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1. Derivation of the ion equation

   Grenier, E (March 2020, Quarterly of applied mathematics)

   We consider the classical Euler-Poisson system for electrons and ions, interacting through an electrostatic field. The mass ratio of an electron and an ion is small and we establish an asymptotic expansion of solutions, where the main term is obtained from a solution to a self-consistent equation involving only the ion variables. Moreover, on R^3, the validity of such an expansion is established even with \ill-prepared" Cauchy data, by including an additional initial layer correction. 

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2. Derivation of the ion equation

   https://doi.org/https://doi.org/10.1090/qam/1558  

   E. Grenier, Y. Guo (September 2019, Quarterly of applied mathematics)

   We consider the classical Euler-Poisson system for electrons and ions, interacting through an electrostatic field. The mass ratio of an electron and an ion $$ m_e/M_i\ll 1$$ is small and we establish an asymptotic expansion of solutions, where the main term is obtained from a solution to a self-consistent equation involving only the ion variables. Moreover, on $$ \mathbb{R}^3$$, the validity of such an expansion is established even with ``ill-prepared'' Cauchy data, by including an additional initial layer correction. 

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3. Analytical Modeling of the Ion Number Fluctuations in Biological Ion Channels

   https://doi.org/10.1166/jnn.2012.5771  

   Pandey, Santosh (March 2012, Journal of Nanoscience and Nanotechnology)
4. Role of Ion Dehydration in Ion–Ion Selectivity of Dense Membranes

   https://doi.org/10.1021/acs.est.5c04303  

   Ershov, Alexander; Xu, Hengyu; Li, Ying; Tong, Tiezheng; Epsztein, Razi (August 2025, Environmental Science & Technology)
5. Ion Selectivity of Intercalation Electrodes in Dual-Ion Electrolytes: Modeling Ion Selectivity as the Sum of Individual Cation Effects

   https://doi.org/10.1149/1945-7111/adf9cc  

   Pothanamkandathil, Vineeth; Feineman, Maureen; Gorski, Christopher_A (August 2025, Journal of The Electrochemical Society)

   Intercalation electrode materials reversibly insert cations into their lattices under applied potentials or currents, which can be used to perform electrochemical separations. Optimizing performance, however, remains challenging due to tradeoffs between selectivity and separation rate being influenced by multiple variables. This study developed a quantitative model to describe the current response and cation selectivity of an intercalation electrode in a binary cation solution during cyclic voltammetry. We hypothesized that current responses and selectivity could be calculated by summing individual ion contributions. Cyclic voltammograms were experimentally measured using nickel hexacyanoferrate electrodes in NaCl, KCl, or mixed solutions. Post-mortem electron probe micro-analysis quantified intercalated Na⁺and K⁺fractions. A one-dimensional finite element model incorporating the Nernst-Frumkin isotherm, Butler-Volmer kinetics, and ion diffusion was developed, parameterized with pure NaCl or KCl solutions, and validated against mixed solutions. The model accurately reproduced experimental cyclic voltammograms and ion partitioning behaviors at ionic strengths ≥0.2 mol·l⁻¹. However, at lower ionic strengths, significant discrepancies arose for reasons still unclear. Results indicate that modeling ion contributions individually effectively captures the electrochemical response of selective intercalation electrodes at sufficiently high ionic strengths. 

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