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Ion size effects on individual fluxes via Poisson-Nernst-Planck systems with Bikerman's local hard-sphere potential: Analysis without electroneutrality boundary conditions

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  • A quasi-one-dimensional steady-state Poisson-Nernst-Planck model with Bikerman's local hard-sphere potential for ionic flows of two oppositely charged ion species through a membrane channel is analyzed. Of particular interest is the qualitative properties of ionic flows in terms of individual fluxes without the assumption of electroneutrality conditions, which is more realistic to study ionic flow properties of interest. This is the novelty of this work. Our result shows that ⅰ) boundary concentrations and relative size of ion species play critical roles in characterizing ion size effects on individual fluxes; ⅱ) the first order approximation $\mathcal{J}_{k1} = D_kJ_{k1}$ in ion volume of individual fluxes $\mathcal{ J}_k = D_kJ_k$ is linear in boundary potential, furthermore, the signs of $\partial_V \mathcal{ J}_{k1}$ and $\partial^2_{Vλ} \mathcal{J}_{k1}$ , which play key roles in characterizing ion size effects on ionic flows can be both negative depending further on boundary concentrations while they are always positive and independent of boundary concentrations under electroneutrality conditions (see Corollaries 3.2-3.3, Theorems 3.4-3.5 and Proposition 3.7). Numerical simulations are performed to identify some critical potentials defined in (2). We believe our results will provide useful insights for numerical and even experimental studies of ionic flows through membrane channels.

    Mathematics Subject Classification: Primary: 34A26, 34B16, 34D15; Secondary: 37D10, 92C35.


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  • Figure 1.  Numerical detection of critical values for $\lambda$ (left graph) and $m$ (right one) in Theorem 3.2.

    Figure 2.  Numerical detection of critical values for $\sigma$, which corresponds to statement (Ⅰ) in Theorem 3.2. The left graph is for $\lambda<\lambda_1^* = 0.072$, and the right one is for $\lambda>\lambda_2^* = 13.93.$

    Figure 3.  Numerical detection of critical values $\sigma$ corresponding to statement (Ⅱ) in Theorem 3.2 with $\lambda_1^*<\lambda<\lambda_2^*$. The left graph is for $0<m<m^* = 0.4934$, and the right one is for $m^*<m<\frac{1}{2}.$

    Figure 4.  Numerical identification of six critical potentials in (15) with $z_1 = -z_2 = 1$. In the left column, the vertical axis actually represents, from top to bottom, ${\mathcal I}(V;\nu, \lambda)-{\mathcal I}_0(V), \ {\mathcal J}_1(V;\nu, \lambda)-{\mathcal J}_{10}(V)$ and ${\mathcal J}_2(V;\nu, \lambda)-{\mathcal J}_{20}(V)$, respectively. In particular, the x-axis for all figures actually represents $\frac{e}{k_BT}V$.

    Figure 5.  Numerical identification of critical potentials $V_{1c}$ and $ V_1^c$ for individual flux ${\mathcal J}_1$ with $z_1 = -z_2 = 1$ and nonzero permanent charge. The x-axis for all figures actually represents $\frac{e}{k_BT}V$.

    Figure 6.  Numerical approximations of critical potentials $V_1^c$ for individual flux ${\mathcal J}_1$ with $z_1 = -z_2 = 1$ and nonzero permanent charge as illustrated in Proposition 3.14. The x-axis for all figures actually represents $\frac{e}{k_BT}V$.

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