Debye length
The Debye length is the distance over which a charge in an electrolyte is screened. Ions of the opposite sign crowd around the charge, and its potential falls off exponentially, by a factor of e (about 2.7) for each Debye length. For a salt of two singly charged ions in water at 25 °C the length is 0.304 nm divided by the square root of the concentration in moles per liter, which makes it 0.30 nm in 1 M potassium chloride and 3.0 nm in 10 mM. In pure water, which holds only 10−7 M each of hydrogen and hydroxide ions, it is 0.96 µm.
Peter Debye and Erich Hückel derived it in 1923 in a paper on how salts lower the freezing point of water.[1] In a nanopore it sets the thickness of the electric double layer on the walls. A pore 5 nm wide is about sixteen Debye lengths across in 1 M salt and less than two in 10 mM, and that ratio decides whether the pore conducts like a tube of ordinary salt water or like a channel filled with the counterions of its own walls.
Definition
Near a charge, ions of the opposite sign outnumber those of the same sign in the proportions that thermal motion allows, which is the Boltzmann distribution. Linearizing the Poisson-Boltzmann equation for that cloud gives a screened Coulomb potential, ψ(r) = (q/4πεrε0r) exp(−r/λD), in which the Debye length is
Here εr is the relative permittivity of the solvent (78.4 for water at 25 °C), kBT the thermal energy, NA the Avogadro constant, e the elementary charge and I the ionic strength, half the sum of cz2 over every ion, with the concentration c in mol/m3.[2] The Bjerrum length lB = e2/(4πεrε0kBT) is the distance at which two elementary charges interact with energy kBT, and it is 0.71 nm in water at 25 °C. In terms of it the Debye length of a 1:1 salt with n formula units per unit volume is (8πlBn)−1/2.
The length follows the ionic strength, so a divalent salt screens more than its molarity suggests. Magnesium chloride at 10 mM has an ionic strength of 30 mM and a Debye length of 1.8 nm, the same as 30 mM potassium chloride.
Warming the water barely changes it. The thermal energy kBT rises, the permittivity of water falls, and the two nearly cancel: at 0.1 M the length is 0.96 nm at 20 °C and 0.95 nm at 37 °C.
Values
| Salt concentration | Debye length | Width of a 5 nm pore in Debye lengths |
|---|---|---|
| 1 M | 0.30 nm | 16 |
| 0.1 M | 0.96 nm | 5.2 |
| 10 mM | 3.0 nm | 1.6 |
| 1 mM | 9.6 nm | 0.52 |
| 0.1 mM | 30 nm | 0.16 |
| Pure water (10−7 M) | 960 nm | 0.005 |
Where the theory stops
The Debye-Hückel theory treats ions as points in a dilute solution, and at 1 M neither assumption holds. The 0.30 nm of that solution is shorter than the radius of 0.331 nm that Nightingale assigned to a hydrated potassium ion, so the screening is finer than the ion that does it.[3] The linearization also needs potentials well below kBT/e, which is 25.7 mV at room temperature, and a charged wall can exceed that.
Surface-force measurements across concentrated salt solutions have found that the decay length grows with the concentration instead of shrinking. That is the reverse of the theory's prediction, and the effect has been called underscreening.[4]
In nanopores
In a pore many Debye lengths wide the electrolyte at the center is neutral and the current follows the bulk conductivity. In a pore about one Debye length wide or less there is no neutral region, because the charged walls fill it with counterions, and the conductance depends on the surface charge more than on the salt. Smeets and colleagues diluted potassium chloride a million-fold, from 1 M to 1 µM, and found that the conductance of solid-state pores fell by three orders of magnitude where bulk conduction would have fallen by six.[5] Silica channels as thin as 70 nm saturated in the same way. Below a certain concentration their conductance stopped depending on both the salt and the height of the channel.[6]
The screening length also sets the sign of a blockade. In the same experiments a DNA molecule lowered the current in potassium chloride above 0.4 M and raised it below, and the two effects cancelled at 370 ± 40 mM, where the formula gives a Debye length of 0.5 nm.[5] A post by the founder of Molecular Reality Corporation on its member site calls salt types and concentrations "an important dial for our Player Scientists to turn."[7]
A pore 200 µm across, like the one on the company's live molecular stream, would be about 660,000 Debye lengths wide in 1 M potassium chloride, and its double layers would be films 0.3 nm thick on the walls of a tube of bulk liquid.[8] The bill of materials of the MR1 Nanopore Reader lists chips with apertures of 2 to 10 nm, which are 7 to 33 Debye lengths wide in the same solution, and the example buffer in its data-exchange specification is 1 M potassium chloride with 10 mM Tris at pH 8.[9]
See also
References
- ^ Debye, P.; Hückel, E. (1923). "Zur Theorie der Elektrolyte. I. Gefrierpunktserniedrigung und verwandte Erscheinungen". Physikalische Zeitschrift 24: 185-206.
- ^ Israelachvili, J. N. (2011). Intermolecular and Surface Forces. 3rd ed. Academic Press. ISBN 978-0-12-375182-9.
- ^ Nightingale, E. R. (1959). "Phenomenological theory of ion solvation. Effective radii of hydrated ions". Journal of Physical Chemistry 63 (9): 1381-1387. doi:10.1021/j150579a011.
- ^ Lee, A. A.; Perez-Martinez, C. S.; Smith, A. M.; Perkin, S. (2017). "Underscreening in concentrated electrolytes". Faraday Discussions 199: 239-259. doi:10.1039/C6FD00250A.
- ^ a b 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.
- ^ 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.
- ^ Kemmishtree (March 29, 2026). "Clarifying why we say "ionic modulation"". EQBio-OS.
- ^ Lenora Ashgrove (July 29, 2026). "Nineteen Billion Ions a Second, Live on the Internet". Signal & Noise.
- ^ "MR1 Nanopore Reader". GitHub. October 16, 2026.