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.

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A geometrically based fundamental-measure free-energy density functional unified the scaled-particle and Percus-Yevick theories for the hard-sphere fluid mixture. It has been successfully applied to the description of simple ͑''atomic''͒ three-dimensional ͑3D͒ fluids in the bulk and in slitlike pores, and has been extended to molecular fluids. However, this functional was unsuitable for fluids in narrow cylindrical pores, and was inadequate for describing the solid. In this work we analyze the reason for these deficiencies, and show that, in fact, the fundamental-measure geometrically based theory provides a free-energy functional for 3D hard spheres with the correct properties of dimensional crossover and freezing. After a simple modification of the functional, as we propose, it retains all the favorable Dϭ3 properties of the original functional, yet gives reliable results even for situations of extreme confinements that reduce the effective dimensionality D drastically. The modified functional is accurate for hard spheres between narrow plates (Dϭ2), and inside narrow cylindrical pores (Dϭ1), and it gives the exact excess free energy in the Dϭ0 limit ͑a cavity that cannot hold more than one particle͒. It predicts the ͑vanishingly small͒ vacancy concentration of the solid, provides the fcc hard-sphere solid equation of state from closest packing to melting, and predicts the hard-sphere fluid-solid transition, all in excellent agreement with the simulations.

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.

A new free energy density functional for hard spheres is presented, along the lines of the fundamental measure theory, which reproduces the Percus-Yevick equation of state and direct correlation function for the fluid, with a tensor weighted density. The functional, based on the zero-dimension limit, is exact for any one-dimensional density distribution of the spheres. The application to the hard sphere crystals gives excellent results, solving all of the qualitative problems of previous density functional approximations, including the unit cell anisotropy in the fcc lattice and the description of the metastable bcc lattice.

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)tile 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 0 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 > Psat)" 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(

We develop and test an operational definition of the intrinsic surface for liquid-vapor interfaces. The application to the microscopic configurations along Monte Carlo computer simulations gives the statistical properties of the intrinsic surfaces and the intrinsic density profiles for simple fluid models. The results open a framework of quantitative description to close the gap between the mesoscopic capillary wave theory and the sharpest level of resolution for the intrinsic density distribution, relative to the first atomic layer in the liquid surface, as done in the interpretation of experimental x-ray reflectivity.The description of the liquid-vapor interface through a smooth density profile, z, goes back to van der Waals, but the delocalization of the free liquid surface by long wavelength capillary waves (CW) sets a problem in the statistical interpretation of z, tied to the appearance of long-range surface fluctuations. The capillary wave theory (CWT) [1-3] incorporates these fluctuations by means of an intrinsic surface, z R P q q e iqR , to represent the instantaneous microscopic boundary between liquid and vapor, at each transverse position R x; y. For large R 12 jR 1 ÿ R 2 j, the two-particle distribution function is represented by the average over the intrinsic surface shapes of two intrinsic profiles, z, shifted by the local values of R,The microscopic structure in the perpendicular direction is included through z, while the long-ranged transverse dependence goes with the intrinsic surface correlations, hR 1 R 2 i. The two key assumptions of the CWTare that z and R are statistically uncorrelated, and that the later follows a simple surface Hamiltonian leading to uncorrelated Gaussian probabilities for each Fourier component and hj q j 2 i o Aq 2 ÿ1 , with the macroscopic surface tension o , in k B T 1= units. The finite transverse size A L 2 x enters through the lower limit of the wave vector, with periodic boundary conditions q 2=L x , while the fixed number of particles, N, restricts the fluctuations of the 0 component. The specific definition of R and, in particular, the upper limit q u q for the level of resolution at which R follows the atomic positions, would determine the shape of the intrinsic profile [2],The link between CWT and the density functional (DF) or other microscopic theories for the liquid-vapor density profile z [4,5] has not been established at a quantitative level because of the uncertainty in the separation of ''bulk'' and ''surface'' fluctuations, tied to the choice of q u and the shape of z. Qualitatively it is expected that q u should not go beyond 2=, in terms of the molecular diameter , and that z is a rather sharp function, since the CWT gives the density profile, with transverse size L x , as the smoother Gaussian convolution:with the CW square width CW L x ; q u P q h 2 q i. Mecke and Dietrich [5] used a DF approximation to check the validity of the CWT assuming that z is the DF profile for a planar surface, without explicit dependence on q u . Such a smooth intrinsic...

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 pressuresp/psrttclose to those predicted by the macroscopic Kelvin equation, even for radii R, 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 T,(p) exhibits a loop; Tmjumps 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/psatin slits than in cylinders and this has repercussions for the shape of hysteresis loops. For a given pore size the loop in Tmshrinks with increasing T and disappears at a capillary critical temperature T r p (< T,).If T > TEaPcondensation no longer occurs and hysteresis of Tm 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/psat 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 super- critical fluids ( T > T,, the bulk critical temperature) in cylinders is mentioned briefly