Access resistance
Access resistance is the part of a pore's resistance that lies outside the pore. Current that has spread through a large volume of electrolyte must funnel into a small opening, and the funnel costs resistance on the way in and again on the way out. For a circular opening of radius a in an insulating wall, with electrolyte of resistivity ρ on both sides, each side contributes ρ/(4a). The two sides together add ρ/d for a pore of diameter d, in series with the resistance of the channel through the wall.[1]
The term decides how a thin membrane behaves. A 5 nm pore through 20 nm of silicon nitride spends 16% of its resistance outside the pore, and a pore through a membrane one atom thick spends most of it there. The expression is credited to J. E. Hall, whose note on ion channels ran to two pages. It is the reason the conductance of a solid-state nanopore cannot be computed from its hole alone, and it enters every estimate of the open-pore current.
Formula
Hall's result is a problem in electrostatics: the resistance between a conducting disk set into an insulating plane and a distant hemispherical electrode. Taking the mouth of the pore to be an equipotential disk gives ρ/(4a) on each side.[2] A cruder estimate, which sums hemispherical shells of electrolyte spreading outward from the mouth, gives ρ/(2πa). A flat disk crowds the current toward its rim, so its resistance is higher than the shell estimate by a factor of π/2.
Adding a cylindrical channel of length L gives the resistance of the whole pore, and the conductance is its reciprocal:
Here σ = 1/ρ is the conductivity of the electrolyte. The first term is a column of electrolyte inside the wall and the second is the access term. They are equal when L = πd/4, about 0.8d, so a wall thinner than that spends most of its resistance outside the pore. Kowalczyk and colleagues measured the conductance of silicon nitride pores 5 to 100 nm wide and found that the access term was needed to describe it, most of all for the wider pores. An exact solution for an hourglass-shaped pore matched their data with no adjustable parameter.[3]
| Membrane thickness | Channel | Access | Share outside the pore | Current at 200 mV |
|---|---|---|---|---|
| 50 nm | 232 MΩ | 18 MΩ | 7% | 0.80 nA |
| 20 nm | 93 MΩ | 18 MΩ | 16% | 1.8 nA |
| 10 nm | 46 MΩ | 18 MΩ | 28% | 3.1 nA |
| 2 nm | 9 MΩ | 18 MΩ | 66% | 7.3 nA |
| 0 | 0 | 18 MΩ | 100% | 11 nA |
The table takes σ = 11 S/m, close to the conductivity of potassium chloride at 1 M and 25 °C. The 10 nm row is a 5 nm pore through 10 nm of membrane, and the 3.1 nA it carries is the figure worked through in Ionic current; leaving the access term out would give 4.3 nA, which is 39% too high. The last row is a wall of zero thickness, where the conductance reduces to σd and everything the current meets is mouth.
Consequences
Most of the voltage across the access region falls within a few pore diameters of the opening, so the zone in which a molecule changes the current reaches out beyond the faces of the membrane. Feng and colleagues, working with single-layer molybdenum disulfide, state the result directly: the size of the sensing region is determined by the size of the pore and the thickness of the membrane.[4] A 2026 review of solid-state pores says that membranes would ideally be thinner than 0.5 nm to tell the bases of DNA apart, and adds: "Only by simultaneously addressing nanopore thickness and access resistance can the goal of single-base resolution be approached."[5]
Graphene shows the effect. The graphene membranes in the first conductance measurements were one or two atoms thick, yet the conductance of pores drilled in them fits an effective insulating thickness of less than 1 nm.[6] The colloquium cited above puts the effective thickness of monolayer graphene at about 1 nm, against the roughly 0.3 nm implied by the van der Waals diameter of a carbon atom, because ions have diameters of about 0.3 nm of their own and are held back from the pore edge by repulsion.[2]
A molecule inside the channel changes only the channel term, so the fractional current blockade of a thin pore is smaller than the ratio of areas alone would suggest. The access term also sets the electric field in the solution outside the pore, and so the rate at which DNA is drawn in.[a]
Wider apertures
The arithmetic is the same at larger scale. For a 100 µm aperture in a 10 µm membrane, πd/4L is 7.9, so the access term is 89% of the total resistance. The design notes of the open MR1 Nanopore Reader give that figure for that geometry.[8] A Coulter counter aperture is a pore in the same sense, and the access term enters the size of its pulses.
Limits of the formula
Hall's expression assumes a homogeneous electrolyte with no concentration gradients or charge accumulation. It further assumes a uniform potential across the mouth of the pore, no electric field perpendicular to the membrane, and a hemispherical electrode at infinity.[2] Surface charge breaks the second assumption, most visibly in dilute salt, where counterions carry current along the membrane faces; see Surface charge (nanopore) and Pore conductance. In pores a nanometer or less across, the size of hydrated ions enters as well.[2]
History
The colloquium credits Maxwell with the expression: he derived it in the nineteenth century for electrical conduction to an orifice, long before single ion channels were recorded. It lists the same mathematics under other names in heat conduction, gas diffusion and electrical contacts, and gives the quantity several names of its own, among them convergence resistance, interfacial resistance and contact resistance.[2] One of its gas-diffusion citations is a 1900 paper by Brown and Escombe on how gases move through small apertures, written with the uptake of carbon dioxide by leaves in mind.[b] Hall's note applied the result to the channels of cell membranes, and the formula now turns up wherever a nanopore is modeled as a resistor.
See also
Notes
- ^ At distances r much larger than d, the current leaves the mouth of the pore as if from a point on the surface of a half-space, and the potential falls as ΔV d2/(8r(L + πd/4)). Wanunu and colleagues use the form with L alone, which holds when the channel dominates, for the field that focuses DNA toward the pore; for a thin membrane the access term replaces L by L + πd/4.[7]
- ^ The paper is H. T. Brown and F. Escombe, "Static diffusion of gases and liquids in relation to the assimilation of carbon and translocation in plants", Philosophical Transactions of the Royal Society B 193:223-291 (1900), doi:10.1098/rstb.1900.0014. "Translocation" there is the movement of sugars in a plant.
References
- ^ Hall, J. E. (1975). "Access resistance of a small circular pore". Journal of General Physiology 66 (4): 531-532. doi:10.1085/jgp.66.4.531. PMID 1181379.
- ^ a b c d e f Sahu, S.; Zwolak, M. (2019). "Colloquium: Ionic phenomena in nanoscale pores through 2D materials". Reviews of Modern Physics 91 (2): 021004. doi:10.1103/RevModPhys.91.021004.
- ^ a b Kowalczyk, S. W.; Grosberg, A. Y.; Rabin, Y.; Dekker, C. (2011). "Modeling the conductance and DNA blockade of solid-state nanopores". Nanotechnology 22 (31): 315101. doi:10.1088/0957-4484/22/31/315101. PMID 21730759.
- ^ Feng, J.; et al. (2015). "Identification of single nucleotides in MoS2 nanopores". Nature Nanotechnology 10 (12): 1070-1076. doi:10.1038/nnano.2015.219. PMID 26389660.
- ^ Chen, X.; et al. (2026). "Addressing the challenges of solid-state nanopores: strategies for performance enhancement". International Journal of Molecular Sciences 27 (6): 2536. doi:10.3390/ijms27062536.
- ^ Garaj, S.; et al. (2010). "Graphene as a subnanometre trans-electrode membrane". Nature 467 (7312): 190-193. doi:10.1038/nature09379. PMID 20720538.
- ^ Wanunu, M.; et al. (2010). "Electrostatic focusing of unlabelled DNA into nanoscale pores using a salt gradient". Nature Nanotechnology 5 (2): 160-165. doi:10.1038/nnano.2009.379. PMID 20023645.
- ^ "MR1 Nanopore Reader: Micropore & Salt Aperture Configuration Matrix". GitHub. Molecular Reality Corporation. October 2026.
