144,$ are in agreement with a ${1/L}^2$ correction in the partially wet regime. However, at complete wetting we find a large range of surface fields and temperatures with a correction to scaling of type ${1/L}^{4/3}.$ We show that this term is generated by a thin wetting layer whose free energy is dominated by the contacts with the wall. For $L$ sufficiently large we expect a crossover to a ${1/L}^{5/3}$ correction as predicted by the theory for a thick wetting layer. This crossover is indeed found in a solid-on-solid model, which provides a simplified description of the wetting layer and allows the study of much larger systems than those available in a density-matrix renormalization calculation.

The Rubinstein-Duke model for polymer reptation is analyzed by means of density matrix renormalization techniques. It is found that the crossover in the scaling behavior of polymer renewal time (or viscosity) arises from the competing effect of the contribution due to tube length fluctuations and higher-order corrections, which are of opposite sign. Experiments which ought to emphasize both contributions are suggested. The exponent describing the subleading scaling behavior of the diffusion coefficient is also investigated.

The solvation force of a simple fluid confined between identical planar walls is studied in two model systems with short ranged fluid-fluid interactions and long ranged wall-fluid potentials decaying as $-Az^{-p}, z\to \infty$, for various values of $p$. Results for the Ising spins system are obtained in two dimensions at vanishing bulk magnetic field $h=0$ by means of the density-matrix renormalization-group method; results for the truncated Lennard-Jones (LJ) fluid are obtained within the nonlocal density functional theory. At low temperatures the solvation force $f_{solv}$ for the Ising film is repulsive and decays for large wall separations $L$ in the same fashion as the boundary field $f_{solv}\sim L^{-p}$, whereas for temperatures larger than the bulk critical temperature $f_{solv}$ is attractive and the asymptotic decay is $f_{solv}\sim L^{-(p+1)}$. For the LJ fluid system $f_{solv}$ is always repulsive away from the critical region and decays for large $L$ with the the same power law as the wall-fluid potential. We discuss the influence of the critical Casimir effect and of capillary condensation on the behaviour of the solvation force.Comment: 48 pages, 12 figure

We argue that in a fluid, or magnet, confined by adsorbing walls which favor liquid, or the $(+)$ phase, the solvation (Casimir) force in the vicinity of the critical point is strongly influenced by capillary condensation which occurs below the bulk critical temperature $T_c$. At $T$ slightly below and above $T_c$, a small bulk field $h<\mathrm{}0\mathrm{}$, which favors gas, or the $(-)$ phase, leads to residual condensation and a solvation force which is much more attractive (at the same large wall separation) than that found exactly at the critical point. Our predictions are supported by results obtained from density-matrix renormalization-group calculations in a two-dimensional Ising strip subject to identical surface fields.