Ion association and dipolar dumbbells: Solutions of the HNC and HNC/MS approximations at L=σ/2 and σ/3 for the sticky electrolyte model

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Abstract
We extend an earlier analytic study of a sticky electrolyte model (SEM) to the case L=σ/3, where L is the distance at which positive and negative ions bind to each other, using the hypernetted chain (HNC) approximation within the spherical core and the mean spherical approximation (MSA) outside. We also present numerical solutions to the HNC approximation alone for σ/3 ≤L≤σ/2. The average number of bonded pairs is found to be essentially the same for the two approximations but the ion–ion correlation functions are very different except at high concentrations when the shielding is large. Small amounts of tetramers are also observed in the HNC correlation functions for σ/3<L≤σ/2, and trimers are found when L=σ/2. An expression for the excess free energy of the SEM electrolyte is derived by turning on the stickiness between oppositely charged ions. The excess energy of the system of dipolar dumbbells with charges at a distance L=σ/3 is obtained in the MSA and the atom–atom correlation functions are compared with the HNC approximation and with recent Monte Carlo simulations. The asymptotic form of the direct correlation functions defined through the Ornstein–Zernike equation is given for dipolar dumbbells.