Closure approximation hypernetted chain

The general equation can be further reduced to the case of infinite dilution limit, a binary mixmre, ionic solutions, and so on. These equations are supplemented by closure relations such as the Percus-Yevick (PY) and hypernetted chain (HNC) approximations. 

One should perhaps mention some other closures that are discussed in the literature. One possibility is to combine the PY approximation for the hard core part of the potential and then use the HNC approximation to compute the corrections due to the attractive forces. Such an approach is called the reference hypernetted chain or RHNC approximation [48,49]. Recently, some new closures for a mixture of hard spheres have been proposed. These include one by Rogers and Young [50] (RY) and the Martynov-Sarkisov [51] (MS) closure as modified by Ballone, Pastore, Galli and Gazzillo [52] (BPGG). The RY and MS/BPGG closure relations take the forms... 

To solve the replica OZ equations, they must be completed by closure relations. Several closures have been tested against computer simulations for various models of fluids adsorbed in disordered porous media. In particular, common Percus-Yevick (PY) and hypernetted chain approximations have been applied [20]. Eq. (21) for the matrix correlations can be solved using any approximation. However, it has been shown by Given and Stell [17-19] that the PY closure for the fluid-fluid correlations simplifies the ROZ equation, the blocking effects of the matrix structure are neglected in this... 

Other closure relations are the hypernetted chain approximation (HNQ, defined by [25,32]... 

The hypernetted chain (HNC) approximation follows from the closure... 

One more relation is required to achieve closure, i.e., to determine the two types of correlation functions. The most commonly used relations are the Percus-Yevick (PY) and the hypernetted chain (HNC) approximations [47-49]. From graph or diagram expansion of the total correlation function in powers of the density n(r) and resummation, an exact relation between the total and direct correlation functions is obtained, namely... 

A closure similar to the RHNC is the modified hypernetted chain (MHNC), introduced by Rosenfeld and Ashcroft [67]. This approximation is based on the empirical observation that the bridge functions for a wide variety of pair potentials belong to the same family of curves. This means that the bridge functions calculated for a suitable reference fluid can be used to a good approximation for another fluid. The reference fluid is usually... 

These equations must be complemented with a closure relation in r-space, for which in this work we have chosen the hypernetted chain (HNC) approximation. This equation is known to give reasonable results for ionic fluids, and consequently we can expect a similar accuracy here. In the present instance the HNC can be written as,... 

For separations outside the hard core, the direct correlation functions have to be approximated. Classic closure approximations recently applied to QA models axe the Percus-Yevick (PY) closure [301], the mean spherical approximation (MSA) [302], and the hypernetted chain (HNC) closure [30]. None of these relations, when formulated for the replicated system, contains any coupling between different species, and wc can directly proceed to the limit n — 0. The PY closure then implies... 

The second commonly used closure relation is the hypernetted chain approximation (HNC). This can be derived by taking the logarithm of both sides of Eq. (2.1.30). 

Here we have included a density factor of in the Fourier transforms of the site-site correlation functions h (r) and which is more convenient for mixture calculations. is the interamolecular distance between sites a and rj. For mixtures, = 0 when sites a and t] are in molecules of different species. The SSOZ equation simply relates the total correlation functions h (k) to the direct correlation functions c, (/c). A second relation is required to obtain a closed system of equations. Two of the commonly used closure relations are the Percus-Yevick (PY) and hypernetted chain (HNC) approximations. The PY closure is ... 

Much more flexible is to describe the solute-solvent system within the supercell technique employing a set of 3D plane waves. The convolution in the OZ equation can be then approximated by means of the 3D fast Fourier transform (3D-FFT) procedure. For the first time, this approach was elaborated by Beglov and Roux [16] for numerical solution of the 3D-OZ integral equation with the hypernetted chain (HNC) closure for the 3D distribution of a simple LJ solvent around non-polar solutes... 

Alternative closure approximations for the repulsive force fluid have been investigated and will be briefly commented on in subsequent sections. Based on the idea that the atomiclike closures are useful by analogy for molecular fluids, there are several alternatives to the PY or MSA for hard core fluids. These include the hypernetted chain (HNC) approximation... 

Simulation results were compared with the predictions of the Ornstein-Zernike (OZ) equation with the hypernetted chain (HNC) closure approximation and the non-linear Poisson-Boltzmann equation, both augmented by pertinent Lifshitz NES potentials. We show in Fig. 1 that there is very good agreement between modified Poisson-Boltzmann theory, MC simulations, and HNC calculations when the counterions and co-ions are monovalent. There is also good agreement between the different approaches with divalent co-ions (not shown here). However, the results from MPBE cannot account for ion correlation effects that occur in Fig. 2 when the counterions are divalent. The reason is simply that the... 

The results summarised here are for pure water at the temperature 25°C and the density 1.000 g cm , and are obtained by solving numerictdly the Ornstein-Zemike (OZ) equation for the pair correlation functions, using a closure that supplements the hypernet-ted chain (HNC) approximation with a bridge function. The bridge function is determined from computer simulations as described below. The numerical method for solving the OZ equation is described by Ichiye and Haymet and by Duh and Haymet. ... 

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