We present a new time-dependent density functional approach to study the relaxational dynamics of an assembly of interacting particles subject to thermal noise. Starting from the Langevin stochastic equations of motion for the velocities of the particles we are able by means of an approximated closure to derive a self-consistent deterministic equation for the temporal evolution of the average particle density. The closure is equivalent to assuming that the equal-time two-point correlation function out of equilibrium has the same properties as its equilibrium version. The changes in time of the density depend on the functional derivatives of the grand canonical free energy functional F[ρ] of the system. In particular the static solutions of the equation for the density correspond to the exact equilibrium profiles provided one is able to determine the exact form of F[ρ]. In order to assess the validity of our approach we performed a comparison between the Langevin dynamics and the dynamic density functional method for a one-dimensional hard-rod system in three relevant cases and found remarkable agreement, with some interesting exceptions, which are discussed and explained. In addition, we consider the case where one is forced to use an approximate form of F[ρ]. Finally we compare the present method with the stochastic equation for the density proposed by other authors [Kawasaki, Kirkpatrick etc.] and discuss the role of the thermal fluctuations.

Phase transitions at fluid interfaces and in fluids confined in pores have been investigated by means of a density functional approach that treats attractive forces between fluid molecules in mean-field approximation and models repulsive forces by hard-spheres. Two types of approximation were employed for the hard-sphere free energy functional: (a) the well-known local density approximation (LDA) that omits short-ranged correlations and (b) a non-local smoothed density approximation (SDA) that includes such correlations and therefore accounts for the oscillations of the density profile near walls. Three different kinds of phase transition were considered: (i) wetting transition. The transition from partial to complete wetting at a single adsorbing wall is shifted to lower temperatures and tends to become first-order when the more-realistic SDA is employed. Comparison of the results suggests that the LDA overestimates the contact angle γ in a partial wetting situation. (ii) capillary evaporation of a fluid confined between two parallel hard walls. This transition, from dense ‘liquid’ to dilute ‘gas’, occurs in a supersaturated fluid (p > p _sat). The lines of capillary coexistence calculated in the LDA and SDA are rather close, suggesting that non-local effects are not especially important in this case. (iii) capillary condensation of fluids confined between two adsorbing walls or in a single cylindrical pore. For a partial wetting situation the condensation pressures p(<p _sat) obtained from the SDA are in remarkably good agreement with the macroscopic Laplace (or Kelvin) prediction for wall separations H or pore radii R _c ≳ 5σ; σ is a molecular diameter. While, because of different packing, the density profiles of the fluid differ considerably between slits and cylinders this has little effect on the coexistence line until H or R _c ∼ σ. In contrast to the LDA the SDA describes two-dimensional-like liquid-gas coexistence for very narrow pores (H < σ) and temperatures below the two-dimensional critical temperature and this has ramifications for the existence of capillary critical points.

By means of a density functional approach the phase equilibria of a simple fluid confined by two adsorbing walls have been investigated as a function of wall separation H and chemical potential μ for temperature T corresponding to both partial and complete wetting situations. For large values of H and small undersaturations Δμ ≡ μ_sat−μ, we recover the macroscopic formulas for the undersaturation at which a first‐ order phase transition (capillary condensation) from dilute ‘‘gas’’ to a dense ‘‘liquid’’ occurs in a single, infinitely long slit. For smaller H we compute the lines of coexistence between gas and liquid in the (Δμ, 1/H) plane at fixed values of T. The adsorption Γ(Δμ), at fixed T and H, is characterized by a loop. At the first order transition Γ jumps discontinuously by a finite amount; however metastable states exist and these could give rise to hysteresis of the adsorption isotherms obtained for the single slit. The loop disappears at a capillary critical point (Δμ_c, 1/H_c) at each T. For H<H_c, or Δμ>Δμ_c, condensation can no longer occur and no metastable states are present. The location of the critical points is described and for a complete wetting situation we find that these lie outside the bulk two phase region. Our theory provides a simple explanation of phase transitions observed in earlier computer simulations and mean‐field lattice gas calculations for confined fluids and suggests that measurements of the forces between plates, either by simulation or in real fluids, should provide rather direct information about capillary condensation and, possibly, capillary critical points. The relevance of our results for adsorption experiments on mesoporous solids is discussed briefly.

The nature of adsorption of simple fluids confined in model pores is investigated by means of a density functional approach. For temperatures T corresponding to a partial wetting situation a first-order phase transition (capillary condensation) from dilute ‘gas’ to dense ‘liquid’ occurs at relative pressures p/p_sat close to those predicted by the macroscopic Kelvin equation, even for radii R_c or wall separations H as small as 10 molecular diameters. In a complete wetting situation, where thick films develop, the Kelvin equation is, in general, not accurate. At fixed T the adsorption Γ_m(p) exhibits a loop; Γ_m jumps discontinuously at the first-order transition, but the accompanying metastable portions of the loop could produce hysteresis similar to that observed in adsorption measurements on mesoporous solids. Metastable thick films persist to larger p/p_sat in slits than in cylinders and this has repercussions for the shape of hysteresis loops. For a given pore size the loop in Γ_m shrinks with increasing T and disappears at a capillary critical temperature T^cap _c(< T_c). If T > T^cap _c condensation no longer occurs and hysteresis of Γ_m will not be observed. Such behaviour is found in experiments. A prewetting (thick–thin film) transition can occur for confined fluids. The transition is shifted to a smaller value of p/p_sat than that appropriate to prewetting at a single planar wall. Whereas the magnitude of the shift is very small for slits, it is substantial for cylinders and this leads to the possibility of finding a triple point, where ‘liquid’ and thick and thin films coexist, in cylindrical pores whose radii may not be too large for investigation by experiment or computer simulation. Adsorption of supercritical fluids (T > T_c, the bulk critical temperature) in cylinders is mentioned briefly.

The thermodynamics of fluids confined between two adsorbing solid substrates (walls) is revisited. Attention is focused on the phase equilibria of an open system characterized by the variables μ (chemical potential), T (temperature), and H (wall separation). Clapeyron equations for the shape of lines of coexistence are derived and used to interpret the results of earlier calculations of two first‐order transitions, namely capillary condensation of an undersaturated ‘‘gas’’ to ‘‘liquid’’ and prewetting (thick–thin film transition) at finite H. At such transitions the adsorption Γ and the solvation force f jump discontinuously. Criticality of a confined fluid is associated with the divergence of the derivatives (∂Γ/∂μ)_T,H and (∂²Γ/∂μ²)_T,H or equivalently, with the divergence of (∂f/∂H)_T,μ and (∂²f/∂H²)_T,μ. The presence of the additional field variable H, and its conjugate density f, implies that the phase equilibria of a confined fluid can be much richer than those of a bulk fluid or of a single interface (H=∞). By extending the formalism to multicomponent systems Clapeyron equations are derived for the coexistence of phases in confined fluid mixtures. An equation for the shift in chemical potential (or concentration) of the phase separation curve of a binary liquid mixture resulting from confinement at constant pressure and temperature is presented. This equation, which becomes exact for large separations H, is the appropriate analog for mixtures of the Kelvin equation used to describe capillary condensation in pure fluids; it can also be regarded as a generalization to nonzero concentrations of the Ostwald–Freundlich formula for the dependence of solubility on particle size. Our analysis provides a framework for interpreting recent solvation force measurements on phase‐separating liquid mixtures.

We derive the stationary probability distribution for a non-equilibrium system composed by an arbitrary number of degrees of freedom that are subject to Gaussian colored noise and a conservative potential. This is based on a multidimensional version of the Unified Colored Noise Approximation. By comparing theory with numerical simulations we demonstrate that the theoretical probability density quantitatively describes the accumulation of active particles around repulsive obstacles. In particular, for two particles with repulsive interactions, the probability of close contact decreases when one of the two particle is pinned. Moreover, in the case of isotropic confining potentials, the radial density profile shows a non trivial scaling with radius. Finally we show that the theory well approximates the “pressure” generated by the active particles allowing to derive an equation of state for a system of non-interacting colored noise-driven particles.