Extended Data Fig. 10: Calculation of the electrophoretic mobilities and diffusion coefficients of ion–water clusters.

a. Variation of the z position of the K⁺ ion in cluster i (see Fig. 3e) as a function of the simulation time obtained in the presence of a 0.25 V·nm^-1 electric field applied along the z direction. The simulation data (solid curve) is obtained from 5 independent MD simulations with different random velocities of the molecules in the system. The dashed curve represents the fit to the data obtained using the terminal velocity of the K⁺ ion, \(v_z = \frac{{dz}}{{dt}}\), which is estimated to be 24.7 ± 3.9 m·s^-1. b. Variation of the z position of the K⁺ ion in cluster iv (see Fig. 3e) as a function of the simulation time obtained in the presence of a 0.25 V·nm^-1 electric field applied along the z direction. Similar to a, the simulation data is obtained from 5 independent MD simulations. The calculated terminal velocity in this case is: v_z = 54.7 ± 5.6 m·s^-1. The terminal velocities obtained at three different electric fields have been used to obtain the data shown in Fig. 3g, h. c. Plot showing the variation of the MSD of cluster i in the absence of any electric field, that is, where E_z = 0. d. Plot showing the variation of the MSD of cluster iv in the absence of any electric field. Additional data on the variation of the terminal velocity and MSD of cluster v is presented in Supplementary Discussion S6.