Smoothed Density Approximation Research Articles

Effect of three-body interactions on Ar adsorption on graphitized carbon black

The Axilrod-Teller equation is incorporated into nonlocal density functional theory (NLDFT) to improve the description of low-temperature argon adsorption isotherm on graphitized carbon black. Using the NLDFT based on the Tarazona smoothed density approximation and the test-particle method, the pair and triplet distribution functions for liquid argon were analyzed. The latter allowed us to quantitatively account for the three-body Axilrod-Teller nonadditive term. An analogous scheme to that of Weeks, Chandler, and Andersen is proposed to incorporate the Axilrod-Teller equation into the NLDFT in the framework of mean field approximation. The three-body nonadditive term appeared to decrease the internal energy of liquid argon by about 5%, which is in agreement with molecular simulation results. The new approach has been applied to the vapor-liquid coexistence and to argon adsorption on graphitized carbon black at 87.3 K. The approach has been shown to fit the experimental adsorption isotherm within the relative error of about 4% as opposed to 37% in the case of the standard NLDFT. This is due to weakening of the intermolecular interaction potential by about 20% in the molecular layer nearest to the graphite surface resulted mainly from the positive Axilrod-Teller contribution of triplets comprising one carbon atom.

Density functional theory and simulation of the columnar phase of a system of parallel hard ellipsoids with attractive interactions

A simple molecular model consisting of parallel hard oblate ellipsoids with superimposed square-well attractive interactions of variable range is considered for the study of the phase behavior of thermotropic discotic molecules. A density functional theory appropriate for nonuniform fluids is formulated in which the hard-core contributions to the free energy are treated within a nonlocal weighted-density approximation (WDA) while the attractive contributions are treated at a mean-field level. It is shown that the columnar phase becomes stable relative to the nematic phase at fluid densities for a range of values of the range of the attractive well. In these cases, the region of stability of the columnar phase is bounded at high temperatures by a nematic-columnar-solid triple point. The calculations show that if the attractions are made too long ranged (lambda/D> or approximately =0.84 for particles of aspect ratio of L/D=0.1, where lambda/D is the range of the attractive interaction in units of the molecular diameter D), columnar ordering becomes unstable and the nematic phase dominates at all fluid densities. It is shown that columnar ordering is also predicted when the density functional theory is supplemented with the smoothed-density approximation (SDA). Computer simulations have also been carried out for a particular choice of model parameters; our simulation data confirm the stabilization of the hexagonal columnar phase between the solid and nematic phases. A comparison with simulation data allows us to conclude that the WDA provides a fairly good description of the columnar phase and very good agreement for the nematic-columnar transition properties. On the other hand, our calculations show that the SDA largely underestimates the transition pressure and predicts a too-strongly first-order nematic-columnar transition

Phase coexistence and interface structure of a two-component Lennard-Jones fluid in porous media: application of Born—Green—Yvon equation

The Born-Green-Yvon equation with smoothed density approximation is used to calculate the liquid-liquid density profiles of a symmetric Lennard-Jones fluid in a hard sphere disordered matrix. The phase diagrams are evaluated for model systems characterized by different matrix densities and compared with the results of theoretical predictions and the Monte Carlo simulations of Gordon, P. A., and Glandt, E. D., 1996, J. chem. Phys., 105, 4257. It was found that increasing the matrix packing fraction reduces the magnitude of the miscibility gap and smooths the density profiles between two coexisting phases.

Density functional theories and molecular simulations of adsorption and phase transitions in nanopores.

The nonlocal density functional theory (NLDFT) of confined fluids is tested against Monte Carlo simulations by using the example of Lennard-Jones (LJ) fluid sorption in slit-shaped and cylindrical nanopores ranging from 0.3 to 10 nm in width. The fluid-fluid and solid-fluid parameters of the LJ potentials were chosen to represent several experimentally important adsorption systems: nitrogen and carbon dioxide in activated carbons, zeolites, and mesoporous molecular sieves of the MCM-41 type. Freezing in nanopores is discussed using the example of methane sorption in carbon at 111 K. Comparison with reference experiments is given when available. Two versions of NLDFT, the smoothed density approximation and the fundamental measure theory, are considered. It is shown that NLDFT approaches with properly chosen parameters provide quantitative agreement with the results of Monte Carlo simulations and reference experiments. Appreciable deviations are found in extremely narrow pores of less than two molecular diameters in width. In wider pores, NLDFT models can be used for quantitative predictions of reversible and hysteretic adsorption isotherms and analyses of the specifics of phase transformations, including the equilibrium and spinodal phase transitions.

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.

Liquid crystal phase diagram of the Gay-Berne fluid by density functional theory

We calculate the liquid crystal phase diagram for a model fluid with Gay-Berne interparticle potential using the Tarazona smoothed-density approximation of density functional theory. Vapour, liquid, nematic and smectic A phases are considered. For length to breadth ratio kappa 3 and energy anisotropy kappa 5, comparison with the simulation data of de Miguel et al. shows reasonable agreement.

A density functional analysis of the restricted orientation model of liquid crystals and its implications for theories of orientational ordering

We compare the predictions of various density functional approximation schemes as applied to the restricted-orientation, hard, rectangular-parallelepiped model of liquid crystals. These calculations illustrate some features of density functional calculations which may aid in understanding other approximate, mean-field theories of orientational ordering phase transitions and their interfaces. The isotropic–nematic coexistence curve of this model is apparently well described by the so-called third-order y expansion, and the analytical expression for the nonideal free energy derived from the simple y expansion thus provides a benchmark against which to compare the density functional approximation schemes. Using standard relations applicable to inhomogeneous fluids, we show how expansions about the bulk isotropic phase compare more favorably with the ‘‘exact’’ y-expansion results when truncated at third order than do second-order truncations. Thus this model behaves somewhat differently than other hard-particle models of fluids, notably the hard-sphere one. We also examine the possibility of expanding the free energy about the ordered phase to obtain the properties of a disordered phase. An expansion about local values of the density leads to the widely used smoothed-density approximation and a hierarchy of systematic extensions. The latter appear more stable than conventional Taylor expansions about bulk disordered phases.

Structure effects and phase equilibria of Lennard-Jones mixtures in a cylindrical pore.

The effect of fluid confinement in a narrow cylindrical pore is studied by means of a density-functional approach. An extension of the smoothed-density approximation to the two-component case is employed to explore the structure and phase behaviour of a mixture in a cylindrical pore. Contact is made with the previous local-density treatment and with the macroscopic thermodynamic approach.

Smectic-A, crystalline and columnar ordering in the system of hard parallel cylinders

We study a system of hard parallel cylinders in the framework of the smoothed-density approximation (SDA). Using a bifurcation analysis, we argue that, apart from the nematic phase, smectic-A, solid (or crystalline smectic-B) and columnar phases should also occur in this system. We predict the following sequence of phase transitions: nematic-smectic-A at η* = η/ηcp = 0·31, smectic-A-solid (or smectic-B) at η* ≈ 0·59 and solid (or smectic-B)-columnar at η* ≈ 0·84, where ηcp = 0·9069 is the close-packing density. In our approach we cannot distinguish between a solid phase and a crystalline smectic-B phase.

Density-functional approach to two-dimensional classical fluids

The density-functional (DF) method is combined with the smoothed-density approximation to calculate the fluid structure in two-dimensional (2D) hard-disc (HD) systems. Density-dependent weighting functions in the DF method proposed by Tarazona for three-dimensional fluids have been applied to 2D systems for the calculation of the radial distribution function of HD fluids and the determination of the static structure of 2D HD fluids around the central hard triatomic molecule. The results compare reasonably well with Monte Carlo data. The radial distribution of a 2D Lennard-Jones fluid has also been calculated and shown to be in satisfactory agreement with Monte Carlo data except for the high-density fluid.

Density-functional theory for systems of hard rods.

We present a density-functional theory, based on the smoothed density approximation, to study systems of hard rods with full translational and orientational freedom. For hard spherocylinders, we find both the nematic-isotropic and the nematic--smectic-A transition in a wide range of length-to-width ratios (L+D)/D. We locate the tricritical point for the nematic--smectic-A transition and also make some predictions about the nematic--smectic-A--smectic-B point. Finally, we calculate the nematic elastic constants. The predictions of our theory are compared with the results of computer simulations and other theories. We also make some comments about application of the theory to systems of hard ellipsoids of revolution and hard cylinders.

Study of the Landau bicritical point in dense systems of hard biaxial molecules

We study two systems of hard biaxial molecules: hard spheroplatelets and hard ellipsoids, using the liquid-crystalline version of the smoothed density approximation (SDA). The first system is studied for all elongations of the spheroplatelet c, whereas the second only for c ⩽ 7. For both systems, we locate the line of Landau bicritical points at which a direct transition from the isotropic phase to the biaxial phase occurs. We find that the density of the isotropic phase at the Landau bicritical point is always higher than that at the isotropic-nematic transition in the limit of uniaxial molecules and the difference ranges from 10 to 30 per cent. For hard ellipsoids, we obtain a similar scaling behaviour at the Landau bicritical point as for spheroplatelets, i.e. b ∼c 1/2, where b denotes the breadth of the ellipsoid.

Packing effects in the adsorption at the liquid-solid interface

We report a study of the dense hard-sphere fluid near a wall with a square well adsorbing potential. We use Monte Carlo simulations, integral equations and density functional theory in the smoothed-density approximation to determine the structure of the fluid as a function of the wall potential. We propose a theory which relates the properties of the adsorbed layer to those of an effective 2-d fluid which takes into account the presence of the bulk liquid. Some consequences for the interpretation of adsorption isotherms at the liquid-solid interface are discussed.

Comparative study of the nematic phase-isotropic phase transition in systems of uniaxial hard cores

We apply the smoothed density approximation to study the nematic phase-isotropic phase transition (N-I) for the following systems: hard ellipsoids of revolution (HE), hard spherocylinders (HSC), and cylinders (HC). We find that the transition is very sensitive to the shape of the hard core modelling a liquid crystal molecule. In the case of HC, the nematic phase can exist in the whole range of elongations x, while for HE and HSC the nematic density at the transition exceeds the close packing density in some range of x, around x = 1. Also the dependence on x of the nematic order parameter, Q, the density, Δη, and entropy, ΔS, jumps at N–I transition is completely different for HC in comparison to HE and HSC.

Nematic-smectic-A transition for perfectly aligned hard spherocylinders: Application of the smoothed-density approximation.

We apply the smoothed-density approximation to study the nematic--smectic-A transition for the system of perfectly aligned hard spherocylinders. We find that the transition occurs for all length-to-width ratios L/D of a spherocylinder and is always continuous. The bifurcation analysis is applied to locate the transition. Our results are in reasonable agreement with those obtained in computer simulations.

Lennard-Jones fluids in cylindrical pores: Nonlocal theory and computer simulation

We present adsorption isotherms, phase diagrams, and density profiles for a Lennard-Jones fluid confined to a cylindrical pore. In particular, we concentrate on the gas–liquid transition in the pore (capillary condensation). We compare simulations for a series of radii and different temperatures with mean field density functional theory (MFT). Two forms of MFT are considered, the simple local density approximation (LDA) and Tarazona’s nonlocal or smoothed density approximation (SDA). We find that the SDA provides a quite accurate description of fluid structure in the pore and that it produces phase diagrams in good agreement with the simulation data. For larger radii and temperatures T/Tc≳0.6 the SDA shows steep rises in adsorption close to the transition. This strongly affects the shape of the coexistence curve in the T, ρ̄ plane. Here ρ̄ is defined as the average density inside the pore. This behavior is confirmed by the simulation. In contrast, LDA gives a poor representation of the fluid structure and this underlies the failure to reproduce the phase diagrams and adsorption isotherms found with SDA or simulation. For extremely small radii (R*≈1) the simulation adsorption isotherms are smooth, and for not too low a temperature they are accurately described by an approach which starts from the potential distribution theorem and uses perturbation theory for the true one-dimensional fluid.

The density profile of a confined fluid

The density profile of a liquid confined in a narrow capillary depends markedly on the ratio of capillary radius to molecular size σ. We present a direct comparison of the results of a non-local density functional theory (smoothed density approximation) for the profiles of coexisting ‘liquid’ and ‘gas’ phases of a Lennard-Jones fluid confined in a structureless cylindrical capillary with those of recent computer simulations. Good quantitative agreement is found for capillary radii in the range 3σ to 6σ, indicating that the theory yields an accurate description of packing effects. Taken with earlier results for other problems this suggests that the smoothed density approximation is a versatile theory of inhomogeneous fluids, providing a realistic account of both the phase equilibria and structure of a wide variety of fluid interfaces.

Phase equilibria of fluid interfaces and confined fluids

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.

Effect of Volume Exclusion on the Dimensions of Polymer Chains

The effect of volume exclusion on the mean-square radius of gyration s2 is treated by use of the empirical expression W0(s)=const s6exp[−(7/2)〈s2〉0−1s2] for the statistical distribution of s for a system of linear chain molecules unperturbed by volume exclusion. Segment interactions within the perturbed molecule are treated in the smoothed-density approximation. The result is α5—α3=Azh(z/α3), where α2=〈s2〉/〈s2〉0, A is a numerical constant, z is the usual parameter proportional to the excluded volume for a segment pair and to the square root of the chain length, and h(ζ) is a function which decreases by almost one-half as ζ increases from 0 to ∞. Asymptotic dependence of α2 on the ⅕th power of the chain length is thus confirmed. Extremely long chains, exceeding 106 bonds, would be required however for realization of this limiting behavior.