Fluid pair theory

Sec. Ill is concerned with the description of models with directional associative forces, introduced by Wertheim. Singlet and pair theories for these models are presented. However, the main part of this section describes the density functional methodology and shows its application in the studies of adsorption of associating fluids on partially permeable walls. In addition, the application of the density functional method in investigations of wettability of associating fluids on solid surfaces and of capillary condensation in slit-like pores is presented. 

A set of equations (15)-(17) represents the background of the so-called second-order or pair theory. If these equations are supplemented by an approximate relation between direct and pair correlation functions the problem becomes complete. Its numerical solution provides not only the density profile but also the pair correlation functions for a nonuniform fluid [55-58]. In the majority of previous studies of inhomogeneous simple fluids, the inhomogeneous Percus-Yevick approximation (PY2) has been used. It reads... 

Pi(l) and the two-partiele partial eorrelation funetions tend to their bulk eounterparts for assoeiating fluid. The teehnique of the solution of this set is essentially the same as for the system of equations of the pair theory dis-eussed in See. II. 

Now we turn our attention to the results obtained from the pair theory for the system of assoeiating Lennard-Jones fluid in eontaet with a hard wall. The nonassoeiative part of the interpartiele potential is given by Eq. (87), whereas the assoeiative interaetion is given by Eq. (60), with d = 0.45 and <3 = 0.1. The diameter of fluid partieles a is taken as the unit of length. 

The proper treatment of ionic fluids at low T by appropriate pairing theories is a long-standing concern in standard ionic solution theory which, in the light of theories for ionic criticality, has received considerable new impetus. Pairing theories combine statistical-mechanical theory with a chemical model of ion pair association. The statistical-mechanical treatment is restricted to terms of the Mayer/-functions which are linear in / , while the higher terms are taken care by the mass action law... 

At the present stage, only MSA- and DH-based theories have been widely applied. From these theories, and their comparison with MC data, there is now a quite consistent picture of the major mechanisms driving these ionic phase transitions, although there are still difficulties to predict the critical point on a quantitative level. The results point toward the crucial importance of ion pairing and of interactions between ion pairs and the remainder of the ionic fluid. Deficits in predicting the critical point quantitatively seem to be closely connected to a substantial overestimate of ionicity in almost all pairing theories. 

We refer to Eq. (92) as a second-order or a pair theory. It is a pair theory in the sense that two fluid particles are considered. The HAB scheme is a singlet theory in the sense that only one fluid particle is considered. 

In coUoid-polymer mixtures there is no direct attraction between the colloids but the attraction enters through repulsion. Attraction is caused by overlap of depletion layers rather than through direct pair interactions. For q > 0.15, multiple overlap of depletion layers (see Fig. 3.8) occurs which is expected to promote the occurrence of a colloidal liquid. The critical point at which the range of attraction is just sufficient for a stable liquid state is termed the critical end point [25]. For a shorter range of attraction no critical point borders a stable fluid phase. Theory and computer simulations point out that the critical end point generally corresponds to a range of attraction close to 1/3 of the particle diameter [25]. 

Much progress in recent years has been made towards understanding the fluid state via the "pair theory" of fluids wherein the series in eq. (2) is truncated after the first term (see, for example, reference ( l)),i.e.,... 

Kirkwood derived an analogous equation that also relates two- and tlnee-particle correlation fiinctions but an approximation is necessary to uncouple them. The superposition approximation mentioned earlier is one such approximation, but unfortunately it is not very accurate. It is equivalent to the assumption that the potential of average force of tlnee or more particles is pairwise additive, which is not the case even if the total potential is pair decomposable. The YBG equation for n = 1, however, is a convenient starting point for perturbation theories of inliomogeneous fluids in an external field. 

Theories based on the solution to integral equations for the pair correlation fiinctions are now well developed and widely employed in numerical and analytic studies of simple fluids [6]. Furtlier improvements for simple fluids would require better approximations for the bridge fiinctions B(r). It has been suggested that these fiinctions can be scaled to the same fiinctional fomi for different potentials. The extension of integral equation theories to molecular fluids was first accomplished by Chandler and Andersen [30] through the introduction of the site-site direct correlation fiinction c r) between atoms in each molecule and a site-site Omstein-Zemike relation called the reference interaction site... 

In perturbation theories of fluids, the pair total potential is divided into a reference part and a perturbation... 

In the theory of the liquid state, the hard-sphere model plays an important role. For hard spheres, the pair interaction potential V r) = qo for r < J, where d is the particle diameter, whereas V(r) = 0 for r s d. The stmcture of a simple fluid, such as argon, is very similar to that of a hard-sphere fluid. Hard-sphere atoms do, of course, not exist. Certain model colloids, however, come very close to hard-sphere behaviour. These systems have been studied in much detail and some results will be quoted below. 

Twin-fluid atomizer Twisted pair cable Twitchell splitting Twitchell s reagents Two-film theory... 

R. Kjellander, S. Sarman. A study of anisotropic pair distribution theories for Lennard-Jones fluids in narrow slits. II. Pair correlations and solvation forces. Mol Phys 74 665-688, 1991. 

B. Application of the singlet-level and pair-level theories for fluids with spherically symmetric associative interactions... 

The present chapter is organized as follows. We focus first on a simple model of a nonuniform associating fluid with spherically symmetric associative forces between species. This model serves us to demonstrate the application of so-called first-order (singlet) and second-order (pair) integral equations for the density profile. Some examples of the solution of these equations for associating fluids in contact with structureless and crystalline solid surfaces are presented. Then we discuss one version of the density functional theory for a model of associating hard spheres. All aforementioned issues are discussed in Sec. II. 

B. Application of the Singlet-level and Pair-level Theories for Fluids with Spherically Symmetric Associative Interactions in Contact with Surfaces... 

We should mention here one of the important limitations of the singlet level theory, regardless of the closure applied. This approach may not be used when the interaction potential between a pair of fluid molecules depends on their location with respect to the surface. Several experiments and theoretical studies have pointed out the importance of surface-mediated [1,87] three-body forces between fluid particles for fluid properties at a solid surface. It is known that the depth of the van der Waals potential is significantly lower for a pair of particles located in the first adsorbed layer. In... 

Far from the surface, the theory reduces to the PY theory for the bulk pair correlation functions. As we have noted above, the PY theory for bulk pair correlation functions does not provide an adequate description of the thermodynamic properties of the bulk fluid. To eliminate this deficiency, a more sophisticated approximation, e.g., the SSEMSA, should be used. 

These two equations represent the assoeiative analogue of Eq. (14) for the partial one-partiele eavity funetion. It is eonvenient to use equivalent equations eontaining the inhomogeneous total pair eorrelation funetions. Similarly to the theory of inhomogeneous nonassoeiating fluids, this equiva-lenee is established by using the multidensity Ornstein-Zernike equation (68). Eq. (14) then reduees to [35]... 

Let us proceed with the description of the results from theory and simulation. First, consider the case of a narrow barrier, w = 0.5, and discuss the pair distribution functions (pdfs) of fluid species with respect to a matrix particle, gfm r). This pdf has been a main focus of previous statistical mechanical investigations of simple fluids in contact with an individual permeable barrier via integral equations and density functional methodology [49-52]. 

FIG. 5 Adsorption isotherms for a hard sphere fluid from the ROZ-PY and ROZ-HNC theory (solid and dashed lines, respectively) and GCMC simulations (symbols). Three pairs of curves from top to bottom correspond to matrix packing fraction = 0.052, 0.126, and 0.25, respectively. The matrix in simulations has been made of four beads (m = M = 4). 

We conclude, from the results given above, that both the ROZ-PY and ROZ-HNC theories are sufficiently successful for the description of the pair distribution functions of fluid particles in different disordered matrices. It seems that at a low adsorbed density the PY closure is preferable, whereas... 

Equilibrium Theory of Fluid Structure. In all the theoretical work reported herein, we assume that the particles Interact with pair additive forces whose pair potentials can be approximated by... 

Consider a fluid of molecules Interacting with pair additive, centrally symmetric forces In the presence of an external field and assume that the colllslonal contribution to the equation of motion for the singlet distribution function Is given by Enskog s theory. In a multicomponent fluid, the distribution function fi(r,Vj,t) of a particle of type 1 at position r, with velocity Vj at time t obeys the equation of change (Z)... 

A quantity of central importance in the study of uniform liquids is the pair correlation function, g r), which is the probability (relative to an ideal gas) of finding a particle at position r given that there is a particle at the origin. All other structural and thermodynamic properties can be obtained from a knowledge of g r). The calculation of g r) for various fluids is one of the long-standing problems in liquid state theory, and several accurate approaches exist. These theories can also be used to obtain the density profile of a fluid at a surface. 

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