Linearized hypernetted-chain approximation

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-... 

The relation is referred to as the hypernetted-chain (HNC) approximation. Further linearizing expt(r, r ) in 1.23, one has the Percus-Yevick (PY) 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... 

---