Hypernetted-chain approximation

When all terms in Eq. (54) are neglected, Eqs. (32)-(34) form a closed set, which can be solved. This kind of approximation has been called by Zhou and Stell the hydrostatic hypernetted chain approximation (HHNCl) [91] a procedure for the numerical solution of the HHNCl equations can be found in Ref. 45. 

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]... 

Another possible approach solving the equilibrium distribution for an electric double layer is offered by integral equation theories [22]. They are based on approximate relationships between different distribution functions. The two most common theories are Percus-Yevick [23] and Hypernetted Chain approximation (HNQ [24], where the former is a good method for short range interactions and the latter is best for long-range interactions. They were both developed around 1960, but are still used. The correlation between two particles can be divided into two parts, one is the direct influence of particle j on particle i and the other originates from the fact that all other particles correlate with particle j and then influence particle i in precisely... 

Recent statistical-mechanical theories [62] and Monte Carlo simulations of the diffuse double layer [63, 64] predict that the potential of the outer Helmholtz plane is generally overestimated in the Gouy-Chapman theory. If this is the case, the potential drop across the inner layer can be even greater, e.g., by c. 15% if a hypernetted-chain approximation is applied to the primitive model of the diffuse layer [12]. 

The DH and MSA theory, that are linear in charge can be considered in the framework of linearized Poisson-Boltzmann (PB) equation. The concept of ion association entails nonlinearity in the treatment of electrostatic interactions by the formulation of appropriate thermodynamic equilibrium constants between free ions and ion clusters [14], In general, this formulation can be considered as the division of ion-ion interaction potentials into an associative part responsible for the ion association, and nonassociative part which is more or less arbitrary. In order to optimize this division in the framework of associative hypernetted chain approximation (AHNC), the division of energy and distance were considered [17] with the parameters calculated from the condition of sta-... 

Lozada-Cassou, M. and Saavedra-Barrera, R. The application of the hypernetted chain approximation to the electrical double layer Comparison with Monte Carlo results for symmetric salts. Journal of Chemical Physics, 1982, 77 (10), p. 5150-5156. 

Estimate the direct correlation function for liquid argon at 85 K using the hypernetted chain approximation with the data given in problem 3. Compare the result with that found using the Percus-Yevick approximation. 

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). 

Gonzales-Tovar E, Lozada-Cassou M, Henderson D. Hypernetted chain approximation for the distribution of ions around a cylindrical electrode. II. Numerical solution for a model cylindrical polyelectrolyte. J. Chem. Phys. 1985 83 361. 

Many attempts have been made to improve the classical Poisson-Boltzmann equation to include discrete charge effects, finite ion size, and so on (see, e.g., Refs. 35-37). At present some fundamental progress is being made on the basis of certain models in modem fluid-state theory, in which the hard-core repulsions of the ions are incorporated in a consistent way. " The Poisson-Boltzmann equation was found to be a limiting case of the hypernetted chain approximation at low densities. Also a computer simulation was reported." ... 

In the second-quality we are identifying the connection between the direct correlation function and the functional inverse of the response function for a uniform simple atomic fluid. The reader can verify this result using the Ornstein-Zernike equation [i.e., the Chandler-Andersen equation (8) for an atomic or single site fluid] and Eq. (11). As p(r) - p, Eq. (22) becomes exact. Its use for all p(r), however, is an approximation — the hypernetted chain approximation. 

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