A sodium ion in bulk water is surrounded by a hydration shell — water molecules oriented by the ion's charge. The energy of this arrangement is the hydration free energy, and it is large: roughly 100 kcal/mol for sodium. In the bulk, the surrounding water has enough room to form a full hydration shell, and the Born equation — a continuum model treating water as a uniform dielectric medium — predicts the energy reasonably well.
Kevin Leung (arXiv:2603.04651, March 2026) shows that confining ions inside carbon nanotubes breaks both the hydration shell and the theory.
In a nanotube with a 7.5 Angstrom radius, the tube walls physically prevent a complete hydration shell from forming. Water molecules that would ordinarily surround the ion from all directions are geometrically excluded. The hydration free energy drops — sodium loses up to 4.3 kcal/mol, chloride loses 7.8 kcal/mol. The confinement penalty is asymmetric: chloride, with its larger and more diffuse electron cloud, suffers more from the restricted geometry. The Born equation, which treats the solvent as structureless, predicts none of this. The penalty is a molecular-geometry effect invisible to continuum theory.
The unexpected finding is what happens when you add background electrolyte. In bulk solution, adding salt screens an ion's electric field over a characteristic length — the Debye length. The Debye-Huckel theory predicts how much the screening reduces interaction energies. In nanotubes, the screening effect is almost an order of magnitude larger than Debye-Huckel predicts for the equivalent bulk concentration.
The mechanism is geometric. In a one-dimensional tube, ions cannot avoid each other. Every ion must pass through the same narrow channel. At 1.0 molar concentration, the average spacing between ions is comparable to the tube diameter. Each ion's electric field is screened not just by the diffuse ion atmosphere of Debye-Huckel theory but by the forced proximity of nearby counterions in a tube too narrow for them to spread out. Confinement compresses the screening atmosphere from three dimensions into one, concentrating the shielding effect. The penalty for losing your hydration shell is partially offset by the bonus of having your neighbors' charges closer than they would ever be in free solution.
Leung, "Anomalous Ion Confinement Penalties and Giant Ion-Screening Effects in One-Dimensional Nanopores," arXiv:2603.04651 (March 2026).