Pair correlation functions work function

Although more complex pair-correlation functions are available, the Debye-Huckel expression is adequate for our present purpose. It is valid when the work required to bring the reactants... 

Radial distribution functions can be determined experimentally using diffraction (i.e., interference) experiments. X-rays or neutrons can be used. If one knows the pair correlation function g ir) for each atom, one can work out the short-range structure in a liquid. The question is then how does one find gj ir) ... 

The material given here then shows how measurement of the diffraction of X-rays (also neutrons, see later discussion) gives the pair correlation function, g p. It can give much more. As shown in Section 3.11, the determination of ga p allows one to calculate a number of properties of the liquid or solution. A property calculated from pair correlation functions does not involve an assumed modeling theory. Instead, the experimentally determined pair correlation functions are the basis of the calculated properties. It is as though one had worked out the structure first and then used the knowledge of that structure to calculate the properties. Is this a higher level approach... 

We implement a modified version of the reconstruction method developed in a previous work to model two porous carbons produced by the pyrolysis of saccharose and subsequent heat treatment at two different temperatures. We use the Monte Carlo g(r) method to obtain the pair correlation functions of the two materials. We then use the resulting pair correlation functions as target functions in our reconstruction method. Our models present structural features that are missing in the slit-pore model. Structural analyses of our resulting configurations are useful to characterize the materials that we model. 

In this work, we explore the use of the pair correlation function as the target function in our reconstruction method [4]. This speeds up the simulations, allowing us to construct models in much larger simulation boxes. We build models for two saccharose-based carbons treated at different temperatures. We compare the exact pore size distributions and perform Grand Canonical Monte Carlo (GCMC) simulations of nitrogen at 77 K in the resulting models. 

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. 

For the RPM, the first term cancels, and the second is dominant. In principle, c (r ) defined in this way may also vary locally. Third, one may also calculate C2g by working backward from the Laplace transformation of the pair correlation function, as done by Lee and Fisher [94] for the pair correlation function of their generalized Debye-Hiickel theory (GDH). 

However, the pair correlation function as well as the potential of average force are finite at this limit. We can think of WAA(R) in the limit of pA — 0 as the work required to bring two A s from infinite separation to the distance R in a pure solvent B at constant Tand V(or T, P depending on the ensemble we use). 

Having information on the Gy, one can compute the thermodynamic quantities. However, the original KB theory could have been used only in rare cases where Gy could be obtained from theoretical work. In principle, having an approximate theory for computing the various pair correlation functions gij(R), it is possible to evaluate the integrals Gy and then compute the thermodynamic quantities through the KB theory. Comparison between the thermodynamic quantities thus obtained, and the corresponding experimental data, could serve as a test of the theory that provides the pair correlation functions. 

The history of the search for an integral equation for the pair correlation function is quite long. It probably started with Kirkwood (1935), followed by Yvon (1935, 1958), Born and Green (1946), and many others. For a summary of these efforts, see Hill (1956), Fisher (1964), Rushbrooke (1968), Munster (1969), and Hansen and McDonald (1976). Most of the earlier works used the superposition approximation to obtain an integral equation for the pair correlation function. It was in 1958 that Percus and Yevick developed an integral equation that did not include explicitly the assumption of superposition, i.e., pairwise additivity of the higher order potentials of mean force. The Percus-Yevick (PY) equation was found most useful in the study of both pure liquids as well as mixtures of liquids. 

In the site-site representation, the virial pressure involves higher spherical harmonic coefficients of the site-site pair correlation functions. The difficulties involved in the standard virial route to the pressure in polymer systems have been well illustrated in recent work. ... 

The Gaussian probability (18) differs from the expression given in (49b) in ref. 2. In ref, 2 the probability distribution by the quadratic terms in (4) of this paper is used to represent T,. p. When this form of is taken to calulate the cell-pair correlation function (see the next section) as a function of p, the series for the cell-pair correlation function contains additional terms (ref. 2) which are not present in this work. 

In Section 3 the formation of the SEFS spectrum was described in the approximation of single scattering of secondary electrons by the nearest atoms to the ionized atom. In the framework of this approximation, the local atomic surroundings of the ionized atom are entirely described by the atomic pair correlation function g(r), which determines the probability of detecting an atom of a specific chemical species at a distance r from the ionized (central) atom (also of a specific chemical species). In the present work we restrict our consideration to one-component systems thus, the PCF g(r) has no indices denoting the chemical species. 

Compared to the effort devoted to experimental work, theoretical studies of the partial molar volume have been very limited [61, 62]. The computer simulations for the partial molar volume were started a few years ago by several researchers, but attempts are still limited. As usual, our goal is to develop a statistical-mechanical theory for calculating the partial molar volume of peptides and proteins. The Kirkwood-Buff (K-B) theory [63] provides a general framework for evaluating thermodynamic quantities of a liquid mixture, including the partial molar volume, in term of the density pair correlation functions, or equivalently, the direct correlation functions. The RISM theory is the most reliable tool for calculating these correlation functions when the solute molecule comprises many atoms and has a complicated conformation. 

A more detailed study of the pair correlation function of this model was undertaken by Urbic et al. (2003). This work was also reviewed by Dill et al. (2005). 

It is known from both experiments and theoretical work that the range of the pair correlation functions gap R) is only a few molecular diameters. This means that there exists a correlation radius Rcorr, such that for R > RcoRRy gap(R) is nearly unity. Alternatively, there is no correlation at distances beyond the correlation radius. 

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. 

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