Short-range electrostatic screening in ionic liquids as inferred by direct force measurements

Abstract

Previous experimental reports of long-range interactions in ionic liquids (ILs) stand in contradiction with theoretical predictions and numerical simulations. To provide insights into the literature discrepancies regarding the experimental ranges of electrostatic screening, claimed with orders of magnitude larger, the interactions between pairs of mica and borosilicate surfaces confining ILs are investigated by two complementary advanced Surface Force Apparatuses. Regardless of differences in confinement geometries (crossed-cylinders, sphere-flat), radii of curvature (cm-mm), and measurement techniques (stepwise vs continuous approach), two ever present force regimes are evidenced. At small surface separations, oscillatory forces reflect IL structuration and layering, while outside this gap, the interaction is monotonic repulsive. In both regimes the spatial extent and force magnitude depend critically on motion conditions, as demonstrated by achieving velocities as low as 9 pm/s with equilibration times up to 90 s. At large separations, fast surface displacements generate long-range interactions (over tens of ion size) creating the illusion of anomalous underscreening, whereas increasingly slow ones shrink both magnitude and range of the repulsion with decay-lengths converging ultimately to a screening length consistent with Poisson-Boltzmann theory with finite ion sizes. The transition from apparent long-range to short-range screening unfolds over nearly two orders of magnitude in time, revealing slow relaxation dynamics reminiscent of aging phenomena. These findings definitely resolve a decade-old controversy on force measurements and reveal rich out-of-equilibrium dynamics. The hydrodynamic contribution to the net force is admittedly crucial to be reduced especially when relaxations span decades in time, but approaching thermodynamic equilibrium during measurements proves essential.

Figures (3)

• Figure 1: Measured forces for 1-ethyl-3-methylimidazolium [C$_2$mim]$^+$ bis-(trifluoromethylsulfonyl)imide [NTf$_2$]$^-$, normalized by the mean radius of curvature of the surfaces between crossed mica cylinders, as a function of surface separation, $D$. Solid symbols correspond to the force on approaching the surfaces, while the open symbols are the data measured upon separation. The force-distance profiles are composed of two distinct regimes with spatial extent and magnitude both depending on velocity, stepwise and resting time conditions for surface movements (all parameters reported in SI Appendix, section 4). At short separations the oscillatory functions of distance are prominently related to the structuration of the IL that may not be reversible upon decompression. At large separations the interactions are all monotonic repulsive and reversible. Fit tests by exponentials (solid lines) give large decay-lengths $\lambda$ in agreement with Gebbie et al. gebbie2015 for the same IL in similar conditions (Top Right). However, increased slowdown leads to a progressive fading of the apparent long-range interaction with its related $\lambda$ being reduced by one-order of magnitude (central graph). The $\lambda$-uncertainties represent the estimated SD inferred from the fits on different force profiles collected during several approach and separation cycles and from possible variations in the repeatability of multiple independent experiments. The travel time segments indicate the $\tau$-ranges of displacement increments used along the measured monotonic repulsion only. At very slow speed (Bottom) the interaction is of short-range with $\lambda = 0.75 \pm 0.05$ nm close to the expected IL screening length suggesting an electrical double-layer interaction at constant potential $22 \pm 1$ mV boundary condition (solid line: DLVO fit).
• Figure 2: Force-distance profiles of butyl-1-methyl-3-imidazolium [C$_4$mim]$^+$ hexafluorophosphate [PF$_6$]$^-$ between borosilicate surfaces in a flat-sphere (radius $R = 3.3$ mm) geometry. (A) Solid symbols are the measured forces at different approaching surface velocities $v = 3.0$ (blue), 0.6 (green), and 0.015 nm/s (black) while dotted lines represent the calculated corresponding hydrodynamic force (SI Appendix, Figs. S2 and S3). (B) Beyond the elastic and solvation forces regime (gray zone), the nonhydrodynamic interactions (open symbols), inferred by subtracting the hydrodynamics from the total measured force at $v = 3.0$ and 0.6 nm/s, could appear as having exponential profiles with decay-lengths $\lambda = 4.0 \pm 0.5$ nm and $2.5 \pm 0.5$ nm respectively. (C) At the slowest velocity $v = 0.015$ nm/s, the total interaction is short-range, exponentially decaying with $\lambda = 0.50 \pm 0.05$ nm close to the expected screening length for the IL, with a double-layer interaction at constant charge boundary condition.
• Figure 3: Evolution of the apparent screening length, $\lambda$, of the monotonic repulsion at large separations (from Figs. \ref{['fig:1']} and \ref{['fig:2']}) as a function of travel time (inverse of approach velocity). Blue symbols represent [C$_2$mim][NTf$_2$] confined between crossed mica cylinders of radii of curvature $R \sim 10$ to 20 mm (all experimental parameters reported in SI Appendix, Fig. S4) using the stepwise protocol set for conventional SFA, while black symbols correspond to [C$_4$mim][PF$_6$] between borosilicate surfaces in a sphere-flat geometry ($R = 3.3$ mm) measured with the dSFA. Regardless of different ILs, surface materials, experimental configurations and protocols, both datasets show a steady decrease of $\lambda$ from $\sim$12 nm at high velocities (short travel times) down to 0.5 to 0.75 nm under quasi-static conditions (long travel times $\tau \sim 100$ s/nm). Error bars are uncertainties on $\lambda$ inferred from the fits and $\tau$-ranges of displacement increments used along the measured monotonic repulsion only (SI Appendix, section 4), while shadowy blue rectangles indicate the full $\tau$-ranges used over the entire force-distance profile; the dotted line serves as a guide to the eye. Through two independent techniques and different ILs, the semilogarithmic representation reveals that the relaxation extends over two orders of magnitude in time scale, underscoring the IL slow progressive approach toward equilibrium. Thus, by allowing sufficient time for the confined systems to approach their equilibrium state, the decay lengths converge toward the theoretically predicted screening length.