| Citation: |
Xijun Deng, Yusheng Jia, Mingji Zhang. STUDIES ON CURRENT-VOLTAGE RELATIONS VIA POISSON-NERNST-PLANCK SYSTEMS WITH MULTIPLE CATIONS AND PERMANENT CHARGES[J]. Journal of Applied Analysis & Computation, 2022, 12(3): 932-951. doi: 10.11948/20210003
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STUDIES ON CURRENT-VOLTAGE RELATIONS VIA POISSON-NERNST-PLANCK SYSTEMS WITH MULTIPLE CATIONS AND PERMANENT CHARGES
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Abstract
We study a one-dimensional Poisson-Nernst-Planck system with multiple cations having the same valences and small permanent charges. Viewing the permanent charge as a small parameter, via regular perturbation analysis, approximations of the current-voltage (I-V) relations are derived explicitly, and this allows us to further study the qualitative properties of ionic flows through membrane channels. Our main interest are small permanent charge and channel geometry effects on the I-V relations, which additionally depend on the nonlinear interactions with other physical parameters involved in the model. Critical potentials are identified and their important roles played in the study of the property of ionic flows are characterized. We perform numerical simulations to provide more intuitive illustrations of our theoretical results. Those non-intuitive observations from analysis of the system provide better understandings of the mechanism of ionic flows through membrane channels, particularly the internal dynamics that is not able to be detected via current technology.
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Keywords:
- PNP /
- diffusion coefficients /
- permanent charges /
- channel geometry /
- electroneutrality conditions
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References
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Xijun Deng, Yusheng Jia, Mingji Zhang. STUDIES ON CURRENT-VOLTAGE RELATIONS VIA POISSON-NERNST-PLANCK SYSTEMS WITH MULTIPLE CATIONS AND PERMANENT CHARGES[J]. Journal of Applied Analysis & Computation, 2022, 12(3): 932-951. doi: 10.11948/20210003
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Figure 1. Illustration of a singular orbit projected to the space of
$ (u, \sum z_kc_k, \tau) $ under current setup. -
Figure 2.
$ {\cal I}_1(V;\varepsilon) $ , the approximation of$ {\cal I}_{1}(V) $ , has two zeros$ V_{g1}^z $ and$ V_{g2}^z $ , and a critical point$ V_{g1}^{c} $ with$ V_{g1}^z<V_{g1}^c<V_{g2}^z $ . -
Figure 3.
$ {\cal I}_1(V;\varepsilon) $ , the approximation of$ {\cal I}_{1}(V) $ , has no zeros, but has three critical point$ V_{g2}^c, V_{g3}^c $ and$ V_{g4}^c $ with$ V_{g2}^c<V_{g3}^c<V_{g4}^c $ . -
Figure 4.
$ {\cal I}_1(V;\varepsilon) $ , the approximation of$ {\cal I}_{1}(V) $ , has a unique zero$ V_{g4}^z $ , and three critical point$ V_{g2}^c, V_{g3}^c $ and$ V_{g4}^c $ with$ V_{g2}^c<V_{g3}^c<V_{g4}^c=V_{g4}^z $ . -
Figure 5.
$ {\cal I}_1(V;\varepsilon) $ , the approximation of$ {\cal I}_{1}(V) $ , has two zeros$ V_{g5}^z $ and$ V_{g6}^z $ and three critical point$ V_{g2}^c, V_{g3}^c $ and$ V_{g4}^c $ with$ V_{g2}^c<V_{g3}^c<V_{g5}^z<V_{g4}^c<V_{g6}^z $ . -
Figure 6.
$ {\cal I}_1(V;\varepsilon) $ , the approximation of$ {\cal I}_{1}(V) $ , has three zeros$ V_{g22}^z, \ V_{g23}^z, \ V_{g24}^z $ , and three critical point$ V_{g2}^c, V_{g3}^c $ and$ V_{g4}^c $ with$ V_{g22}^z<V_{g2}^c<V_{g3}^c=V_{g23}^z<V_{g4}^c<V_{g24}^z $ . -
Figure 7.
$ {\cal I}_1(V;\varepsilon) $ , the approximation of$ {\cal I}_{1}(V) $ , has four zeros$ V_{g18}^z, \ V_{g19}^z, \ V_{g20}^z, \ V_{g21}^z $ and three critical point$ V_{g2}^c, V_{g3}^c $ and$ V_{g4}^c $ with$ V_{g18}^z<V_{g2}^c<V_{g19}^z<V_{g3}^c<V_{g20}^z<V_{g4}^c<V_{g21}^z $ .