Modelling Inhomogeneous Fluids

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 

While the DGT-based SAFT approach is easy to implement and can provide a good representation of the surface tension of pure fluids and mixtures, it typically requires the use of empirical adjustable parameters, the so-called influence parameters, which limit the predictive ability of the method. In contrast, DFT treatments, although more complex and numerically more demanding, do not rely on adjustable parameters to provide information on the interfacial properties. Chapman was the first to suggest the possibility of incorporating a SAFT-like description of associating fluids within a DFT 

Recently the DFT method combined with SAFT equations of state has been used to predict the interfacial properties of real fluids. LDA methods are accurate enough to treat liquid-liquid and liquid-liquid interfaces where the density profiles are usually smooth functions, and have been used in combination with the SAFT-VR approach to predict the surface-tension of real fluids successfully. The intermolecular model parameters required to treat real substances are determined by fitting to experimental vapour-pressure and saturated liquid density data in the usual way (see section 8.5.1) and the resulting model is found to provide accurate predictions of the surface tension. A local DFT treatment has also been combined with the simpler SAFT-HS approach, but in this case only qualitative agreement with experimental surface tension data is found due to the less accurate description of the bulk properties provided by the SAFT-HS equation. Kahl and Winkelman have followed a perturbation approach similar to the one proposed with the SAFT-VR equation and have coupled a local DFT treatment with a Lennard-Jones based SAFT equation of state. They predict the surface tension of alkanes from methane to decane and of cyclic and aromatic compounds in excellent agreement with experimental data. 

The most sophisticated DFT approaches incorporate weighted densities, which depend on several weighting factors, often based on the FMT. The implementation of WDA in DFT provides accurate oscillatory profiles, such as those found in solid-fluid interfaces (near walls, in confined fluids) or in solid and other structured phases. The DFT for associating fluids of 

Segura el al. combines Tarazona s WDA DFT for hard-spheres with Wertheim s thermodynamic perturbation theory and has been used in a number of studies of associating fluids in pores and with functionalized walls in the limit of complete association a DFT for polymeric fluids is obtained in this method. Based on these works, Chapman and co-workers have presented the interfacial-SAFT (iSAFT) equation, which is a DFT for polyatomic fluids formulated by considering the polyatomic system as a mixture of associating atomic fluids in the limit of complete association this approach allows the study of the microstructure of chain fluids. Interfacial phenomena in complex mixtures with structured phases, including lipids near surfaces, model lipid bilayers, copolymer thin films and di-block copolymers, have all been studied with the iSAFT approach. 

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To the best of our knowledge, there was only one attempt to consider inhomogeneous fluids adsorbed in disordered porous media [31] before our recent studies [32,33]. Inhomogeneous rephca Ornstein-Zernike equations, complemented by either the Born-Green-Yvon (BGY) or the Lovett-Mou-Buff-Wertheim (LMBW) equation for density profiles, have been proposed to study adsorption of a fluid near a plane boundary of a disordered matrix, which has been assumed uniform in a half-space [31]. However, the theory has not been complemented by any numerical solution. Our main goal is to consider a simple model for adsorption of a simple fluid in confined porous media and to solve it. In this section we follow our previously reported work [32,33]. 

Local Average Density Model (LADM) of Transt)ort. In the spirit of the Flscher-Methfessel local average density model. Equation 4, for the pair correlation function of Inhomogeneous fluid, a local average density model (LADM) of transport coefficients has been proposed ( ) whereby the local value of the transport coefficient, X(r), Is approximated by... 

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. 

The application of these techniques to the simulation of adsorbed phases began to appear in the 1960s. The great majority of such studies are based on the idea that the solid adsorbent can be treated as a rigid, inert field of force that produces an inhomogeneous fluid when adsorbate molecules are in its vicinity. Clearly, the success of such simulations depends on the quality of the model used to represent the adsorbate-adsorbent (gas-solid) interactions, These models are discussed in Sec. III. 

Over the p t several years we and our collaborators have pursued a continuous space liquid state approach to developing a computationally convenient microscopic theory of the equilibrium properties of polymeric systems. Integral equations method [5-7], now widely employed to understand structure, thermodynamics and phase transitions in atomic, colloidal, and small molecule fluids, have been generalized to treat macromolecular materials. The purpose of this paper is to provide the first comprehensive review of this work referred to collectively as Polymer Reference Interaction Site Model (PRISM) theory. A few new results on polymer alloys are also presented. Besides providing a unified description of the equilibrium properties of the polymer liquid phase, the integral equation approach can be combined with density functional and/or other methods to treat a variety of inhomogeneous fluid and solid problems. 

Rosenfeld, Y. 1993. Eree energy model for inhomogeneous fluid mixtures Yukawa-charged hard spheres, general interactions, and plasmas. The Journal of Chemical Physics 98, no. 10 8126. doi 10.1063/1.464569. 

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... 

Tang Y, Wu J Modeling inhomogeneous Van Der Waals fluids using an analytical direct correlation function, Phys Rev E 70(1) 011201, 2004. 

Constructed model of blood flow with variable viscosity and density allows to get the patterns of leaflet deformation and admixture distribution effected by inhomogeneous fluid flow. 

Dolgov, D. A., Zakharov, Y.N. Modeling of viscous inhomogeneous fluid flow in large blood vessels. Vestnik Kemerovo State University 2(62) T.l, pp. 30-35 (2015) (in Russian)... 

The calculations in ref. 25 for model micropores only consider interactions between a single adsorptive molecule and the walls of the model micropore. They do not account for interactions between adsorptive molecules and so cannot model the process of micropore filling. Recently (ref. 43) results from molecular modelling studies were reported for the adsorption of nitrogen on porous carbons in which both adsorptive-adsorbent and inter-adsorptive interactions were considered. Using an approximate theory of inhomogeneous fluids known as mean-field theory, a function p(p, w) was derived (ref. 43) which relates the... 

Recently we proposed a weight and poljrmer liquids. Thig theory was developed by com-bininggt j lattice fluid (LF) model with the density gradient theory of inhomogeneous fluids. The main objective of this paper is to illustrate that this method of estimating polymer melt surface tensions is superior to existing empirical or semi-empirical methods. 

The theory of inhomogeneous associating fluids evidently has benefited from the developments available for bulk associating models. The theory of... 

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