# Interdisciplinary Applied Mathematics

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Here we introduce the concept of electrochemical potential correction to account for the interactions neglected in the classical Poisson-Boltzmann equation. At thermodynamic equilibrium, the electrochemical potential of an ion should be constant in the entire system, i.e.,

pi = Ziq’-ф + квT log Ci +    = квT log coy,    (12.3)

where z* is the valency of ion i, ф is the electric potential in the system, c* is the ion concentration, феху is the electrochemical potential correction of ion i, and    coy    is    the    concentration    of ion i    when the    electric    potential    and

the electrochemical potential correction terms are zero. Based on equation (12.3), the ion concentration can be expressed as

_ Sjqip _ Фех.,г

Ci = c0y e квт e квт .    (12-4)

Substituting equation (12.4) into the Poisson equation (12.5b), we have

У2ф

У2ф

q

e

q

e

N

y^ZjCj,

i=1

N

^ ^ zic0,i e i=1

kBT e

Фех,г kBT

(12.5a)

(12.5b)

where N is the total number of ionic species, which is equal to 1 in all our simulations presented in this section.

The electrochemical potential correction term accounts for the deviation of the ion-water and ion-wall molecular interactions from their values at the channel center. Since the wall-ion interaction via the Lennard-Jones potential is short-ranged, and the water-ion interaction would not deviate significantly in the entire system except at positions very close to the channel wall where the water concentration is not constant, the electrochemical potential correction term is nonzero only at positions close to the channel wall. The precise contribution of ion-water and ion-wall molecular interactions to the electrochemical potential correction term depends on the specific case under study. For example, for case 1, where the ion-wall (Cl_ —Si) interaction is strong, the contribution of the ion-wall molecular interactions to the electrochemical potential correction term is significant, whereas for case 3 the contribution of ion-wall (Na+ —Si) molecular interactions will be minor.

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