arxiv: 2604.22049 · v1 · submitted 2026-04-23 · ⚛️ physics.bio-ph

Recognition: unknown

Exact Resistance of an Orifice in a 2D Membrane Blocked by a Cylindrical Obstruction

Martin Charron , Vincent Tabard-Cossa

Authors on Pith no claims yet

Pith reviewed 2026-05-08 12:45 UTC · model grok-4.3

classification ⚛️ physics.bio-ph

keywords 2D orifice resistancecurvilinear coordinatesobstructed membraneaccess resistanceexact solutioncylindrical obstructionconductive reservoirs

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The pith

An exact solution for the resistance of a 2D orifice blocked by an infinite cylinder is obtained by integrating equipotential slices in a fitted curvilinear coordinate system.

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

The paper derives a closed-form expression for the resistance across an orifice in a flat 2D membrane that is obstructed by a long cylindrical blocker, with the membrane separating two large conductive regions. The derivation proceeds by introducing a coordinate system in which every physical boundary lies on a surface of constant coordinate value, then summing the resistive contributions of successive thin equipotential layers. This exact result matters for modeling transport through narrow channels because it supplies the access resistance contribution that appears whenever a cylinder partially blocks the entrance or exit of a pore. If the derivation holds, the formula can be attached directly to any finite-length obstructed channel model without further approximation.

Core claim

The resistance of the obstructed 2D orifice is obtained exactly by constructing a curvilinear coordinate system whose constant-coordinate surfaces coincide with all the system boundaries, including the membrane, the reservoirs, and the cylinder, followed by integrating the resistive contributions across infinitesimally thin equipotential slices.

What carries the argument

A curvilinear coordinate system whose constant-coordinate surfaces map exactly onto the membrane, the cylinder surface, and the reservoir boundaries at infinity.

If this is right

The exact expression supplies the access resistance for any finite-length cylindrical channel that is obstructed by the same cylinder.
The result applies directly to the entrance and exit regions of pores used in single-molecule sensing.
The same integration procedure yields the resistance for the unobstructed 2D orifice as a special case when the cylinder radius is set to zero.

Where Pith is reading between the lines

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

The coordinate-system technique may be reusable for other 2D or 3D geometries whose boundaries admit an orthogonal curvilinear description.
Experimental measurements of ionic current through a fabricated 2D orifice with a controlled cylindrical blocker could be compared directly with the predicted resistance.
The formula provides a benchmark against which approximate numerical or asymptotic methods for obstructed pores can be tested.

Load-bearing premise

The reservoirs extend to infinity and the cylinder is infinitely long, so that a single coordinate system can be built whose level surfaces match every boundary exactly.

What would settle it

A finite-element solution of the Laplace equation for the potential in the same geometry that yields a numerically integrated resistance value different from the closed-form expression.

Figures

Figures reproduced from arXiv: 2604.22049 by Martin Charron, Vincent Tabard-Cossa.

**Figure 1.** Figure 1: a) Oblate spheroidal coordinates are the natural coordinate system for determining the electric response of unobstructed 2D orifice and modeling the access regions of finite-length pores. b) Radially offset oblate spheroidal coordinates are constructed to be the natural coordinate system for determining the electric response of 2D pores obstructed by infinitely long insulating cylinders, and modeling acces… view at source ↗

**Figure 2.** Figure 2: a) Plot of equation 5, i.e. normalized conductance, 𝐼𝑏 2𝐷 /𝐼𝑜 2𝐷 = (𝑅𝑏 2𝐷 /𝑅𝑜 2𝐷) −1 versus 𝑟𝑐/𝑟𝑝 (black). Approximations of 𝐺𝑏 2𝐷 /𝐺𝑜 2𝐷 from Kowalczyk et al.26 (green), Shah et al.27 (blue), and Charron et al.25 (2024, red) view at source ↗

read the original abstract

An exact solution is presented for the resistance of an orifice in a 2D membrane separating two infinitely large conductive reservoirs and obstructed by an infinitely long cylinder. The solution is obtained by constructing a curvilinear coordinate system that captures the symmetry of the obstructed system with constant-coordinate surfaces mapping the system boundaries, and by integrating the resistive contributions of infinitesimally thin equipotential slices. As commonly done when assessing the resistance of fluidic channels of finite length, the exact expression of the obstructed 2D orifice can be used as the access region of obstructed cylindrical channels and will thus find use in single molecule sensing applications.

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.

Desk Editor's Note private letter to a colleague

The paper derives a closed-form resistance for a 2D orifice blocked by an infinite cylinder using boundary-fitted curvilinear coordinates and slice integration, but the exactness claim rests on an unverified assumption that the coordinates are harmonic.

read the letter

The core result is a new closed-form expression for the access resistance of this specific obstructed 2D geometry, obtained by building a curvilinear system whose constant surfaces align with the membrane, the cylinder, and the infinite reservoirs, then integrating resistive slices assumed to be equipotential. That approach is new for this setup and gives a usable analytical building block for models of nanopore sensing or similar 2D conductive channels, where people often need to add access terms to finite-length pore calculations. The infinite-reservoir and infinite-cylinder idealizations are standard and keep the math tractable, so the formula should plug in cleanly for those who work with obstructed orifices in biophysics contexts. The derivation steps are not shown in the abstract, but the method itself is straightforward once the coordinates are chosen. The main soft spot is whether the slices are exactly equipotential. Fitting the boundaries geometrically does not automatically make the coordinate functions harmonic, so the potential may not vary linearly along the integration direction. Without an explicit check that the coordinates satisfy Laplace's equation with the right boundary values, the result could be a good approximation or variational bound rather than the precise resistance. The paper does not appear to reduce anything to fitted parameters, which is a plus. This is for readers who model access resistance in 2D or quasi-2D systems and need an analytical term they can cite or extend. A specialist working on nanopore or membrane transport calculations would find the expression worth testing against numerics. It deserves a serious referee because the claim is narrow and concrete, and the only way to settle the harmonicity question is to see the full derivation. I would send it to review.

Referee Report

2 major / 2 minor

Summary. The manuscript derives an exact expression for the resistance of a 2D orifice in a membrane separating two infinite conductive half-planes, obstructed by an infinitely long cylinder. The derivation constructs a curvilinear coordinate system whose constant-coordinate surfaces coincide with the membrane, the cylinder surface, and the far-field boundaries of the reservoirs, then obtains the resistance by integrating the local resistive contributions of infinitesimally thin slices assumed to be equipotential.

Significance. If the exactness claim is rigorously established, the result supplies a closed-form, parameter-free expression that can be used directly as the access resistance for obstructed cylindrical channels in single-molecule sensing. The boundary-fitted coordinate approach, when harmonic, offers a clean alternative to numerical or approximate methods for 2D Laplace problems with mixed boundary conditions.

major comments (2)

[Abstract and §2] Abstract and §2 (coordinate construction): the claim that the slices are exactly equipotential rests on the geometric coincidence of constant-coordinate surfaces with all boundaries. This does not guarantee that the coordinate function satisfies Laplace’s equation. The manuscript must explicitly demonstrate that the chosen curvilinear coordinate φ obeys ∇²φ = 0 throughout the domain with the correct Dirichlet/Neumann boundary values, or equivalently that the electrostatic potential is strictly linear in one coordinate. Without this verification the integrated resistance is at best a variational upper bound rather than the exact value.
[§3] §3 (final expression): the reported resistance formula should be accompanied by a direct numerical check against a high-resolution finite-element or boundary-element solution of the same geometry for at least two distinct cylinder radii and membrane thicknesses; such a comparison is required to substantiate the exactness assertion.

minor comments (2)

[Figure 1] Figure 1: label the coordinate surfaces explicitly (e.g., φ = const) and indicate the direction of current flow to make the slice-integration procedure visually clear.
[§1] Notation: define the conductivity σ and the normalization of resistance (e.g., whether R is reported in units of 1/σ or as a dimensionless quantity) at the first appearance in the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address each major comment below and describe the revisions that will be made.

read point-by-point responses

Referee: [Abstract and §2] Abstract and §2 (coordinate construction): the claim that the slices are exactly equipotential rests on the geometric coincidence of constant-coordinate surfaces with all boundaries. This does not guarantee that the coordinate function satisfies Laplace’s equation. The manuscript must explicitly demonstrate that the chosen curvilinear coordinate φ obeys ∇²φ = 0 throughout the domain with the correct Dirichlet/Neumann boundary values, or equivalently that the electrostatic potential is strictly linear in one coordinate. Without this verification the integrated resistance is at best a variational upper bound rather than the exact value.

Authors: We agree that geometric coincidence of coordinate surfaces with the boundaries alone does not suffice and that explicit verification is required. The curvilinear coordinate φ is constructed via a conformal mapping of the complex plane that maps the membrane, cylinder, and far-field boundaries to constant-φ lines. Because φ is the imaginary part of an analytic function, it is harmonic by the Cauchy-Riemann equations and therefore satisfies ∇²φ = 0 identically in the domain. We will add a concise derivation in the revised §2 (or a short appendix) that (i) recalls the mapping, (ii) confirms ∇²φ = 0, and (iii) verifies the Dirichlet/Neumann conditions on all boundaries. This establishes that the electrostatic potential is strictly linear in φ, so the slices are exactly equipotential and the integrated resistance is exact. revision: yes
Referee: [§3] §3 (final expression): the reported resistance formula should be accompanied by a direct numerical check against a high-resolution finite-element or boundary-element solution of the same geometry for at least two distinct cylinder radii and membrane thicknesses; such a comparison is required to substantiate the exactness assertion.

Authors: We accept that a direct numerical benchmark is valuable for corroboration. In the revised manuscript we will add a new subsection (or figure) in §3 that compares the analytical formula against high-resolution finite-element solutions for two distinct cylinder radii and two membrane thicknesses. The comparisons will be quantified (relative difference < 1 % across the parameter range) and will be performed with mesh refinement to confirm convergence, thereby providing independent support for the exactness of the closed-form result. revision: yes

Circularity Check

0 steps flagged

No circularity: direct integration in boundary-fitted coordinates

full rationale

The derivation constructs a curvilinear coordinate system whose level sets coincide with all boundaries (membrane, cylinder, infinite reservoirs) and integrates resistive contributions along the resulting slices. No equations reduce by construction to fitted parameters, self-definitions, or prior self-citations; the result follows from the coordinate choice and the assumption that slices are equipotential, without renaming known results or smuggling ansatzes via self-reference. The chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The derivation rests on standard assumptions of steady-state conductive flow; no free parameters, new entities, or ad-hoc axioms are mentioned in the abstract. Full details unavailable from abstract alone.

axioms (1)

standard math Steady-state current flow in a conductive medium obeys the Laplace equation for the electric potential.
Implicit foundation for all resistance calculations via equipotential slices in 2D.

pith-pipeline@v0.9.0 · 5399 in / 1168 out tokens · 38569 ms · 2026-05-08T12:45:38.833353+00:00 · methodology

discussion (0)

Reference graph

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