Numerical study of a modified BBGY integral equation for the primitive model of an electrolyte solution

A modified Poisson-Boltzmann equation derived from the BBGY hierarchy of equations is solved numerically for the restricted primitive model of an electrolyte solution. Computations are carried out for 1 : 1 and 2 : 2 electrolytes, and results compared with 1 : 1 Monte Carlo data and results from the Poisson-Boltzmann extensions arising from the Kirkwood hierarchy. Satisfactory agreement with the Monte Carlo results are found in the 1 : 1 case, and the correct qualitative behaviour of thermodynamic properties obtained in the 2 : 2 case. At low concentrations in the 2 : 2 case the modified Poisson-Boltzmann equations from the Kirkwood and BBGY hierarchies predict the anomalous behaviour found in the thermodynamic properties of real electrolytes. The behaviour of the mean electrostatic potential changes from a damped exponential to a damped oscillatory form for 1·2 < κa < 1·3 (where κ is the Debye-Hückel constant and a is an ionic diameter) in the 1 : 1 case and at κa ∼ 0·8 in the 2 : 2 case for the parameters considered.