Pair correlation function fluid models

However, before proceeding with the description of simulation data, we would like to comment the theoretical background. Similarly to the previous example, in order to obtain the pair correlation function of matrix spheres we solve the common Ornstein-Zernike equation complemented by the PY closure. Next, we would like to consider the adsorption of a hard sphere fluid in a microporous environment provided by a disordered matrix of permeable species. The fluid to be adsorbed is considered at density pj = pj-Of. The equilibrium between an adsorbed fluid and its bulk counterpart (i.e., in the absence of the matrix) occurs at constant chemical potential. However, in the theoretical procedure we need to choose the value for the fluid density first, and calculate the chemical potential afterwards. The ROZ equations, (22) and (23), are applied to decribe the fluid-matrix and fluid-fluid correlations. These correlations are considered by using the PY closure, such that the ROZ equations take the Madden-Glandt form as in the previous example. The structural properties in terms of the pair correlation functions (the fluid-matrix function is of special interest for models with permeabihty) cannot represent the only issue to investigate. Moreover, to perform comparisons of the structure under different conditions we need to calculate the adsorption isotherms pf jSpf). The chemical potential of a... 

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... 

There have been several attempts to treat the RPM on an analogous basis. To this end, Leote de Carvalho and Evans [281] used the GMSA, Lee and Fisher [283] used the GDH, and Weiss and Schroer [239,280,284] examined several DH-based models that approximate the direct correlation function or the pair correlation function. In some cases the results depended significantly on details of the approximations. In total, none of these studies, whatever theory used, gave evidence that Nqi may be significantly smaller than observed for simple nonionic fluids. Rather the opposite seems to be the case. From this perspective, the experimental results for some ionic systems remain a mystery. 

There have been a number of modeling efforts that employ the concept of clustering in supercritical fluid solutions. Debenedetti (22) has used a fluctuation analysis to estimate what might be described as a cluster size or aggregation number from the solute infinite dilution partial molar volumes. These calculations indicate the possible formation of very large clusters in the region of highest solvent compressibility, which is near the critical point. Recently, Lee and coworkers have calculated pair correlation functions of solutes in supercritical fluid solutions ( ). Their results are also consistent with the cluster theory. 

We propose the study of Lennard-Jones (LJ) mixtures that simulate the carbon dioxide-naphthalene system. The LJ fluid is used only as a model, as real CO2 and CioHg are far from LJ particles. The rationale is that supercritical solubility enhancement is common to all fluids exhibiting critical behavior, irrespective of their specific intermolecular forces. Study of simpler models will bring out the salient features without the complications of details. The accurate HMSA integral equation (Ifl) is employed to calculate the pair correlation functions at various conditions characteristic of supercritical solutions. In closely related work reported elsewhere (Pfund, D. M. Lee, L. L. Cochran, H. D. Int. J. Thermophvs. in press and Fluid Phase Equilib. in preparation) we have explored methods of determining chemical potentials in solutions from molecular distribution functions. 

The so-called product reactant Ornstein-Zernike approach (PROZA) for these systems was developed by Kalyuzhnyi, Stell, Blum, and others [46-54], The theory is based on Wertheim s multidensity Ornstein-Zernike (WOZ) integral equation formalism [55] and yields the monomer-monomer pair correlation functions, from which the thermodynamic properties of the model fluid can be obtained. Based on the MSA closure an analytical theory has been developed which yields good agreement with computer simulations for short polyelectrolyte chains [44, 56], The theory has been recently compared with experimental data for the osmotic pressure by Zhang and coworkers [57], In the present paper we also show some preliminary results for an extension of this model in which the solvent is now treated explicitly as a separate species. In this first calculation the solvent molecules are modelled as two fused charged hard spheres of unequal radii as shown in Fig. 1 [45],... 

The correlation functions provide an alternate route to the equilibrium properties of classical fluids. In particular, the two-particle correlation function of a system with a pairwise additive potential determines all of its thermodynamic properties. It also determines the compressibility of systems with even more complex three-body and higher-order interactions. The pair correlation functions are easier to approximate than the PFs to which they are related they can also be obtained, in principle, from x-ray or neutron diffraction experiments. This provides a useful perspective of fluid stmcture, and enables Hamiltonian models and approximations for the equilibrium stmcture of fluids and solutions to be tested by direct comparison with the experimentally determined correlation functions. We discuss the basic relations for the correlation functions in the canonical and grand canonical ensembles before considering applications to model systems. 

Chiew and Glandt46 modeled the structure of a dispersion of identical spheres as an equilibrium hard sphere fluid. They used pair correlation functions to compute the contributions of pairs of spheres to the effective thermal conductivity of the dispersions. Their results, derived for arbitrary values of the conductivity of the two phases, reduce to the following equation for the case of gas bubbles in electrolyte ... 

We shall now use topological reductions to manipulate the cluster expansion for the pair correlation function of such a model fluid. 

Throughout this chapter, we use very simple models to introduce a number of concepts that are frequently used in the context of the theory of fluids. Examples are the pair-correlation function, direct and indirect correlations, potential of average force, nonadditivity of the triplet correlation function, and so on. All these will be introduced again in Chapter 5. However, it is easier to grasp these concepts within the simple models. This should facilitate understanding them in more complex systems. 

When the smoothed or nonlocal density approximation (or NL-DFT model) is used, the weighting function is chosen so that the hard-sphere direct pair-correlation function is well described for the uniform fluid over a wide range of densities. One example of such a weighting function is the model proposed by Tarazona [69], which uses the Percus-Yevick theory for approximating the correlation function over a wide range of density. In this case, the weighting function is expanded as a power series of the smoothed density. The use of a smoothed density in NL-DFT provides an oscillating density profile expected of a fluid adjacent to a sohd surface, the existence of which is corroborated by molecular simulation results [17,18]. 

More modem approaches borrow ideas from the liquid state theory of small molecule fluids to develop a theory for polymers. The most popular of these is the polymer reference interaction site model (PRISM) theory " which is based on the RISM theory of Chandler and Andersen. More recent studies include the Kirkwood hierarchy, the Bom-Green-Yvon hierarchy, and the perturbation density functional theory of Kierlik and Rosinbeig. The latter is based on the thermodynamic perturbation theory of Wertheim " where the polymeric system is composed of very sticky spheres that assemble to form chains. For polymer melts all these liquid state approaches are in quantitative agreement with simulations for the pair correlation functions in short chain fluids. With the exception of the PRISM theory, these liquid state theories are in their infancy, and have not been applied to realistic models of polymers. 

Statistical thermodynamics already provide an excellent framework to describe and model equilibrium properties of molecular systems. Molecular interactions, summarized for instance in terms of a potential of mean force, determine correlation functions and all thermodynamic properties. The (pair) correlation function represents the material structure which can be determined by scattering experiments via the scattering function. AU macroscopic properties of pure and mixed fluid systems can be derived by weU-estabhshed multiphase thermodynamics. In contrast, a similar framework for particulate building blocks only partly exists and needs to be developed much further. Besides equibbrium properties, nonequilibrium effects are particularly important in most particulate systems and need to be included in a comprehensive and complete picture. We will come back to these aspects in Section 4. 

In the case of a single test particle B in a fluid of molecules M, the effective one-dimensional potential f (R) is — fcrln[R gBM(f )]. where 0bm( ) is th radial distribution function of the solvent molecules around the test particle. In this chapter it will be assumed that 0bm( )> equilibrium property, is a known quantity and the aim is to develop a theory of diffusion of B in which the only input is bm( )> particle masses, temp>erature, and solvent density Pm- The friction of the particles M and B will be taken to be frequency indep>endent, and this should restrict the model to the case where > Wm, although the results will be tested in Section III B for self-diffusion. Instead of using a temporal cutoff of the force correlation function as did Kirkwood, a spatial cutoff of the forces arising from pair interactions will be invoked at the transition state Rj of i (R). While this is a natural choice because the mean effective force is zero at Rj, it will preclude contributions from beyond the first solvation shell. For a stationary stochastic process Eq. (3.1) can then be... 

The RSOZ equations Eqs. (7.41) and (7.42) still involve both the total and the direct correlation functions. Therefore, appropriate closure expressions relating the correlation functions to the pair potentials are needed to calculate the correlation functions at given densities and temperatures. Typically, one uses standard closure expreasions familiar from bulk liquid state theory [30]. One should note, however, that the performance of these closures for disordered fluids can clearly not be taken for granted. Instead, they need to be tested for each new model system under consideration. 

The task of extending the pair distribution function based on theoretical considerations has been addressed many times (Verlet 1968 Galam and Hansen 1976 Jolly, Freasier, and Bearman 1976 Ceperley and Chester 1977 Dixon and Hutchinson 1977 Foiles, Ashcroft, and Reatto 1984). Often the goal has been to study the correlation functions themselves or to calculate structure factors, not to obtain properties. Here we will emphasize applications aimed toward representing thermodynamic properties of molecular fluids that do not have conformational variations. While many publications have been confined to atomic model fluids, such as LJ particles, we focus here on applications for real molecular systems and their mixtures. 

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