Fluids inhomogeneous

Henderson, D. Fundamentals of Inhomogeneous Fluids. Marcel Dekker, New York (1992). 

P. Attard. Spherically inhomogeneous fluids. II. Hard-sphere solute in a hard-sphere solvent. J Chem Phys 97 3083-3089, 1989. 

Integral equations have been developed for inhomogeneous fluids. One such integral equation is that of Henderson, Abraham and Barker (HAB) [88] who assumed the OZ equation for a mixture and regarded the surface as a giant particle. For planar geometry they obtained... 

Recently, the HAB approach plus the MV closure has been applied both to hard spheres near a single hard wall [24,25] and in a slit formed by two hard walls. Some results [99] for the latter system are compared with simulation results in Fig. 7. The results obtained from the HAB equation with the HNC and PY closures are not very satisfactory. However, if the MV closure is used, the results are quite good. There have been a few apphcations of the HAB equation to inhomogeneous fluids with attractive interactions. The results have not been very good. The fault hes with the closure used and not Eq. (78). A better closure is needed. Perhaps the DHH closure [27,28] would yield good results, but it has never been tried. 

Integral equations provide a satisfactory formalism for the study of homogeneous and inhomogeneous fluids. If the usual OZ equation is used, the best results are obtained from semiempirical closures such as the MV and DHH closures. However, this empirical element can be avoided by using integral equations that involve higher-order distribution functions, but at the cost of some computational complexity. 

D. Henderson. In D. Henderson, ed. Fundamentals of Inhomogeneous Fluids. New York Marcel Dekker, 1992. 

Eq. (5) is useful when analyzing different approximations in the theory of inhomogeneous fluids. In particular, if all the terms involving third- and higher-order correlations in the right-hand side of Eq. (5) are neglected, and if Pi(ro))P2( o)i )Pv( o) are chosen as the densities of species for a uniform system at temperature T and the chemical potentials p,, the singlet hypemetted chain equation (HNCl) [50] results... 

The multidensity Ornstein-Zernike equation (70) and the self-consistency relation (71) actually describe a nonuniform system. To solve these equations numerically for inhomogeneous fluids one needs only an appropriate generalization of the Lowett-Mou-Buff-Wertheim equation (14). Such a generalization, employing the concept of the partial correlation function has been considered in Refs. 34,35. 

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]. 

Equation 3 is exact for fluids obeying Equations 1 and 2. However, in order to compute the density n(r) from the YBG equation one must know the relationship between density distribution and the pair correlation function of Inhomogeneous fluid. Such a relationship Is not available in general. However, an approximation introduced by Fischer and Methfessel (1.) has been shown to give fairly accurate predictions of the density... 

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 attractive feature of LADM Is that once the fluid structure Is known (e.g., by solution of the YBG equations given In the previous section or by a computer simulation) then theoretical or empirical formulas for the transport coefficients of homogeneous fluids can be used to predict flow and transport In Inhomogeneous fluid. For diffusion and Couette flow In planar pores LADM turns out to be a surprisingly good approximation, as will be shown In a later section. 

As expected from continuum theory, the friction and diffusion coefficients are replaced In Inhomogeneous fluid by tensors whose symmetry reflects that of the Inhomogeneous media. 

In Figure 10, we present flow velocity predictions of the high density approximation, Equations 32 - 33, 38 and 39, of Davis extension of Enskog s theory to flow In strongly Inhomogeneous fluids (1 L). The velocity profile predicted In this way Is also plotted In Figure 10. The predicted profile, the simulated profile, and the profile predicted from the LADM are quite similar. 

A final comment has to do with the concept of effective viscosity In strongly Inhomogeneous fluids. For these systems the definition of the effective viscosity depends on the type flow, hence different effective viscosities will be measured for different flow situations In the same system with the same density profile. Therefore, the effective viscosity Is a concept of limited value and measurements of this quantity do not provide much information about the effects of density structure on the flow behavior. 

In accordance with Equation (2.338) the determination of the figure of fiuid equilibrium is reduced to the following problem we have to find such a surface of the fluid, S(x,y,z), that its partial derivatives should be proportional to the corresponding components of the acting force. As we pointed out, when a fluid rotates uniformly around the same axis the total force can be represented as a sum of the attraction and centrifugal forces, and the former depends on the shape of the fluid mass in a rather complicated way. Besides, in the case of an inhomogeneous fluid the potential of the attraction field depends on the distribution of a density of a fiuid and for this reason this problem becomes even more complicated. 

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. 

Both a uniform bulk fluid and an inhomogeneous fluid were simulated. The latter was in the form of a slit pore, terminated in the -direction by uniform Lennard-Jones walls. The distance between the walls for a given number of atoms was chosen so that the uniform density in the center of the cell was equal to the nominal bulk density. The effective width of the slit pore used to calculate the volume of the subsystem was taken as the region where the density was nonzero. For the bulk fluid in all directions, and for the slit pore in the lateral directions, periodic boundary conditions and the minimum image convention were used. 

The Laplace-Young equation refers to a spherical phase boundary known as the surface of tension which is located a distance from the center of the drop. Here the surface tension is a minimum and additional, curvature dependent, terms vanish (j ). The molecular origin of the difficulties, discussed in the introduction, associated with R can be seen in the definition of the local pressure. The pressure tensor of a spherically symmetric inhomogeneous fluid may be computed through an integration of the one and two particle density distributions. 

---