Fluid Density Functional Theory Research Articles

Density-functional theory for fluids in porous media.

As models for substances adsorbed within amorphous solid matrices, we consider mixtures of spheres with either hard or ideal interactions where several (matrix) components are quenched and the remaining (adsorbate) components are equilibrated. We propose a density-functional theory, based on the exact zero-dimensional limit, which treats both matrix and adsorbate components on the level of the respective one-body density profiles. As a test, we calculate pair correlation functions for hard spheres adsorbed in either a hard sphere or an ideal sphere matrix, and find good agreement with our computer simulation results.

Density functional theory of single-file classical fluids

An overcomplete density functional formalism is derived to represent the thermal behaviour of confined classical fluids under single-file conditions. Strictly one-dimensional hard rod systems are reanalysed from this viewpoint, and the context is then broadened to encompass hard spheres in cylindrical enclosures. Applications to non-single-file conditions, modulated enclosures, and confined polymers are mentioned.

The density functional theory of classical fluids revisited

We reconsider the density functional theory of nonuniform classical fluids from the point of view of convex analysis. From the observation that the logarithm of the grand-partition function $\log \Xi [\phi]$ is a convex functional of the external potential $\phi$ it is shown that the Kohn-Sham free energy ${\cal A}[\rho]$ is a convex functional of the density $\rho$. $\log \Xi [\phi]$ and ${\cal A}[\rho]$ constitute a pair of Legendre transforms and each of these functionals can therefore be obtained as the solution of a variational principle. The convexity ensures the unicity of the solution in both cases. The variational principle which gives $\log \Xi [\phi]$ as the maximum of a functional of $\rho$ is precisely that considered in the density functional theory while the dual principle, which gives ${\cal A}[\rho]$ as the maximum of a functional of $\phi$ seems to be a new result.

Density functional theory and Monte Carlo simulations for hard sphere fluids in square and rectangular channels

The density distribution of hard spheres in square and rectangular channels is studied using density functional theory and grand canonical ensemble Monte Carlo simulations. The theory uses the weighted density approximation for the excess Helmholtz free energy functional with the Curtin–Ashcroft approximation for the weighting function. The theoretical predictions are in good quantitative agreement with the simulations except for very high densities. The theory predicts pronounced layering in the channel as the density is increased, with high values of density at the surfaces and even higher densities at the corners where any two surfaces meet. Interesting interference effects are observed in the density profiles when compared to the case of a hard sphere fluid in slitlike pores.

High-Order Perturbative Density Functional Theory for Nonuniform Fluids with an Attractive Tail near Surfaces

The high-order perturbative density functional theory (DFT) for a nonuniform hard-sphere fluid (S. Zhou and E. Ruckenstein, Phys. Rev. E, 61, 2704, (2000)) was truncated at the third order and reformulated to be adapted for nonuniform non-hard-sphere interaction potential fluids. The only required input parameter (the second-order direct correlation function of a uniform fluid) can be obtained from the Ornstein–Zernike equation numerically or analytically. The constant relating to the bulk fluid parameter and measuring the strength of the third-order term was determined by the sum rule, which specifies the bulk pressure by the hard-wall contact density. The present third-order perturbative DFT predicted density distribution profiles for a Lennard–Jones fluid and an adhesive hard-sphere fluid confined between two hard walls that are in very good agreement with the simulation data. Furthermore, the present third-order perturbative DFT predicted very well the contact density as a function of the gap width for the confined adhesive hard-sphere fluid.

Renormalization-Group Calculation of the Dependence on Gravity of the Surface Tension and Bending Rigidity of a Fluid Interface

The surface tension and the bending rigidity of a planar liquid-vapor interface in the presence of vanishing gravity are analyzed using the renormalization group. Based on the density functional theory of inhomogeneous fluids, we show that a term, quartic in the density fluctuations, can be added to the classical capillary-wave model so that a renormalization-group calculation can be performed. By comparing the outcome of such a calculation with rigorous results relating the direct correlation function with surface tension and bending rigidity, we find the scaling dependence of the latter on gravity. The results agree with the expected fact that the interface should become unstable as gravity vanishes.

Weighted density functional theory for simple fluids: supercritical adsorption of a Lennard-Jones fluid in an ideal slit pore.

The adsorption of a Lennard-Jones fluid in an ideal slit pore is studied using weighted density functional theory. The intrinsic Helmholtz free-energy functional is separated into repulsive and attractive contributions. Rosenfeld's accurate fundamental measure functional is employed for the repulsive functional while another weighted density functional method is employed for the attractive functional. This other method requires an accurate equation of state for the bulk fluid and an accurate pair-direct correlation function for a uniform fluid, determined analytically or numerically. The results for this theory are compared against mean-field density functional theory and grand canonical ensemble simulation results, modeling the adsorption of ethane in a graphite slit. The results indicate that the weighted density functional method applied to the attractive functional can offer a significant increase in accuracy over the mean-field theory.

Weighted density functional theory of spherically inhomogeneous hard spheres.

The weighted density functional theory of hard sphere fluids proposed by Tarazona is applied to spherically inhomogeneous hard sphere fluids. The density profile of a hard sphere fluid around a hard sphere particle with structureless hard wall and varying radii is obtained. Our results are compared with previously obtained computer simulation with good agreement. We also calculate the density profile of a hard sphere fluid confined to spherical pores. We compare these results with those obtained by Calleja et al., in which both theory and computer simulation are used. In this case the results are also in agreement.

Entropic selectivity of microporous materials

The absorption of hard spheres into narrow pores is examined in the framework of Rosenfeld's “fundamental measure” formulation of density functional theory (DFT) for inhomogeneous fluids. The influence of the dimensionality of the confining geometry is assessed by considering the cases of a spherical cavity, an infinite cylindrical channel and an infinite slit. The pores are assumed to be in chemical equilibrium with a reservoir which fixes the chemical potentials of the various species. The hard sphere mixture is considered as a highly simplified model of aqueous solutions, involving a majority component (solvent) and solutes competing for absorption into the pores. It is shown that excluded volume effects alone can lead to very strong selectivities of the pores, for certain ratios of the solute and solvent to pore diameters. The selectivity is strongest for spherical cavities, and is least pronounced in the slit geometry. More complex geometries, including pore edge effects, with dimensions typical of simple ion channels through membranes, are also examined within the same DFT framework. The DFT predictions for the density profiles inside the pores, and the resulting absorbances and selectivities, are tested by grand-canonical Monte Carlo (GCMC) simulations, and good agreement is found.

Two- and Three-Dimensional Nonlocal Density Functional Theory for Inhomogeneous Fluids: II. Solvated Polymers as a Benchmark Problem

In a previous companion paper, we presented the details of our algorithms for performing nonlocal density functional theory calculations in complex two- and three-dimensional geometries. We discussed scaling and parallelization, but did not discuss other issues of performance. In this paper, we detail the precision of our methods with respect to changes in the mesh spacing. This is a complex issue because given a Cartesian mesh, changes in mesh spacing will result in changes in surface geometry. We discuss these issues using a series of rigid solvated polymer models including square rod polymers, cylindrical polymers, and bead–chain polymers. In comparing the results of the various models, it becomes clear that surface curvature or roughness plays an important role in determining the strength of structural solvation forces between interacting solvated polymers. The results in this paper serve as benchmarks for future application of these algorithms to complex fluid systems.

Two- and Three-Dimensional Nonlocal Density Functional Theory for Inhomogeneous Fluids: I. Algorithms and Parallelization

Fluids adsorbed near surfaces, near macromolecules, and in porous materials are inhomogeneous, exhibiting spatially varying density distributions. This inhomogeneity in the fluid plays an important role in controlling a wide variety of complex physical phenomena including wetting, self-assembly, corrosion, and molecular recognition. One of the key methods for studying the properties of inhomogeneous fluids in simple geometries has been density functional theory (DFT). However, there has been a conspicuous lack of calculations in complex two- and three-dimensional geometries. The computational difficulty arises from the need to perform nested integrals that are due to nonlocal terms in the free energy functional. These integral equations are expensive both in evaluation time and in memory requirements; however, the expense can be mitigated by intelligent algorithms and the use of parallel computers. This paper details our efforts to develop efficient numerical algorithms so that nolocal DFT calculations in complex geometries that require two or three dimensions can be performed. The success of this implementation will enable the study of solvation effects at heterogeneous surfaces, in zeolites, in solvated (bio)polymers, and in colloidal suspensions.

Quantum fluid density functional theory of chemical reactivity in a two-state ensemble

A quantum fluid density functional approach is adopted in monitoring the dynamics of various global and local reactivity parameters of a helium atom in a two-state equiensemble, having an interaction with an external laser field as well as an incoming proton. The effect of an increase in the excited-state contribution in a two-state ensemble is more pronounced in global reactivity parameters than the local ones. Global reactivity indices are found to be more sensitive to the external perturbation, for the whole time-dependent process, than the local indices. Most of the local quantities show drastic changes only at the beginning of the processes. In the atom–field interaction problem, electronegativity, polarizability and entropy oscillate with the field and hardness attains a steady value after the initial transients die out. The ion–atom collision process can be divided into three distinct regimes in terms of the time-dependent electronegativity profile. In the encounter regime, hardness and entropy maximize and polarizability minimizes. Dynamical variants of the principles of maximum hardness, maximum entropy and minimum polarizability are shown to be operative.

Dynamic density functional theory of fluids

We present a new time-dependent density functional approach for studying 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 approximate 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 over time of the density depend on the functional derivatives of the grand canonical free-energy functional F[] of the system. 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. In addition, we consider the case where one is forced to use an approximate form of F[].

Effective Hamiltonian for liquid-vapor interfaces.

Starting from a density functional theory for inhomogeneous fluids we derive an effective Hamiltonian for liquid-vapor interfaces of simple fluids which goes beyond the common phenomenological capillary-wave description. In contrast to other approaches we take into account the long-ranged power-law decay of the dispersion forces between the fluid particles which changes the functional form of the wave-vector-dependent surface tension qualitatively. In particular, we find two different forms of the bending rigidity for the capillary waves, a negative one for small wave vectors determined by the long-ranged dispersion forces and a positive rigidity for large wave vectors due to the distortions of the intrinsic density profile in the vicinity of the locally curved interface. The differences to the standard capillary-wave theory and the relevance of these results for the interpretation of scattering experiments are discussed.

Dynamic density functional theory of fluids

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.

Density functional theory: X-ray reflectivity studies of pure fluid liquid/vapour interfaces

Although there are numerous model independent methods for the extraction of a density profile from X-ray reflectivity data, for many systems there are fundamental limitations to the uniqueness of profiles so determined owing to the limited range of scattering data and the nature of the inverse transformation. Based on density functional theory (DFT) of inhomogeneous fluids we model the liquid/vapour interface of some pure simple liquids (Ar, N2, Kr, CCl4). Attractive contributions to the Helmholtz free energy are treated by a mean-field approximation (MF). Both local density approximation (LDA) and smoothed density approximation (SDA) are applied to the repulsive reference system described by a hard sphere equation of state. A cut-off and shifted Lennard-Jones potential divided according to the Weeks–Chandler–Andersen (WCA)-prescription is used as intermolecular interaction force model. The potential parameters were determined by modelling saturated liquid densities of a pure fluid. Computed theoretical density profiles are used to predict X-ray specular reflectivities. Capillary waves models of Braslau etal. (Phys. Rev. A, 1998, 38, 2457), Evans etal. (Mol. Phys., 1981, 42, 1169) and of Meunier (J. Phys. (Paris), 1987, 48, 1819) are adopted to approximate these contributions to both specular reflectivities and surface tension. We propose a parameter free procedure based on the intrinsic interfacial thickness to calculate these terms. The agreement of predicted X-ray reflectivities with experimental data is good in the case where these terms are included. Similarly the surface tension is calculated from the intrinsic part plus contributions due to capillary wave roughening. All the calculations are based exclusively on potential parameters determined from saturated liquid densities.

Femtosecond quantum fluid dynamics of helium atom under an intense laser field

A comprehensive, nonperturbative, time-dependent quantum mechanical (TDQM) approach is proposed for studying the dynamics of a helium atom under an intense, ultrashort (femtoseconds) laser pulse. The method combines quantum fluid dynamics (QFD) and density functional theory. It solves a single generalized nonlinear Schrödinger equation of motion (EOM), involving time and three space variables, which is obtained from two QFD equations, namely, a continuity equation and an Euler-type equation. A highly accurate finite difference scheme along with a stability analysis is presented for numerically solving the EOM. Starting from the ground-state Hartree–Fock density for He at t=0, the EOM yields the time-dependent (TD) electron density, effective potential surface, difference density, difference effective potential, ground-state probability, 〈r〉, magnetic susceptibility, polarizability, flux, etc. By a Fourier transformation of the TD dipole moment along the linearly polarized-field direction, the power and rate spectra for photoemission are calculated. Eleven mechanistic routes for photoemission are identified, which include high harmonic generation as well as many other spectral transitions involving ionized, singly excited, doubly excited (autoionizing), and continuum He states, based on the evolution of the system up to a particular time. Intimate connections between photoionization and photoemission are clearly observed through computer visualizations. Apart from being consistent with current experimental and theoretical results, the present results offer certain predictions on spectral transitions which are open to experimental verification. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 441–474, 1998

Stripped ion-helium atom collision dynamics within a time-dependent quantum fluid density functional theory

A nonperturbative, time-dependent (TD) quantum mechanical approach is described for studying the collision dynamics between the He atom and a fully stripped ion. The method combines quantum fluid dynamics and density functional theory to solve two coupled equations: one for the trajectory of the projectile nucleus and the other for the electronic charge distribution of the target atom. The computed TD and frequency-dependent properties provide detailed features of the collision process. Inelastic and ionization cross sections are also reported. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 251–271, 1998

Quantum fluid density functional theory of time‐dependent processes

A quantum fluid density functional theory has been developed through an amalgamation of the quantum fluid dynamics and the time-dependent density functional theory. It is used in studying typical time-dependent processes like ion–atom collisions and atom–field interaction. Temporal evolution of chemical reactivity parameters as electronegativity, hardness, entropy, and polarizability is monitored for a He atom in its ground and excited states interacting with an external electric field and an incoming proton. It is observed that these reactivity parameters either remain static or oscillate with the external field in the atom–field interaction case, whereas during the collision process, hardness and entropy maximize and polarizability minimizes for both the electronic states of the atom. The possibility of a quantum theory of motion within the purview of this quantum fluid density functional framework is also explored. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 279–291, 1998

Quantum fluid density functional theory of time-dependent processes

A quantum fluid density functional theory has been developed through an amalgamation of the quantum fluid dynamics and the time-dependent density functional theory. It is used in studying typical time-dependent processes like ion–atom collisions and atom–field interaction. Temporal evolution of chemical reactivity parameters as electronegativity, hardness, entropy, and polarizability is monitored for a He atom in its ground and excited states interacting with an external electric field and an incoming proton. It is observed that these reactivity parameters either remain static or oscillate with the external field in the atom–field interaction case, whereas during the collision process, hardness and entropy maximize and polarizability minimizes for both the electronic states of the atom. The possibility of a quantum theory of motion within the purview of this quantum fluid density functional framework is also explored. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 279–291, 1998