Membranes

15 pages, 293 KB

Open AccessArticle

Relaxed Boundary Conditions in Poisson–Nernst–Planck Models: Identifying Critical Potentials for Multiple Cations

by Xiangshuo Liu, Henri Ndaya, An Nguyen, Zhenshu Wen and Mingji Zhang

Membranes 2025, 15(11), 339; https://doi.org/10.3390/membranes15110339 - 13 Nov 2025

Ion channels are protein pores that regulate ionic flow across cell membranes, enabling vital processes such as nerve signaling. They often conduct multiple ionic species simultaneously, leading to complex nonlinear transport phenomena. Because experimental techniques provide only indirect measurements of ion channel currents, [...] Read more.

Ion channels are protein pores that regulate ionic flow across cell membranes, enabling vital processes such as nerve signaling. They often conduct multiple ionic species simultaneously, leading to complex nonlinear transport phenomena. Because experimental techniques provide only indirect measurements of ion channel currents, mathematical models—particularly Poisson–Nernst–Planck (PNP) equations—are indispensable for analyzing the underlying transport mechanisms. In this work, we examine ionic transport through a one-dimensional steady-state PNP model of a narrow membrane channel containing multiple cation species of different valences. The model incorporates a small fixed charge distribution along the channel and imposes relaxed electroneutrality boundary conditions, allowing for a slight charge imbalance in the baths. Using singular perturbation analysis, we first derive approximate solutions that capture the boundary-layer structure at the channel—reservoir interfaces. We then perform a regular perturbation expansion around the neutral reference state (zero fixed charge with electroneutral boundary conditions) to obtain explicit formulas for the steady-state ion fluxes in terms of the system parameters. Through this analytical approach, we identify several critical applied potential values—denoted

V_{k}^{a}

(for each cation species k),

V^{b}

, and

V^{c}

—that delineate distinct transport regimes. These critical potentials govern the sign of the fixed charge’s influence on each ion’s flux: depending on whether the applied voltage lies below or above these thresholds, a small positive permanent charge will either enhance or reduce the flux of each ion species. Our findings thus characterize how a nominal fixed charge can nonlinearly modulate multi-ion currents. This insight deepens the theoretical understanding of nonlinear ion transport in channels and may inform the interpretation of current–voltage relations, rectification effects, and selective ionic conduction in multi-ion channel experiments. Full article

17 pages, 4105 KB

Open AccessArticle

Ion Exchange Membrane-like Deposited Electrodes for Capacitive De-Ionization: Performance Evaluation and Mechanism Study

by Siyue Xue, Chengyi Wang, Tianxiao Leng, Chenglin Zhang, Long-Fei Ren and Jiahui Shao

Membranes 2025, 15(11), 338; https://doi.org/10.3390/membranes15110338 - 11 Nov 2025

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Membranes

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