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

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 $\frac{e}{k_BT}V$.