# Interdisciplinary Applied Mathematics

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To analyze the ion concentrations across the channel, the z-direction mean force acting on the ions is computed in the lower portion of the channel. The    mean    force    fi(z)    acting    on    an    ion    i, located    at position    z,    is

computed as the total force on ion i from all other particles in the system averaged over all configurations. The force fi(z) is considered negative if it drives the ion toward the lower channel wall, and positive otherwise. The potential of mean force (PMF), denoted by wi(z), for an ion i at a position z is computed by rf

Wi (z)

fi(z’) dz ,

where rf is the reference plane (taken as the channel center plane here) at which the PMF is taken as zero. Within the limit of classical statistical mechanics, the concentration of an ion i at a position z, denoted by ci(z), is related to the PMF by the Boltzmann distribution

Ci(z) = crif exp(-Wi(z)/квT),

where cd, kB, and T are the concentration of ion i at the reference plane, the Boltzmann constant, and the temperature, respectively. To facilitate discussion, we decompose the total mean force into an electrostatic mean force and a nonelectrostatic mean force. In certain cases, we further decompose the nonelectrostatic mean force into several components arising from the interactions    of    the molecule    (e.g.,    wall    atoms,    ion,    or    water)    with    the

ion.

To measure the screening of the surface charge by the ions, we define a screening factor

Sf (z)=[ F [cNa+ (z)cc— (z)V К | dz,    (12.14)

J 0

where F is the Faraday constant. The screeing factor Sf (z) > 1 corresponds to an overscreening of the surface charge.

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