| Citation: |
Yiwei Wang, Lijun Zhang, Mingji Zhang. STUDIES ON INDIVIDUAL FLUXES VIA POISSON-NERNST-PLANCK MODELS WITH SMALL PERMANENT CHARGES AND PARTIAL ELECTRONEUTRALITY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2022, 12(1): 87-105. doi: 10.11948/20210045
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STUDIES ON INDIVIDUAL FLUXES VIA POISSON-NERNST-PLANCK MODELS WITH SMALL PERMANENT CHARGES AND PARTIAL ELECTRONEUTRALITY CONDITIONS
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Abstract
We study a one-dimensional Poisson-Nernst-Planck system for ionic flows through membrane channels with two ion species, one positively charged and one negatively charged. Nonzero but small permanent charges are included. The cross-section area of the channel is included in the system, which provides certain information of the geometry of the three-dimensional channel. This is crucial for our analysis. Of particular interest is to analyze the qualitative properties of the individual fluxes with partial neutral boundary conditions, which provides complementary insights and better understanding of the ionic flow properties. Our study shows that the individual fluxes depend sensitively on multiple system parameters such as permanent charges, channel geometry, boundary conditions (concentrations and potentials) and boundary layers caused by the violation of electroneutrality conditions. Numerical simulations are further performed to provide a more intuitive illustration of our analytical results, and it turns out that numerical results are consistent with our analytical ones.
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Keywords:
- PNP, permanent charges /
- channel geometry /
- individual fluxes /
- boundary layers
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References
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Yiwei Wang, Lijun Zhang, Mingji Zhang. STUDIES ON INDIVIDUAL FLUXES VIA POISSON-NERNST-PLANCK MODELS WITH SMALL PERMANENT CHARGES AND PARTIAL ELECTRONEUTRALITY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2022, 12(1): 87-105. doi: 10.11948/20210045
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Figure 1. Identification of critical potentials
$ V_{kp}^* $ with boundary layers and$ V_q^k $ under electroneutrality boundary conditions for small$ \varepsilon = 0.01 $ . We also point out that$ V_{cp}^{k*} $ is a common zero with$ J_k(V;0) $ with boundary layers while$ V_{c}^{kq} $ is a common zero with$ J_k(V;0) $ under electroneutrality boundary conditions, see Figure 2. One can easily see that for$ (\sigma, \rho) = (1.005, 1.0), $ $ V_q^1-V_{1p}^* = 0.024618 $ and$ V_q^2-V_{2p}^* = -0.022409 $ . This shows clearly the important role played by the boundary layers in the study of ionic flows through membrane channels. -
Figure 2. Numerical simulations to
$ J_{k0}(V) $ for both the case with boundary layers and the one under electroneutrality boundary conditions by$ J_k(V;0) $ with$ \varepsilon = 0.01 $ . Critical potentials$ V_{cp}^{k0*} $ and$ V_c^{k0q} $ are identified. From Figure 1, one can see that$ V_{cp}^{k0*} $ is close to$ V_{cp}^{k*} $ , and$ V_c^{k0q} $ is close to$ V_c^{kq} $ (see the discussion in (ⅲ) for more details). -
Figure 3. Orders of the critical potentials
$ V_{kp}^* $ with boundary layers and$ V_q^{k} $ under electroneutrality boundary conditions, respectively for$ k = 1, 2 $ with$ \varepsilon = 0.01 $ . Under our set-up, the result with boundary layers is consistent with statement (ⅱ) in Theorem 3.1, and the one under electroneutrality boundary conditions is also consistent with statement (ⅱ) in Theorem 4.8 of [30] with$ \beta_1 $ replaced by$ \beta_2 $ under current set-up.