<p>We consider a quasi-one-dimensional Poisson-Nernst-Planck system for two oppositely charged particles with nonuniform ion sizes under relaxed neutral boundary conditions. Of particular interest is the effect on zero-current ionic flows from finite ion sizes under this new setups. The existence and local uniqueness result is established in the frame work of geometric singular perturbation theory, together with the specific structures of the model. Viewing finite ion sizes as small parameters, regular perturbation analysis is employed to obtain approximations of individual fluxes, from which detailed characterization of the qualitative properties of zero-current ionic flows is provided. Additionally, the effects on ionic flows from boundary layers due to the relaxation of electroneutrality conditions are examined. Critical potentials are identified, and they split the potential region into different subregions, over which distinct dynamical behaviors of ionic flows through membrane channels are observed. Some critical potentials can be estimated experimentally, which provides efficient ways to adjust boundary concentrations and boundary electric potentials to observe different ionic flow properties. The study provides deep insights into the internal dynamics of ionic flows through membrane channels, which cannot be detected via current technology.</p>