Water desalination and energy harvesting technologies often utilize ion-selective nanoporous materials. These materials facilitate the preferential transport of counterions (ions with opposite charge to the surface charge), generating an electrical current driven by a potential drop or concentration gradient (or both). The system's electrical behavior is characterized by its current-voltage (*I*-*V*) response, specifically its ohmic conductance (*G*_ohmic), zero-voltage current (*I*_V=0), and zero-current voltage (*V*_I=0). Currently, a unified theoretical framework explaining these characteristics is lacking. This research aims to address this gap by developing a self-consistent theoretical model that provides analytical expressions for each characteristic. Understanding the fundamental physics of ion transport is crucial for improving the efficiency of electrodialysis (ED) and reverse electrodialysis (RED) processes used in water desalination and energy harvesting. These processes typically involve ion-selective materials separating reservoirs with asymmetric salt concentrations. The inherent surface charge of the material leads to permselectivity, where counterions are preferentially transported while coions are restricted. Many materials exhibit this permselectivity, including carbon nanotubes, boron nitride, silicon nanochannels, graphene oxide membranes, and various biological pores. This study focuses on a simplified nanochannel system for theoretical analysis, although the results are applicable to a wider range of permselective materials. The *I*-*V* curve is central to understanding the electrical characterization. With symmetric concentrations, the curve passes through the origin, and the slope represents the ohmic conductance. Asymmetric concentrations shift the curve, introducing *I*_V=0 and *V*_I=0 as key parameters.

Existing literature describes the square root law for ohmic conductance in systems with symmetric concentrations. This law, however, does not account for asymmetric concentrations or the full range of selectivity. Previous attempts to extend these expressions to asymmetric systems have been largely phenomenological, lacking a rigorous theoretical basis. Studies have explored ion transport in various nanoporous materials, but a comprehensive theoretical framework unifying these observations has remained elusive. This paper builds upon previous work on the electrical circuit modeling of nanofluidic systems and addresses the limitations of existing empirical approximations for ohmic conductance in both symmetric and asymmetric concentration scenarios. The Goldman-Hodgkin-Katz (GHK) equation, widely used in biological systems, is also discussed. The GHK equation assumes a constant electric field and thus a linear potential profile, which under certain conditions, simplifies the calculations of conductance, but may not correctly represent the system's behavior in many scenarios, particularly in the presence of significant surface charge.

The authors utilize the Poisson-Nernst-Planck (PNP) equations to derive a theoretical model for the *I*-*V* response of nanofluidic systems with asymmetric concentrations. The PNP equations describe the coupled transport of ions under the influence of electric fields and concentration gradients. The derivation is detailed in the supplementary material. The model provides explicit expressions for *G*_ohmic, *V*_I=0, and *I*_V=0. These expressions account for the effects of both the surface charge density (Σ) and the bulk concentrations (c_left and c_right) on either side of the nanochannel. The theoretical model is validated through comparison with non-approximated numerical simulations. These simulations were performed using methods described in the supplementary methods section, demonstrating excellent agreement between the theoretical predictions and the numerical results. The detailed calculations and derivations from the PNP equations are found in the supplementary information. This theoretical approach enables a comprehensive understanding of the interplay between various parameters affecting ionic transport. The model is designed to be adaptable to various geometries and conditions, making it a robust tool for nanofluidic system design and analysis. The model's applicability to systems with different geometries is discussed, demonstrating that the derived equations hold for a range of nanochannel cross-sections, including parallel plates and cylindrical geometries.

The key findings of the paper include the derivation of analytical expressions for the three key electrical characteristics of nanofluidic systems under asymmetric salt concentrations. 1. **Ohmic Conductance (G_Ohmic):** The derived expression for *G*_ohmic [Eq. (7)] reveals its dependence on the surface charge density (Σ), and the bulk concentrations (c_left and c_right). In the limit of symmetric concentrations (c_left = c_right), the expression reduces to the well-known square root law [Eq. (1)]. The analysis shows a transition between bulk-dominated and surface-charge-dominated regimes, which depends on the ratio Σ/c_bulk. Figure 3a illustrates this behavior, depicting the conductance versus bulk concentration for different concentration gradients. 2. **Zero-Current Voltage (V_I=0):** The expression for *V*_I=0 [Eq. (9)] provides a more accurate description than previously proposed models. It considers the effect of both surface charge and bulk concentrations. In the limit of ideal selectivity (Σ << c_left, c_right), the expression simplifies to the known expression for ideally selective systems [Eq. (6)]. Figure 3b shows the zero-current voltage as a function of bulk concentration, highlighting the differences between the newly derived expression and previous approximations. 3. **Zero-Voltage Current (I_V=0):** The zero-voltage current is calculated using the derived expression for *I*_V=0 [Eq. (12)]. Figure 3c compares this calculation to the approximation |*I*_V=0| ≈ *G*_ohmic *V*_I=0, showing that while this approximation is relatively accurate, it overpredicts the current, and consequently the power output, by a few percent. This indicates that the approximation, while useful for rough estimations, is not suitable for precise calculations. The power output P is not accurately calculated from the approximation P = I_V=0V_I=0, instead requiring using the local slope obtained from the I-V curve around V = 0. 4. **Comparison to GHK Equation:** The paper contrasts the derived conductance expression with the Goldman-Hodgkin-Katz (GHK) equation, commonly used in biological ion transport studies. The GHK equation's assumption of a constant electric field leads to discrepancies, particularly in systems with significant surface charge. The authors argue that their model, based on the PNP equations and corroborated by numerical simulations, provides a more accurate representation of ion transport under these conditions.

The results of this study provide a significant advancement in understanding and modeling the electrical behavior of nanofluidic systems. The accurate expressions for *G*_ohmic, *V*_I=0, and *I*_V=0 offer valuable insights into the underlying physics of ion transport and enable more precise predictions of system performance. The improved accuracy in calculating power output is crucial for the design and optimization of RED systems. The comparison with the GHK equation highlights the limitations of existing models in scenarios with significant surface charge density and emphasizes the importance of using more accurate and comprehensive theoretical models. The discrepancies with the GHK equation suggest that the traditional approach used in biological contexts could benefit from a revision using more rigorous physical models, especially in cases where the surface charge plays a major role.

This work provides a robust theoretical framework for characterizing the electrical behavior of nanofluidic systems, offering accurate analytical expressions for crucial parameters. The validated model allows for improved design and optimization of nanofluidic devices for water desalination and energy harvesting applications. Future research could focus on extending the model to include microchannel effects, concentration-dependent surface charge, and multi-species electrolytes. Investigating the combined effects of pressure, potential, and concentration gradients would further enhance the model's applicability.

The current model assumes a constant surface charge density, which might not be valid in all scenarios, particularly in the presence of surface charge regulation. The simplified nanochannel geometry also represents a limitation, although the results are expected to be qualitatively applicable to more complex geometries. The model, in its current form, does not account for advection effects, which could be important in some systems.