Simple Mechanism for the Observed Breakdown of the Nernst-Einstein Relation for Ions in Carbon Nanotubes

In this Letter, we present a simple mechanism that explains the recent experimental observation of the breakdown of the Nernst-Einstein (NE) relation for an ion moving in a carbon nanotube of subnanometer diameter. We argue that the friction acting on the ion is largely independent of the ion velocity, i.e., dry friction, and demonstrate, based on the Langevin equation for a particle subject to both dry and viscous friction, that the NE relation is violated when dry friction dominates. We predict that the ratio of the diffusion constant to the mobility of the ion is a few orders of magnitude smaller than the value predicted by the NE relation, in quantitative agreement with experiment.

Figure 1

A cartoon of an ion-water cluster translocating through a carbon nanotube with a subnanometer diameter. The electrostatic interaction between the ion and the carbon atoms gives rise to a force of friction that is independent of the velocity of the ion, i.e., dry friction. If the dry friction dominates the usual viscous friction, the Einstein-Nernst relation breaks down for the motion of the ion.

Figure 2

A plot of the velocity $⟨\tilde{v}⟩\equiv ⟨v⟩/v_T$ vs the external force, $f\equiv F/(\gamma v_T)$ for different strength of the dry friction, $\delta \equiv \mathrm{\Delta }/(\gamma v_T)=0.01$, viscous dominated (the blue line) and $\delta =5$, dry friction dominated (the black line). For the latter, there are two regimes: small mobility in which the mobility $\mu \sim 1/(\gamma {\delta }^2)$ when $f\ll \delta$, and a high mobility in which $\mu \sim 1/\gamma$ when $f\gg \delta$, where $\gamma$ is the viscous damping constant and $v_T=\sqrt{k_{B}T/M}$ is the thermal velocity of the ion-water cluster of mass $M$.