Abstract
In this paper, a stationary one-dimensional Poisson-Nernst-Planck model with permanent charge is studied under the assumption that n - 1 positively charged ion species have the same valence and the permanent charge is small. By expanding the singular solutions of Poisson-Nernst-Planck model with respect to small permanent charge, the explicit formulae for the zeroth order approximation and the first order approximation of individual flux can be obtained. Based on these explicit formulae, the effects of small permanent charges on individual flux are investigated.
Highlights
The cell membrane is a biological membrane that separates the interior of all cells from the outside environment which protects the cell from its environment
A stationary one-dimensional Poisson-Nernst-Planck model with permanent charge is studied under the assumption that n −1 positively charged ion species have the same valence and the permanent charge is small
Under the assumption that Q ( x) is a piecewise constant function, it is basically difficult to obtain the explicit formula for individual flux with respect to permanent charges, so it is not easy to analyze the effects of permanent charges on individual flux
Summary
The cell membrane is a biological membrane that separates the interior of all cells from the outside environment which protects the cell from its environment. It is assumed that the constant Q is small, by expanding the solutions of (1.4) and (1.5) with respect to small Q, the explicit formulae for the zeroth order approximation and the first order approximation of individual flux can be obtained. Based on these explicit formulae, the effects of small permanent charges on individual flux are investigated. As n = 2 in (1.4) and (1.5), namely, only one positively charged ion and one negatively charged ion are involved in the Poisson-Nernst-Planck model, the effects of small permanent charges on individual flux has been analyzed in [22]. First order terms of the solution in Q of (2.9)-(2.11) are given by ( ) ( ) c1a1 + += cna−1,1 znα zn − z φ0a − φ0b
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