A microscopic origin for the breakdown of the Stokes Einstein relation in ion transport

Abstract

Ion transport underlies the operation of biological ion channels and governs the performance of electrochemical energy-storage devices. A long-standing anomaly is that smaller alkali metal ions, such as Li$^+$, migrate more slowly in water than larger ions, in apparent violation of the Stokes-Einstein relation. This breakdown is conventionally attributed to dielectric friction, a collective drag force arising from electrostatic interactions between a drifting ion and its surrounding solvent. Here, combining nanopore transport measurements over electric fields spanning several orders of magnitude with molecular dynamics simulations, we show that the time-averaged electrostatic force on a migrating ion is not a drag force but a net driving force. By contrasting charged ions with neutral particles, we reveal that ionic charge introduces additional Lorentzian peaks in the frequency-dependent friction coefficient. These peaks originate predominantly from short-range Lennard-Jones (LJ) interactions within the first hydration layer and represent additional channels for energy dissipation, strongest for Li$^+$ and progressively weaker for Na$^+$ and K$^+$. Our results demonstrate that electrostatic interactions primarily act to tighten the local hydration structure, thereby amplifying short-range LJ interactions rather than directly opposing ion motion. This microscopic mechanism provides a unified physical explanation for the breakdown of the Stokes-Einstein relation in aqueous ion transport.

Figures (4)

• Figure 1: Measuring ion transport through a solid-state nanopore under a wide range electric field intensity. (a), Schematic of the solid-state nanopore system. The membrane has a 20-nm-thick Si$_3$N$_4$ film, with a single nanopore at its center. A bias voltage ($\delta V$) applied across the membrane drives ions through the nanopore, generating an ionic current ($I$). The voltage drop concentrates inside the nanopore, creating an strong electric field ($E$). (b), Scanning electron microscopy (SEM) micrograph of a 250 nm diameter nanopore. (c), Current-voltage ($I$-$V$) curves for a 250 nm diameter nanopore filled with 0.4 M LiCl, 0.4 M NaCl, and 0.4M KCl solutions, respectively. (d), Nanopore conductance $\sigma$ (derived from data in (c)) as a function of the electric field intensity (E). The conductance remains constant below field 9$\times{10}^7$ V/m and shows a nonlinear increase at higher fields due to Joule heating. Across the entire electrical field range, the conductance follows the order KCl > NaCl > LiCl. In (c) and (d), Data points represent the mean values, and the shaded regions denote standard deviation from 12 independent measurements.
• Figure 2: Time averaged force $\left\langle F_z\right\rangle$ acting on ions from ion-water interactions calculated from molecular dynamics (MD) simulation at different applied elect ric fields. The external electric field was applied along the positive z-direction, and the values plotted represent the force component along this axis. Results are shown for (a) Li$^+$, (b) Na$^+$, and (c) K$^+$. The total force is decomposed into the contribution from Lennard-Jones (LJ) interaction and the electrostatic (ELE) interaction between ion and water. For all three ions, the ELE component consistently acts as a net driving force, while LJ component acts as a net drag force. Data points represent the mean, and the shaded regions denote standard deviation from 5 independent simulations.
• Figure 3: Effect of electrostatic interactions between ion and water on the friction memory kernel and frequency-dependent friction coefficient. (a-c), Friction memory kernel, $\gamma(t)$, for (a) Li$^+$, (b) Na$^+$, and (c) K$^+$. (d-f), Corresponding frequency-dependent friction coefficient, $\tilde{\gamma}(\omega)$, obtained from the Fourier transform of $\gamma(t)$, for (d) Li$^+$, (e) Na$^+$, and (f) K$^+$. In each panel, the result for the standard ion (blue line) is compared with an uncharged counterpart (orange line) where the charge of ion was set to zero to exclude electrostatic interactions. The inclusion of electrostatics introduces significant, decaying fluctuations in the friction memory kernel, most notably for Li$^+$. These fluctuations correspond to the distinct Lorentzian peaks observed in the terahertz regime of the frequency-dependent friction coefficient.
• Figure 4: Decomposition of the Li$^+$ frequency-dependent friction coefficient by spatial region and interaction type. (a-d), Spatially decomposed frequency-dependent friction coefficient, $\tilde{\gamma}(\omega)$ , showing the contribution from interaction between ion and (a) 1$^\text{st}$ hydration layer, (b) 2$^\text{nd}$ hydration layer, (c) remaining bulk solvent, and (d) the total water environment. (e-h), Further decomposition of each spatial component into its Lennard-Jones (LJ, blue) and electrostatic (ELE, orange) contributions: (e) 1$^\text{st}$ hydration layer, (f) 2$^\text{nt}$ hydration layer, (g) remaining bulk solvent, and (h) the total water environment.