Ion channels, large proteins embedding in cell membranes, play a significant role in the exchange of ion species between cells and surroundings [1,2,3,4,5]. For example, a type of calcium channel located in the fungi’s mitochondria controls the process of ATP synthesis, the transportation of calcium and apoptosis [6]. For human beings, ion channels are also crucial to cell functioning. For instance, sodium and potassium channels are widely distributed in neurons and cardiac tissues. They are responsible for the sharp switch between the action and resting potentials when the stimuli propagate through the corresponding cells. In muscle cells, a group of ion channels cooperate to trigger muscle contractions [7]. On the other hand, malfunctioning channels result in many intractable diseases such as cholera and Alzheimer’s [8]. Therefore, exploring the working mechanisms of ion channels is not only promising in theoretical studies but has many practical meanings in disease treatment. The two main subjects related to ion channels, the structure of ion channels and the properties of ion flow, are the primary concerns in ion channel research. Once the structure is provided, the main research direction for open ion channels is to analyze their electric diffusion characteristics.

Ion flow follows the basic physical laws of electric diffusion. The macroscopic characteristics of ions passing through membrane channels depend on external driving forces, mainly boundary potential and concentration [9,10], as well as specific structural features [11,12]. These structural features include factors such as pore shape and size. Permeability and selectivity are two important biological properties of ion channels, which can be characterized by experimentally measuring the current–voltage (I–V) relationship under different ionic conditions.

The PNP system, as a basic macroscopic model for electrodiffusion of charges, particularly for ionic flows through ion channels ([13,14,15,16,17], etc.), can be derived, under various reasonable conditions, from the more fundamental models of the Langevin–Poisson system ([18,19,20,21]) and the Maxwell–Boltzmann equations ([9] and the references therein), and from variational analysis ([11,22,23,24]). The classical PNP system is the simplest PNP system, which has been extensively studied both numerically ([13,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39]) and analytically ([3,10,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70]). Particularly, in [70], the authors employed the method of matched asymptotic expansions to study the I–V relations and obtained the I–V relation up to the second order in the small singular parameter $\epsilon$ (see (6) for definition), which is a cubic function of the potential V that has three distinct real roots. The observation is consistent with the cubic-like feature of the average I–V relation of a population of channels in the FitzHugh–Nagumo simplification of the Hodgkin–Huxley model. In [42,48,65], the authors focused on the small permanent charge effects on ionic flows. Viewing the small permanent charge as a regular parameter, in the discussion of regular perturbation, the authors found that to optimize the effects of (small) permanent charge, the channel neck within which the permanent charge distributes should be “short” and “narrow”. This observation is consistent with the typical structure of an ion channel.

However, a major weak point of the classical PNP is that it treats ions as point charges and ignores ion-to-ion interaction, which is reasonable only in near-infinite dilute situations. A lot of critical properties of ion channels rely heavily on ion sizes. The effects of finite ion size on ionic flows play key roles in the study of the selectivity of ion channels. The PNP system with finite ion size effects has been investigated computationally ([11,22,23,24,71,72,73,74,75,76], etc.) and analytically ([77,78,79,80,81,82,83,84,85,86]) for ion channels.

We focus on the quasi-one-dimensional PNP system first proposed by [36],

where $X\in [0,1]$ is the normalized coordinate along the channel axis, and $A\left(X\right)$ denotes the cross-sectional area at point X. e is the elementary charge, $k_B$ is the Boltzmann constant, and T is the absolute temperature. The electric potential is represented by $\mathrm{\Phi }$, while $Q\left(X\right)$ denotes the permanent charge of the channel. ${\epsilon }_r\left(X\right)$ is the relative dielectric coefficient, and ${\epsilon }_0$ is the vacuum permittivity. For the i-th ion species, $C_i$ represents the concentration, $z_i$ is the valence, ${\mu }_i$ is the electrochemical potential, ${\mathcal{J}}_i$ is the flux density, and ${\mathcal{D}}_i\left(X\right)$ is the diffusion coefficient.

The boundary conditions imposed on the system (1) are as follows ([44]):

For a solution of the PNP system (1)–(2), the total current, $\mathcal{I}$, through a cross-section is defined as

which is the well-known current–voltage (I–V) relation.

We further point out that the electrochemical potential ${\mu }_i\left(X\right)$ in (1) consists of two components: the ideal component ${\mu }_i^{\mathrm{id}}\left(X\right)$ and the excess component ${\mu }_i^{\mathrm{ex}}\left(X\right)$:

where the ideal component is defined by

with $C_0$ being a characteristic number density. The PNP system, considering only the ideal component, is known as the classical PNP, and its major weak point in studying ionic flow properties is discussed above. To better understand the mechanism of ionic flows through membrane channels, one should consider the excess component. A strategic first step is to include hard-sphere potentials of the excess electrochemical potential in the PNP system. In this work, we consider the following Bikerman’s local hard-sphere model ([87]) accounting for finite ion size effects on ionic flows

where ${\nu }_j$ is the volume of the j-th ion species.

The rest of this paper is organized as follows. In Section 2, we set up our problem, briefly recall some results from [88], which is the starting point of our study, and describe the mathematical method to be employed. Section 3 consists of four subsections. Section 3.1 provides the finite ion size effects on the individual fluxes; Section 3.2 deals with the finite ion size effects on the I–V relations; Section 3.3 provides orders of the critical potentials identified in Definition 1 while the boundary layer effects on ionic flows are characterized in Section 3.4. Concluding remarks are provided in Section 4.

We study the finite ion size impacts on ionic flows under relaxed boundary conditions to better understand the dynamics of ionic flows via a one-dimensional PNP model. Ion sizes play vital roles in the characterization of the selectivity phenomena of ion channels. The detailed discussion, particularly, the argument of the relative ion size effects, could provide important insights into the selectivity phenomena of ion channels. Our study is under more realistic setups of the boundary conditions, a state that is not neutral but close to, and not surprisingly, the richer dynamics of ionic flows which are observed compared to the work conducted in [88] under the assumption of electroneutrality boundary conditions, that is, $\sigma =\rho =1$ in current setup. The boundary layer effects on ionic flows due to the relaxation of the electroneutrality conditions are further characterized. Critical potentials are identified under different setups, which play crucial roles in our discussion of the ionic flow properties. Most importantly, those critical potentials can be experimentally identified as stated in Remark 4. The study provides an efficient way to control/adjust the boundary conditions to observe distinct dynamics of ionic flows through membrane channels. This is important for future analytical studies and critical for future numerical and even experimental studies of ion channel problems.