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arxiv: 2601.01536 · v2 · submitted 2026-01-04 · ❄️ cond-mat.mes-hall · physics.bio-ph

The transmembrane potential across a charged nanochannel subjected to asymmetric electrolytes

Ramadan Abu-Rjal , Yoav Green This is my paper

Pith reviewed 2026-05-16 17:54 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall physics.bio-ph
keywords transmembrane voltagenanochannelasymmetric electrolytesPoisson-Nernst-Planckzero-current potentialion transportcharged nanoporesdesalination
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The pith

Two analytical expressions for the zero-current transmembrane voltage in charged nanochannels with asymmetric electrolytes are derived from the Poisson-Nernst-Planck equations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives closed-form expressions for the voltage V that must be applied across a charged nanochannel to produce zero net current when the two ends are bathed in different salt solutions. For the case of two ion species an exact solution is obtained without additional approximations. For an arbitrary number of species a second expression is obtained by assuming a linear concentration profile along the channel length. Both expressions recover known limiting cases, match numerical solutions of the one-dimensional Poisson-Nernst-Planck system, and demonstrate that the resulting voltage depends on the full set of ionic diffusivities and valences rather than on a single effective parameter.

Core claim

We derive two expressions for V_{i=0}. First, we consider the much simpler scenario of two species. Then, we can consider an electrolyte comprised of an arbitrary number of species. The difference in the models is that the latter solution utilizes an ad-hoc assumption of a linear concentration profile, while the former solution does not require such an ad-hoc assumption. We analyze both models and show how to reduce them to several known models. We also verify both models with numerical simulations of the one-dimensional Poisson-Nernst-Planck equations. We show how the interplay between diffusion coefficients and ionic valencies significantly varies the system response and why it isessential

What carries the argument

The zero-current transmembrane voltage V_{i=0} obtained by setting the total ionic current to zero in the Nernst-Planck flux equations under Poisson electrostatics.

If this is right

  • The expressions reduce to several previously published models in appropriate limits.
  • The voltage depends on the full set of diffusion coefficients and ionic valences rather than on a single effective parameter.
  • The models can be used to interpret experimental ion-transport measurements in nanoporous membranes.
  • The two-species exact solution provides a benchmark against which the linear-profile approximation can be tested.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The exact two-species solution may serve as a test case for developing higher-order approximations that avoid the linear-profile assumption entirely.
  • Because the voltage enters energy-harvesting and desalination efficiency calculations, small changes in predicted V_{i=0} translate directly into changes in predicted power density.
  • The same framework could be applied to physiological ion channels once surface-charge and geometry parameters are supplied.

Load-bearing premise

The multi-species expression relies on assuming a linear concentration profile across the channel without detailed justification or bounds on its validity.

What would settle it

A direct numerical solution of the one-dimensional Poisson-Nernst-Planck equations for a multi-species electrolyte with non-linear concentration profiles that deviates from the analytical V_{i=0} prediction by more than numerical discretization error.

read the original abstract

The transmembrane voltage, $V$, which is the potential drop required to nullify the electrical current ($i=0$), is a key characteristic of water desalination and energy harvesting systems that utilize macroscopically large nanoporous membranes, as well as for physiological ion channels subjected to asymmetric salt concentrations. To date, existing analytical expressions for $V_{i=0}$ have been limited to simple scenarios or under simplifying assumptions. In this work, we derive two expressions for $V_{i=0}$. First, we consider the much simpler scenario of two species. Then, we can consider an electrolyte comprised of an arbitrary number of species. The difference in the models is that the latter solution utilizes an ad-hoc assumption of a linear concentration profile, while the former solution does not require such an ad-hoc assumption. We analyze both models and show how to reduce them to several known models. We also verify both models with numerical simulations of the one-dimensional Poisson-Nernst-Planck equations. We show how the interplay between diffusion coefficients and ionic valencies significantly varies the system response and why it is essential to account for all system parameters. Ultimately, this model can be used to improve experimental interpretation of ion transport measurements

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript derives closed-form expressions for the zero-current transmembrane voltage V_{i=0} in a charged nanochannel with asymmetric electrolytes. For binary electrolytes the expression follows directly from the steady-state PNP current-null condition. For an arbitrary number of species an ad-hoc linear concentration profile is imposed to close the integrated flux equations, yielding an explicit formula. Both expressions are reduced to known limits and compared to numerical solutions of the one-dimensional Poisson-Nernst-Planck equations. The work stresses the interplay between diffusion coefficients and ionic valences.

Significance. If the linear-profile approximation holds in the regimes of interest, the results supply practical analytical tools for interpreting transmembrane potentials in nanoporous membranes used for desalination and energy harvesting as well as in physiological ion channels. The binary result is free of extra assumptions and directly verifiable; the multi-species result extends generality but its accuracy is conditional on the fidelity of the imposed profile. Numerical checks are provided, yet the absence of quantified error bounds restricts immediate use for arbitrary electrolytes.

major comments (2)
  1. [§3.2] §3.2 (multi-species derivation): the linear concentration profile c_i(x) is introduced without derivation, asymptotic analysis (e.g., small-Debye-length or weak-charge limits), or error estimates as a function of surface charge, concentration ratio, or valence. Because this assumption closes the integrated Nernst-Planck fluxes for the arbitrary-species case, the headline claim that the model works for “an electrolyte comprised of an arbitrary number of species” rests entirely on the unquantified accuracy of this profile.
  2. [Numerical verification] Numerical verification section: comparisons with 1D PNP simulations are shown, but the deviation between the linear-profile formula and the full numerical solution is not mapped versus Debye length, surface charge density, or valence combinations. Consequently the practical range of validity of the multi-species expression remains uncharted.
minor comments (2)
  1. [Abstract] Abstract: the statement that both models “reduce to several known models” is not accompanied by explicit citations or section references; the reductions should be identified.
  2. [§3.2] Notation: the definition of the effective diffusion coefficient or averaged mobility used in the multi-species formula should be stated explicitly rather than left implicit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and have revised the manuscript to strengthen the presentation of the multi-species approximation and its validation.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (multi-species derivation): the linear concentration profile c_i(x) is introduced without derivation, asymptotic analysis (e.g., small-Debye-length or weak-charge limits), or error estimates as a function of surface charge, concentration ratio, or valence. Because this assumption closes the integrated Nernst-Planck fluxes for the arbitrary-species case, the headline claim that the model works for “an electrolyte comprised of an arbitrary number of species” rests entirely on the unquantified accuracy of this profile.

    Authors: We agree that the linear profile is an ad-hoc closure assumption rather than an exact solution of the PNP system. It is motivated by the expectation that, for channels much longer than the Debye length, the concentration variation between reservoirs is approximately linear when the net current is zero. In the revised manuscript we add a dedicated paragraph in §3.2 that (i) derives the linear profile as the leading-order solution in the thin-double-layer limit, (ii) provides the corresponding asymptotic expansion of V_{i=0}, and (iii) reports relative-error bounds obtained from the existing PNP simulations for representative values of surface charge, concentration ratio, and valence. The abstract and introduction are also updated to state explicitly that the multi-species formula is approximate. revision: yes

  2. Referee: [Numerical verification] Numerical verification section: comparisons with 1D PNP simulations are shown, but the deviation between the linear-profile formula and the full numerical solution is not mapped versus Debye length, surface charge density, or valence combinations. Consequently the practical range of validity of the multi-species expression remains uncharted.

    Authors: We accept this criticism. The revised numerical section now includes two additional figures: one showing the relative error of the multi-species formula versus normalized Debye length (spanning two orders of magnitude in bulk concentration) at fixed surface charge, and a second contour plot of the same error over the plane of surface charge density versus valence asymmetry. These maps are generated from the same 1D PNP solver already used in the manuscript and are accompanied by a short table of maximum errors for the parameter ranges typical of desalination and biological channels. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations start from PNP equations and remain independent of target result

full rationale

The paper derives the two V_{i=0} expressions directly from the steady-state Poisson-Nernst-Planck current-null condition. The two-species case closes without extra assumptions. The arbitrary-species case introduces an explicit ad-hoc linear c_i(x) profile to integrate the flux equations, but this is stated as an assumption rather than obtained by re-arranging the target voltage or by fitting. Verification uses separate numerical PNP solutions. No self-citations, fitted parameters, or self-definitional steps appear; the expressions are not equivalent to their inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The derivations rest on the standard Poisson-Nernst-Planck continuum description plus one ad-hoc linear concentration profile introduced specifically for the arbitrary-species case.

axioms (1)
  • ad hoc to paper Linear concentration profile across the nanochannel for the multi-species model
    Explicitly stated as the key difference from the two-species exact solution and required to close the arbitrary-species expression.

pith-pipeline@v0.9.0 · 5512 in / 1299 out tokens · 40341 ms · 2026-05-16T17:54:57.857640+00:00 · methodology

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