| Citation: |
Yanyu Bao, Jianing Chen, Lijun Zhang, Mingji Zhang. HIGHER ORDER EXPANSIONS IN FINITE ION SIZE VIA POISSON-NERNST-PLANCK SYSTEMS WITH BIKERMAN'S LOCAL HARD-SPHERE POTENTIAL[J]. Journal of Applied Analysis & Computation, 2022, 12(3): 907-931. doi: 10.11948/20220001
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HIGHER ORDER EXPANSIONS IN FINITE ION SIZE VIA POISSON-NERNST-PLANCK SYSTEMS WITH BIKERMAN'S LOCAL HARD-SPHERE POTENTIAL
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Abstract
Finite ion sizes play significant roles in characterizing ionic flow properties of interest, such as the selectivity of ion channels. As an extension of the work done in [Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1775-1802], we further investigate the higher order (in the volume of the cation), mainly the second order, contributions from finite ion sizes to ionic flows in terms of both the total flow rate of charges and the individual fluxes. This is particularly important since the first-order terms approach zero as the left boundary concentration is close to the right one for the same ion species. The interaction between the first-order terms and the second-order terms is characterized in detail. Moreover, several critical potentials are identified, and they play critical roles in examining the qualitative properties of ionic flows. Some can be estimated experimentally. The analysis in this work could provide complementary information and better understanding of the mechanism of ionic flows through ion channels. Numerical simulations are performed to provide intuitive illustration of our analytical results.
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Keywords:
- I-V relations /
- individual fluxes /
- critical potentials /
- finite ion sizes
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References
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Yanyu Bao, Jianing Chen, Lijun Zhang, Mingji Zhang. HIGHER ORDER EXPANSIONS IN FINITE ION SIZE VIA POISSON-NERNST-PLANCK SYSTEMS WITH BIKERMAN'S LOCAL HARD-SPHERE POTENTIAL[J]. Journal of Applied Analysis & Computation, 2022, 12(3): 907-931. doi: 10.11948/20220001
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Figure 1. Numerical identifications of the critical potentials
$ V_k^c, \ V^c_{1k} $ and$ V_{2k}^c $ for$ k=0, 1, 2. $ The monotonicity of the functions$ I_k(V) $ ,$ J_{1k}(V) $ and$ J_{2k}(V) $ for$ k=0, 1, 2 $ is shown clearly in each figure. This provides intuitive illustration of Theorem 4.1 with$ 1<x<x^*=339.75691 $ . -
Figure 2. Numerical identifications of the critical potentials
$ V_0^c, \ V^b, V_{10}^c, V_1^b, V_{20}^c $ and$ V_2^b $ with small$ \nu $ . The monotonicity of the functions$ I_0(V), \ J_{10}(V) $ and$ J_{20}(V) $ (solid line in each figure) and the functions$ I^b(V), \ J_1^b(V) $ and$ J_2^b(V) $ (dashed line with star in each figure) can be observed clearly. This provides intuitive illustration of Theorem 4.2. -
Figure 3. Numerical identifications of the critical potentials
$ V_k^c, \ V^c_{1k} $ and$ V_{2k}^c $ for$ k=0, 1, 2. $ The monotonicity of the functions$ I_k(V) $ ,$ J_{1k}(V) $ and$ J_{2k}(V) $ for$ k=0, 1, 2 $ is shown clearly in each figure for the case with$ L $ close to$ R. $ It is clear that, for$ k=1, 2 $ , the values of$ V_0^c $ and$ V_{k0}^c $ become very small, the values of$ V_k^c\ V_{1k}^c $ and$ V_{2k}^c $ for$ k=1, 2 $ become very large, and the curves$ I_k, \ J_{1k} $ and$ J_{2k} $ become very small as$ L $ close to$ R $ . The observation is consistent with our analytical results stated in Section 4.3.