Associating fluids density functional theory

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. 

Modern theory of associative fluids is based on the combination of the activity and density expansions for the description of the equilibrium properties. The activity expansions are used to describe the clusterization effects caused by the strongly attractive part of the interparticle interactions. The density expansions are used to treat the contributions of the conventional nonassociative part of interactions. The diagram analysis of these expansions for pair distribution functions leads to the so-called multidensity integral equation approach in the theory of associative fluids. The AMSA theory represents the two-density version of the traditional MSA theory [4, 5] and will be used here for the treatment of ion association in the ionic fluids. 

RISM reference interacting site model SAFT statistical associating fluid theory SOFT site density functional theory SPC simple point charged model (for water)... 

Here, we will describe in more detail the interesting example of an application of density functional theory to the very comphcated system, namely the four-bonding Lennard-Jones associating fluid confined in slitlike pore [255]. The fluid is confined in the slitlike pore of width H. The total external field is the sum of contributions from two pore walls, located at z = 0 and z = H ... 

Gloor, G.J. Jackson, G. Bias, F.J. del Rio, E.M. de Miguel, E. (2004). An accurate density functional theory for the vapor-liquid interface of associating chain molecules based on the statistical associating fluid theory for potentials of variable range. ]. Chem. Phys. 121,12740-12759. 

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. 

Second-Order Integral Equations for Associating Fluids As mentioned above in Sec. II A, the second-order theory consists of simultaneous evaluation of the one-particle (density profile) and two-particle distribution functions. Consequently, the theory yields a much more detailed description of the interfacial phenomena. In the case of confined simple fluids, the PY2 and HNC2 approaches are able to describe surface phase transitions, such as wetting and layering transitions, in particular see, e.g.. Ref. 84. 

We apply the singlet theory for the density profile by using Eqs. (101) and (103) to describe the behavior of associating fluids close to a crystalline surface [120-122], First, we solve the multidensity OZ equation with the Percus-Yevick closure for the bulk partial correlation functions, and next calculate the total correlation function via Eq. (68) and the direct correlation function from Eq. (69). The bulk total direct correlation function is used next as an input to the singlet Percus-Yevick or singlet hypernetted chain equation, (6) or (7), to obtain the density profiles. The same approach can be used to study adsorption on crystalline surfaces as well as in pores with walls of crystalline symmetry. 

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

We would like to recall that Xa p) is the fraction of molecules not bonded at an associative site now it is a function of the averaged density p(r). A generalization of the perturbational theory allows us to define Xa p) similar to the case of bulk associating fluids. Namely... 

The remaining chapters in this book are organized as follows. Chapter 2 provides a brief introduction to the mesoscale description of polydisperse systems. There, the mathematical definition of a number-density function (NDF) formulated in terms of different choices for the internal coordinates is described, followed by an introduction to population-balance equations (PBE) in their various forms. Chapter 2 concludes with a short discussion on the differences between the moment-transport equations associated with the PBE and those arising due to ensemble averaging in turbulence theory. This difference is very important, and the reader should keep in mind that at the mesoscale level the microscale turbulence appears in the form of correlations for fluid drag, mass transfer, etc., and thus the mesoscale models can have non-turbulent solutions even when the microscale flow is turbulent (i.e. turbulent wakes behind individual particles). Thus, when dealing with turbulence models for mesoscale flows, a separate ensemble-averaging procedure must be applied to the moment-transport equations of the PBE (or to the PBE itself). In this book, we are primarily... 

Figure 3. The equilibrium vapor and liquid densities of an associating fluid with one square-well bonding site. The circles are data from RCMC-Gibbs ensemble simulations, and the lines are calculations from three different implementations of a theory for associating fluids. The solid line uses exact values of the reference fluid radial distribution function the dashed and long dashed-short dashed lines use the WCA and modified WCA approximations to the radial distribution function, respectively. (Reprinted with permission from Muller et al. [43]. Copyright 1995 American Institute of Physics.)...

In Section V it was shown how Wertheim s multi-density approach could be used to develop an equation for associating fluids with an arbitrary number of association sites provided a number of assumptions were satisfied. The simplicity of the TPTl solution results from the fact that the solution is that of an effective two-body problem. Only one bond at a time is considered. This allows the theory to be written in terms of pair correlation functions only, as well as obtain analytical solutions for the bond volume. Moving beyond TPTl is defined as considering irreducible graphs that contain more than one association bond. 

This strategy is common in atomic fluid theory at low to moderate densities, and for Coulombic systems, and corresponds to the reference idea ubiquitous in liquid state theory [5], The purely hard core problem is treated using the accurate [27] PY closure. (2) The construction of a closure approximation for the tail part of the potential is subject to the constraint of exactly describing the weak coupling limit. In physical terms, for fractal-like interpenetrating molecules these indirect processes may strongly couple the direct correlation functions associated with those pairs of sites which are in simultaneous contact The number of such two-molecule pair contacts. Me. scales with N as [23] ... 

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