Inhomogeneous fluids density-function theory

Density-functional theory is best known as the basis for electronic structure calculations. A variant of this theory can be used to calculate the structure of inhomogeneous fluids [35] the free energy of the fluid is expressed as a functional of the density of the various components a theorem asserts that this functional attains its minimum for the true density profiles. 

An alternative approach is by the application of an approximate theory. At present, the most useful theoretical treatment for the estimation of the equilibrium properties is generally considered to be the density functional theory (DFT). This involves the derivation of the density profile, p(r), of the inhomogeneous fluid at a solid surface or within a given set of pores. Once p(r) is known, the adsorption isotherm and other thermodynamic properties, such as the energy of adsorption, can be calculated. The advantage of DFT is its speed and relative ease of calculation, but there is a risk of oversimplification through the introduction of approximate forms of the required functionals (Gubbins, 1997). 

In recent years, a number of investigators have studied the phase equilibria of simple fluids in pores by the application of density functional theory. Semina] studies were carried out by Evans and his co-workers (1985,1986). Their approach was considered to be the simplest realistic model for an inhomogeneous three-dimensional fluid . The starting point was a model intrinsic Helmholtz free energy functional F(p), with a mean-field approximation for the attractive forces and hard-sphere repulsion. As explained in Section 7.6, the equilibrium density profile of the fluid in a pore was obtained by minimizing the grand potential functional. 

For further reading, see Fundamentals of Inhomogeneous Fluids. D. Henderson, Ed. Marcel Dekker (1992). (Chapter 5 of this book, by R. Evans, describes the application of density functional theory) The Liquid-Solid Interface at High Resolution, Faraday Discuss. Roy. Soc. Chem. (London) (1992).)... 

The choice of the weighting function w depends on the version of density functional theory used. For highly inhomogeneous confined fluids, a smoothed or nonlocal density approximation is introduced, in which the weighting function is chosen to give a good description of the hard sphere direct pair correlation function for the uniform fluid over a... 

The novel approach for calculation of pore size distributions, which is reported in the current study is based on recent developments in the materials science and in the theory of inhomogeneous fluids. First, an application of experimental adsorption data for well-characterized MCM-41 silicas enabled proper calibration of the pore size analysis. Second, an application of a modem theory to describe the behavior of inhomogeneous fluids in confined spaces, that is the non-local density functional theory [6], allowed the numerical calculation of model isotherms for various pore sizes. In addition, a practical numerical deconvolution method that provides a "best fit" solution representing the pore distribution of the sample was implemented [7, 8]. In this paper we describe a deconvolution method for estimating mesopore size distribution that explicitly allows for unfilled large pores, and a method for creating composite, or hybrid, models that incorporate both theoretical calculations and experimental observations. Moreover, we showed the applicability of the new approach in characterization of MCM-41 and related materials. 

One can prove that in thermal equilibrium, in a grand canonical ensemble (i.e., volume, chemical potential, and temperature are fixed), the grand canonical free energy Q(p(r)) of a system can be written as a functional of the one-body density p(r) alone, which will depend on the position r in inhomogeneous systems. The density distribution peq(r) which minimizes the grand potential functional is the equilibrium density distribution. This statement is the basis of the equilibrium density functional theory (DFT) for classical fluids which has been used with great... 

Lowen H (2002) Density functional theory of inhomogeneous classical fluids recent perspectives and new perspectives. J Phys Condens Matter 14 11897-11905... 

In its simplest form, density functional theory involves writing the Helmholtz energy of an inhomogeneous fluid as a sum of two terms. 

The classical models of adsorption processes like Langmuir, BET, DR or Kelvin treatments and their numerous variations and extensions, contain several uncontrolled approximations. However, the classical theories are convenient and their usage is very widespread. On the other hand, the aforementioned classical theories do not start from a well - defined molecular model, and the result is that the link between the molecular behaviour and the macroscopic properties of the systems studied are blurred. The more developed and notable descriptions of the condensed systems include lattice models [408] which are solved by means of the mean - field or other non-classical techniques [409]. The virial formalism of low -pressure adsorption discussed above, integral equation method and perturbation theory are also useful approaches. However, the state of the art technique is the density functional theory (DFT) introduced by Evans [410] and Tarazona [411]. The DFT method enables calculating the equilibrium density profile, p (r), of the fluid which is in contact with the solid phase. The main idea of the DFT approach is that the free energy of inhomogeneous fluid which is a function of p (r), can be... 

Douglas Frink LJ, Salinger AG Two- and three-dimensional nonlocal density functional theory for inhomogeneous fluids I. Algorithms and parallelization, J Comput Phys 159(2) 407 24, 2000. 

Rosenfeld Y Free-energy model for the inhomogeneous hard-sphere fluid mixture and density functional theory of freezing, Phys Rev Lett 63(9) 980—983, 1989. 

The use of SAFT within theories for inhomogeneous fluids such as the square-gradient theory (usually called density-gradient theory, DGT) and density-functional theory" (DFT), provides a route towards the... 

Tang, Y. P, and Wu, J.-Z. 2003. J. Chem. Phys. A density-functional theory for bulk and inhomogeneous Lennard-Jones fluids from the energy route. 119 7388. 

Recent progress in the theory of adsorption on porous solids, in general, and in the adsorption methods of pore structure characterization, in particular, has been related, to a large extent, to the application of the density functional theory (DFT) of Inhomogeneous fluids [1]. DFT has helped qualitatively describe and classify the specifics of adsorption and capillary condensation in pores of different geometries [2-4]. Moreover, it has been shown that the non-local density functional theory (NLDFT) with suitably chosen parameters of fluid-fluid and fluid-solid interactions quantitatively predicts the positions of capillary condensation and desorption transitions of argon and nitrogen in cylindrical pores of ordered mesoporous molecular sieves of MCM-41 and SBA-15 types [5,6]. NLDFT methods have been already commercialized by the producers of adsorption equipment for the interpretation of experimental data and the calculation of pore size distributions from adsorption isotherms [7-9]. 

A third class of new polymer integral equation theories have been proposed by Kierlik and Rosinberg. Their work is an extension of a density functional theory of inhomogeneous polyatomic fluids to treat the homogeneous phase. The Wertheim thermodynamic pertubation theory of polymerization is employed in an essential manner. Applications to calculate the intermolecular structure of rather short homopolymer solutions and melts have been made. Good results are found for short chains at high densities, but the authors comment that their earlier theory appears to be unsuited for long chains at low to moderate (semidilute) densities. ... 

Wichert, J. M. Gulati, H. S. Hall, C. K. J. (1996). Binary hard chain mixtures. I. Generalized Flory equations of state. /. Chem. Phys., Vol. 105, 7669-7682 Woodward, C. E. Forsman, J. (2008). Density functional theory for polymer fluids with molecular weight polydispersity. Phys. Rev. Lett., Vol. 100,098301 Woodward, C. E. (1991). A density functional theory for polymers Application to hard chain-hard sphere mixtures in slitlike pores. /. Chem. Phys., Vol. 94,3183-3191 Woodward, C. E. Forsman, J. (2009). Interactions between surfaces in polydisperse semiflexible polymer solutions. MacromoL, Vol. 42, 7563-7570 Woodward, C. E. Yethiraj A. (1994). Density functional theory for inhomogeneous polymer solutions. /, Chem. Phys., Vol. 100,3181-3186... 

A different and more sophisticated approach to the theory of the nematic-smectic A transition is based on the nonlocal density functional theory developed for inhomogeneous hard-core fluids [68]. The nonlocal free energy functional is defined in the following way [69-71] ... 

IV. DENSITY FUNCTIONAL APPROACHES IN THE THEORY OF INHOMOGENEOUS ASSOCIATING FLUIDS... 

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