Pair correlation function fluid properties

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. 

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... 

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. 

In the study of structural properties of fluids of particles interacting through orientation-dependent potentials, it is convenient to decompose the full pair-correlation function A(l,2) = h ri2,Qi, Q2) ... 

The primary quantity of interest in studies of fluids is their structure from which all other properties can be derived [19]. All necessary information is provided by the full pair correlation function g(l. 2) whose complete experimental determination is however practically impossible. The usual way is to characterize the structure of fluids by the complete set of site-site correlation functions f/ij,... 

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 above analysis demonstrates the importance of the pair correlation function in estimation of the thermodynamic properties of simple liquids. In the following section, the properties of the simplest fluid, namely, one based on non-interacting hard spheres, are developed on the basis of the relationships presented in this section. 

All of the approximate DFTs of melting assume that the properties of the uniform (liquid) phase are well characterized, and, in particular, that the free energy and pair correlation function of the liquid are known. The crystalline phase is viewed as a highly inhomogeneous fluid with a spatially varying density p(r) having the symmetry of the crystalline... 

In this section, we review some of the important formal results in the statistical mechanics of interaction site fluids. These results provide the basis for many of the approximate theories that will be described in Section III, and the calculation of correlation functions to describe the microscopic structure of fluids. We begin with a short review of the theory of the pair correlation function based upon cluster expansions. Although this material is featured in a number of other review articles, we have chosen to include a short account here so that the present article can be reasonably self-contained. Cluster expansion techniques have played an important part in the development of theories of interaction site fluids, and in order to fully grasp the significance of these developments, it is necessary to make contact with the results derived earlier for simple fluids. We will first describe the general cluster expansion theory for fluids, which is directly applicable to rigid nonspherical molecules by a simple addition of orientational coordinates. Next we will focus on the site-site correlation functions and describe the interaction site cluster expansion. After this, we review the calculation of thermodynamic properties from the correlation functions, and then we consider the calculation of the dielectric constant and the Kirkwood orientational correlation parameters. 

The structure of a fluid is characterized by the spatial and orientational correlations between atoms and molecules determined through x-ray and neutron diffraction experiments. Examples are the atomic pair correlation functions (g, gj j ) in liquid water. An important feature of these correlation functions is that the thermodynamic properties of a... 

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. 

The pair correlation function g(xi,X2) is another quantity of interest because it is related to the X-ray and neutron scattering properties of the fluid and to many other measurable properties such as the dielectric constant. It is the basic quantity one uses to discuss the structure of a fluid. For atomic fluids, it is defined so that... 

The blip function theory was originally developed as a way of calculating the thermodynamic properties and the pair correlation function of a fluid whose potential is continuous, positive, and repulsive. For an atomic liquid. 

The central idea of this review is that the technique of topological reduction is a powerful one for the formal manipulation of cluster series and for generating useful approximate theoretical expressions for the thermodynamic properties and pair correlation functions of fluids. We have described just enough of the theory of graphs to have a meaningful discussion of topological reduction and have then discussed a number of types of theories in which topological reductions are applied. A detailed discussion of the approximate theories that result is outside the scope of this article, and the reader is referred to the literature cited in the text. 

At the bases of the second basic assumption made, e.g., that the fluids behave classically, there is the knowledge that the quantum effects in the thermodynamic properties are usually small, and can be calculated readily to the first approximation. For the structural properties (e.g., pair correlation function, structure factors), no detailed estimates are available for molecular liquids, while for atomic liquids the relevant theoretical expressions for the quantum corrections are available in the literature. 

One clever approach to obtaining better convergence is to include asymptotic properties of the pair correlation functions (Lebowitz and Percus 1963). In particular, exact asymptotic expressions have been obtained by Attard and coworkers (Attard 1990 Attard et al. 1991), such as for dipolar fluids. Other work has extended simulation results for a system with a truncated potential to give those for the full potential (Lado 1964). The effects on pair distribution functions of potential truncations are important. 

During World War II, little or no work was done on solution theory, but after the war, activity began again. Now, the emphasis of many theories began to fall on the properties and usefulness of molecular distribution functions, in particular the pair correlation function. This was due, in part, I believe, to the thesis of Jan de Boer (De Boer 1940,1949). As an aside, I once asked J. E. Mayer why he used the canonical ensemble in his early work on statistical mechanics and the grand ensemble in his later works. He replied, Oh, I switched after I read de Boer s thesis and saw how easy the grand ensemble made things. De Boer s work was for pure fluids, not solutions, and other authors, in particular John G. Kirkwood (Kirkwood 1935), also developed the correlation function method. 

Structural Properties. A good way to define the concept of "Structure of fluids is through the pair correlation function g j (r)(17). Recently Blum and H ye(18), found the analytical solution for the Laplace transform of the correlation function g j(r)... 

Now, consider mixtures of A and B (with Gaa ctbb) at different compositions but constant p. If we study the dependence of, say, gAsiB) on the mole fraction Xa, we find that 2ii Xa, gAB R) behaves as in the case of a high-density fluid, whereas at Xa 0, we observe the behavior of the low-density fluid. In order to stress those effects specific to the properties of the mixtures, it is advisable to study the behavior of the pair correlation function when the total volume density is constant. The latter is defined as follows. In a one-component system of particles with effective diameter cr, the ratio of the volume occupied by the particles to the total volume of the system is... 

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 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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