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of ions of species i and j, the number of ions in this solution that are separated by a distance between r and r+dr is given by n (r)=4πr2g (r)c ̄c ̄ (6.5) ij ijij where gij(r) is the pair correlation function. Using the appropriate correlation func- tion and the pair interaction energy, Kij(r), we can then define the excess chemical potential of species i as j0 In order to calculate this excess chemical potential, we must define both the pair interaction energy and the correlation function. Here, we use the same approach used by Debye and Huckel and we will simply modify the Poisson equation that is used. The pair interaction energy in an electrolyte is basically the bare Coulomb interaction energy for point charges, which is defined as Kij = (zie)(zje) (6.7) 4πε0εcr To calculate the correlation function of a spherical central ion of species i, we use the Debye-Huckel approximation: −zj eψ zj eψ gij(r)=e kBT ≈1− kBT (6.8) where ψ = φ − φ ̄ is the electrostatic potential perturbation around the central ion. The above linearization is valid, since the potential perturbations due to one single ion are always going to be small. We now need to find the potential perturbations to be able to define the correlation functions, which can be done using the Poisson- Boltzmann equation. Here, we will specifically use the 4th order Poisson-Boltzmann equation described by Bazant et al [8] to specifically take in account electrostatic 1∞ iBi2 jijij μex = k T ln γ = 4πr2c ̄ K (r)g (r)dr + many-body terms (6.6) 144PDF Image | Shock Electrodialysis for Water Purification and Electrostatic Correlations
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