Osmotic pressure of ionic liquids in an electric double layer: Prediction based on a continuum model

An analysis has been performed for the osmotic pressure of ionic liquids in the electric double layer (EDL). By using the electromechanical approach, we first derive a differential equation that is valid for computing the osmotic pressure in the continuum limit of any incompressible fluid in EDL. Then a specific model for ionic liquids proposed by Bazant et al. [M. Z. Bazant, B. D. Storey, and A. A. Kornyshev, Phys. Rev. Lett. 106, 046102 (2011)] is adopted for more detailed computation of the osmotic pressure. Ionic liquids are characterized by the correlation and the steric effects of ions and their effects are analyzed. In the low voltage cases, the correlation effect is dominant and the problem becomes linear. For this low voltage limit, a closed form formula is derived for predicting the osmotic pressure in EDL with no overlapping. It is found that the osmotic pressure decreases as the correlation effect increases. The osmotic pressures at the nanoslit surface and nanoslit centerline are also obtained for the low voltage limit. For the cases of moderately high voltage with high correlation factor, approximate formulas are derived for estimating osmotic pressure values based on the concept of a condensed layer near the electrode. In order to corroborate the results predicted by analytical studies, the full nonlinear model has been solved numerically.