A theoretical characterization of osmotic power generation in nanofluidic systems

The work addresses the fundamental question of how to self-consistently describe ion transport and the resulting electrical response (I–V characteristics) of ion-selective nanofluidic systems under arbitrary concentration gradients. In desalination (ED) and reverse electrodialysis (RED), permselective materials with surface charge allow counterion-dominated transport, producing currents driven by voltage and/or concentration differences. Despite extensive experimental and numerical studies, a unified theory for the three key electrical metrics—ohmic conductance Gohmic, zero-voltage current IV=0, and zero-current (open-circuit) voltage VI=0—has been lacking. The authors aim to derive explicit expressions for these metrics directly from first principles to enable accurate interpretation of experiments and optimization of nanofluidic devices for water desalination and osmotic energy harvesting. The study emphasizes the universality of permselectivity across many materials when channels are sufficiently small and electrolyte concentrations low, and highlights how concentration asymmetry shifts the I–V curve away from the origin, necessitating a general theoretical treatment.

The paper situates the problem within decades of research on permselective transport and nanofluidics, listing numerous materials and systems that exhibit permselectivity: carbon nanotubes and porins, boron nitride nanotubes, silicon nanochannels/nanopores, graphene and graphene oxide membranes, conducting hydrogels, wood-based cellulose membranes, biological pores, single-layer MoS2, vermiculite-based membranes, and metal-organic framework membranes, among others. Prior work commonly characterized systems by I–V curves, noted the ohmic regime and shifts under concentration asymmetry, and proposed empirical or phenomenological formulas for conductance and open-circuit voltage. A widely used empirical conductance expression GOhmic ≈ DF^2(Σ + 2Cbulk)A/(RTL) (sum of bulk and surface contributions) has been criticized for lacking rigorous derivation. Classic biological membrane literature, notably the Goldman-Hodgkin-Katz (GHK) equation, provided I–V relations under the “no fixed charge” assumption (vanishing selectivity), often adopting a constant-field approximation that implies exponential concentration profiles. Recent modeling has incorporated microchannel and access resistances, concentration polarization, and surface charge regulation, but a closed-form, self-consistent theory for Gohmic, IV=0, and VI=0 under asymmetric concentrations for permselective nanochannels has remained missing.

The authors derive the electrical response from the Poisson–Nernst–Planck (PNP) equations for a nanochannel-only system connecting two reservoirs with possibly asymmetric bulk concentrations Cleft and Cright. Assumptions include: binary 1:1 electrolyte (KCl-like) with equal diffusion coefficients (D+ = D− = D), isothermal conditions (T uniform), and a constant, spatially uniform surface charge density yielding an average excess counterion concentration Σ = σP/(FA), with P the channel perimeter and A its cross-sectional area. The analysis applies to arbitrary cross-sections (e.g., 2D parallel plate or cylindrical) provided L ≫ A characteristic dimension to ensure a quasi-1D description. The authors obtain closed-form expressions for the ohmic conductance under asymmetric concentrations, the open-circuit voltage, and the zero-voltage current by solving the PNP system and identifying limits corresponding to ideal selectivity (Σ ≫ C) and vanishing selectivity (Σ ≪ C). They verify the analytical results against non-approximated numerical simulations (details in Supplementary Methods), showing excellent agreement across parameter ranges and validating the theoretical expressions.

• Symmetric concentrations (Cbulk = Cleft = Cright): The well-known non-empirical “square-root” law for the nanochannel-only conductance is recovered (Eq. (1)), contrasting with the frequently used but incorrect additive empirical form. Defining Σ = σP/(FA), the two limits are: • Vanishing selectivity: GOhmic^(vanishing-selectivity) = (2DF^2 Cbulk A)/(RTL) [Eq. (3)] • Ideal selectivity: GOhmic^(ideal-selectivity) = (DF^2 Σ A)/(RTL) [Eq. (4)] Conductance transitions near Σ ≈ 2Cbulk, with slope vs concentration of 1 in the vanishing selectivity limit and 0 in the ideal selectivity limit (Fig. 2 inset). The additive empirical model GOhmic ≈ DF^2(Σ + 2Cbulk)A/(RTL) [Eq. (5)] is shown to be an incorrect approximation and should be discontinued.
• Asymmetric concentrations (Cleft ≠ Cright): • General ohmic conductance: GOhmic = −(2DF^2 A)/(RTL) · (Sleft − Sright)/ln(Sleft/Sright), with Sleft = √(Σ^2 + Cleft^2), Sright = √(Σ^2 + Cright^2) [Eqs. (7)–(8)]. This reduces to the symmetric result when Cleft = Cright. The transition from bulk- to surface-charge-dominated transport depends strongly on the larger of Cleft or Cright (Fig. 3a). • Zero-current voltage: VI=0 = Vth [ ln(Cright/Cleft) + ln( (Sleft + Σ)/(Sright + Σ) ) ] (as presented in Eq. (9); second term vanishes in the ideal selectivity limit Σ ≪ Cleft, Cright), recovering VI=0^(ideal-selectivity) = Vth ln(Cright/Cleft) [Eq. (6)] at low concentrations (Fig. 3b). • Zero-voltage current: The exact IV=0 (or current density iV=0) follows from the derived I–V relation (implicit Eq. (12)) and related expressions [Eqs. (10)–(13)]. Common experimental approximation |iV=0| ≈ GOhmic · |VI=0| overpredicts the current and thus the extracted power by a few percent; the correct value must use the full I–V relation (Fig. 3c).
• Power estimation: The correct small-signal power near V = 0 depends on the local differential conductance at the operating point (P = I^2/GOhmic = V^2 GOhmic for local slopes; see Eqs. (14)–(15)). Using |iV=0| ≈ GOhmic|VI=0| leads to an overestimation of RED power due to the incorrect current approximation.
• Biological transport implications (GHK comparison): For Z_i = 0 (vanishing selectivity), the derived conductance differs from the GHK constant-field result. The paper presents GGHK (Eq. (16)) versus the present Gohmic(Z_i = 0) = (2DF^2 A)/(RTL) · (Cleft − Cright)/ln(Cleft/Cright) (Eq. (17)), showing that a rigorous PNP-based derivation with linear concentration and logarithmic potential (for Z_i = 0) yields a different, correct conductance without ad hoc constant-field assumptions.
• Numerical validation: Across parameter sweeps in Fig. 3, analytical predictions for GOhmic, VI=0, and iV=0 align closely with full numerical simulations, supporting the model’s accuracy.

The derived expressions provide a long-sought self-consistent framework for interpreting the I–V behavior of permselective nanochannels under arbitrary concentration gradients. By unifying the conductance and the two osmotic metrics (IV=0 and VI=0) in closed form, the theory explains how surface charge (Σ) and asymmetric reservoir concentrations jointly set the electrical response. It resolves ambiguities introduced by empirical additive models and phenomenological corrections to the ideal-selectivity open-circuit voltage. The model delineates the transition from bulk-dominated to surface-charge-dominated regimes and quantifies how the larger of the two reservoir concentrations controls the conductance crossover. Importantly, it corrects common experimental practices that approximate osmotic current and power via GOhmicVI=0, demonstrating systematic overprediction and prescribing the use of the full I–V relation or local differential conductance for accurate power estimation. Compared to classical biological membrane theory (GHK), the work clarifies that constant-field assumptions distort conductance predictions even in the vanishing selectivity limit, and that a PNP-consistent approach yields the appropriate logarithmic potential and linear concentration profiles (for Zi = 0), matching numerical simulations. These insights are directly relevant to optimizing nanofluidic desalination and energy harvesting devices and to reinterpreting past measurements.

The study presents an analytical, self-consistent characterization of nanofluidic ion transport under combined potential and concentration gradients, yielding explicit formulas for: (i) ohmic conductance under symmetric and asymmetric concentrations (Eqs. (1), (7)), (ii) open-circuit voltage VI=0 valid beyond ideal selectivity (Eq. (9)), and (iii) zero-voltage current IV=0 from the general I–V relation (Eq. (12)). The results unify limiting behaviors (ideal and vanishing selectivity), supersede empirical additive conductance models, correct common current/power approximations, and reconcile discrepancies with constant-field approaches in biological transport. These contributions enable reliable interpretation of experiments and guide design of enhanced RED/osmotic power and desalination systems. Future research should incorporate microchannel/access effects, surface charge regulation under asymmetric concentrations, advection and full 3×3 transport coefficient matrices, multicomponent electrolytes with unequal diffusivities, and thermally driven (Soret) effects to broaden applicability.

• The analysis considers a nanochannel-only model and neglects microchannel/access/entrance effects, which can significantly affect total system conductance in practice.
• Assumes a binary 1:1 electrolyte with equal diffusion coefficients (KCl-like); multicomponent electrolytes and unequal mobilities are not treated.
• Surface charge/excess counterion concentration Σ is taken constant and spatially uniform; surface charge regulation and spatial dependence under asymmetric concentrations are not included.
• Advection and pressure-driven contributions are neglected (no coupling to hydrodynamics in this work).
• Thermal gradients and thermoelectric (Soret) effects are not considered; temperature is uniform.
• Some results are validated numerically for model geometries; experimental verification across diverse materials/geometries is implied but not reported within this text.