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Electric double layer

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From Molecupedia, the encyclopedia of very small things
Pair of charged regions at the boundary between a surface and an electrolyte

An electric double layer is the pair of oppositely charged regions that forms where a charged surface meets an electrolyte: the charge on the surface, and a cloud of ions of the opposite sign in the solution that screens it. The cloud is a few Debye lengths thick, which is 0.3 nm in 1 M potassium chloride and 3 nm in 10 mM. The interface therefore acts as a capacitor with a molecular gap, and the double layer of a flat metal electrode in salt water has a capacitance of some 10 to 40 µF per square centimeter.[1]

The standard picture was assembled between 1879 and 1947 by Helmholtz, Gouy, Chapman, Stern and Grahame. It has two parts, a compact layer of ions at the surface and a diffuse cloud beyond it. Inside a nanopore the layer can fill most of the volume, and it governs the pore's surface charge conduction, its electroosmotic flow and its selectivity between ions.

Models

Helmholtz

Hermann von Helmholtz described the interface in 1879 as two sheets of charge a molecular distance apart, one on the surface and one of ions in the liquid, like the plates of a capacitor.[2] The capacitance per unit area of such a pair is the permittivity divided by the gap, and it does not depend on the potential or the salt concentration. Measured double layers do depend on both.[1]

Gouy and Chapman

Louis Georges Gouy in 1910 and David Leonard Chapman in 1913 let the counter-charge spread out under thermal motion.[3][4] The ions form a diffuse cloud whose density follows the Boltzmann distribution in the local potential, and the Poisson-Boltzmann equation for a flat wall has the solution

tanheψ(x)4kBT=tanheψ04kBT e−x/λD.

Far from the wall, and close to it at small potentials, the potential decays by a factor of e for each Debye length λD. The wall charge per unit area follows from the potential ψ0 through the relation that bears Grahame's name, σ = (8εrε0kBTn)1/2 sinh(eψ0/2kBT), where n is the bulk number density of each ion.[5] At small potentials the diffuse layer is a capacitor of εrε0/λD per unit area, 0.23 F/m2 (23 µF/cm2) in 10 mM salt and 2.3 F/m2 in 1 M.

Because the model treats ions as points, it fails at the wall. At a potential of 200 mV it puts the counterion concentration there at 1 M × exp(0.2 V / 25.7 mV), about 2,400 M, more than forty times the concentration of molecules in liquid water. The authors of a 2007 analysis wrote that the theory "predicts absurdly large ion concentrations (exceeding close packing) for surface potentials of only a few tenths of a volt".[6]

Stern and Grahame

Otto Stern's paper of 1924 gave the ions a size.[7] The centers of the ions cannot come closer to the wall than their radius, so a compact layer with a Helmholtz-like gap separates the wall from the diffuse layer of Gouy and Chapman, and the potential falls roughly linearly across the compact layer and exponentially beyond it. The two layers are capacitors in series, 1/C = 1/Ccompact + 1/Cdiffuse, and the smaller one dominates. In 1 M salt the diffuse layer has 2.3 F/m2 and the compact layer decides the total, which is why the measured capacitance stops growing with concentration.[1] Grahame refined the compact layer in 1947. The inner Helmholtz plane passes through the centers of ions that stick to the wall, and the outer Helmholtz plane through the centers of hydrated ions at their nearest approach.[5]

The potential at the plane where the liquid begins to slip past the wall is the zeta potential, ζ. It lies below the wall potential in magnitude and is the quantity that electrophoresis and electroosmotic flow measure.[a]

In nanopores

At pH 8 the surfaces of silicon nitride and silica are negatively charged, so a pore in either holds a layer of positive ions against its walls. In a pore of radius R the counterions needed to cancel a wall charge σ are at a concentration 2|σ|/(eR). For an assumed wall charge of −0.02 C/m2, one elementary charge per 8 nm2, and a radius of 2.5 nm, that is 0.17 M of counterions, whatever the bath. That is about seventeen times the concentration of potassium ions in a 10 mM bath.

Smeets and colleagues found that the conductance of solid-state pores fell by three orders of magnitude while the salt fell by six, and Stein and colleagues found that silica channels as thin as 70 nm reached a conductance that no longer depended on the salt.[9][10] A pore that is mostly counterions also lets one sign of ion through more easily than the other, and the countercharge in the layer moves with the field and drags the water along, which is electroosmotic flow. In a conical pore with charged walls the current at one polarity can exceed the current at the other, an effect called ion current rectification that was described for quartz pipettes in 1997 and reviewed in 2006.[11][12]

The molecule has a double layer too. DNA carries a charge for every 0.17 nm of its length and keeps a sheath of counterions close enough to move with it, so its effective charge falls to about a quarter of the bare value, as described under electrophoresis.

A metal electrode in salt water charges its own double layer like a capacitor, so an inert electrode would charge to the applied voltage and pass no steady current. The silver chloride electrode that nanopore cells use exchanges chloride ions with the solution, so charge crosses the interface continuously and the double-layer capacitance becomes a side path. Outside nanopores the double layer is where electrochemical capacitors store their charge, on carbon electrodes with a large internal surface, and it keeps colloidal particles apart until added salt shortens it and they clump.[13][14]

At Molecular Reality

The company's essay The Nanopore World Model has a slider that drags a pore from 200 µm down to 2 nm. As the pore narrows the essay lists "Electric double layer dominates" among the effects that appear, and it warns that the parts of a model that recognize molecules "must be RE-TRAINED".[15]

See also

Notes

  1. ^ Stern published the model two years after the experiment with Walther Gerlach that carries both their names. He received the Nobel Prize in Physics in 1943 "for his contribution to the development of the molecular ray method and his discovery of the magnetic moment of the proton".[8]

References

  1. ^ a b c Bard, A. J.; Faulkner, L. R. (2001). Electrochemical Methods: Fundamentals and Applications. 2nd ed. Wiley. ISBN 0-471-04372-9.
  2. ^ Helmholtz, H. (1879). "Studien über electrische Grenzschichten". Annalen der Physik 243 (7): 337-382. doi:10.1002/andp.18792430702.
  3. ^ Gouy, M. (1910). "Sur la constitution de la charge électrique à la surface d'un électrolyte". Journal de Physique Théorique et Appliquée 9 (1): 457-468. doi:10.1051/jphystap:019100090045700.
  4. ^ Chapman, D. L. (1913). "A contribution to the theory of electrocapillarity". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 25 (148): 475-481. doi:10.1080/14786440408634187.
  5. ^ a b Grahame, D. C. (1947). "The electrical double layer and the theory of electrocapillarity". Chemical Reviews 41 (3): 441-501. doi:10.1021/cr60130a002.
  6. ^ Kilic, M. S.; Bazant, M. Z.; Ajdari, A. (2007). "Steric effects in the dynamics of electrolytes at large applied voltages. I. Double-layer charging". Physical Review E 75 (2): 021502. doi:10.1103/PhysRevE.75.021502.
  7. ^ Stern, O. (1924). "Zur Theorie der elektrolytischen Doppelschicht". Zeitschrift für Elektrochemie und angewandte physikalische Chemie 30 (21-22): 508-516. doi:10.1002/bbpc.192400182.
  8. ^ "Otto Stern: Facts". NobelPrize.org. Nobel Prize Outreach. Retrieved October 25, 2026.
  9. ^ Smeets, R. M. M.; Keyser, U. F.; Krapf, D.; Wu, M.-Y.; Dekker, N. H.; Dekker, C. (2006). "Salt dependence of ion transport and DNA translocation through solid-state nanopores". Nano Letters 6 (1): 89-95. doi:10.1021/nl052107w.
  10. ^ Stein, D.; Kruithof, M.; Dekker, C. (2004). "Surface-charge-governed ion transport in nanofluidic channels". Physical Review Letters 93 (3): 035901. doi:10.1103/PhysRevLett.93.035901.
  11. ^ Wei, C.; Bard, A. J.; Feldberg, S. W. (1997). "Current rectification at quartz nanopipet electrodes". Analytical Chemistry 69 (22): 4627-4633. doi:10.1021/ac970551g.
  12. ^ Siwy, Z. S. (2006). "Ion-current rectification in nanopores and nanotubes with broken symmetry". Advanced Functional Materials 16 (6): 735-746. doi:10.1002/adfm.200500471.
  13. ^ Simon, P.; Gogotsi, Y. (2008). "Materials for electrochemical capacitors". Nature Materials 7 (11): 845-854. doi:10.1038/nmat2297.
  14. ^ Israelachvili, J. N. (2011). Intermolecular and Surface Forces. 3rd ed. Academic Press. ISBN 978-0-12-375182-9.
  15. ^ "Keltar's Critique v2: The Nanopore World Model (Hardened)". Molecular Reality Corporation. June 2026.
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