Nanoconfined ionic liquids: Disentangling electrostatic and viscous forces

Recent reports of surface forces across nanoconfined ionic liquids have revealed the existence of an anomalously long-ranged interaction apparently of electrostatic origin. Ionic liquids are viscous, and therefore it is important to inspect rigorously whether the observed repulsive forces are indeed equilibrium forces or, rather, arise from the viscous force during drainage of the fluid between two confining surfaces. In this paper we present our direct measurements of surface forces between mica sheets approaching in the ionic liquid $\left[{\mathrm{C}}_2{\mathrm{C}}_1\mathrm{Im}\right]\left[{\mathrm{NTf}}_2\right]$, exploring three orders of magnitude in approach velocity. Trajectories are systematically fitted by solving the equation of motion, allowing us to disentangle the viscous and equilibrium contributions. First, we find that the drainage obeys classical hydrodynamics with a negative slip boundary condition in the range of the structural force, implying that a $\mathrm{nanometer}$-thick portion of the liquid in the vicinity of the solid surface is composed of ordered molecules that do not contribute to the flow. Second, we show that a long-range static force must indeed be invoked, in addition to the viscous force, in order to describe the data quantitatively. This equilibrium interaction decays exponentially and with decay length in agreement with the screening length reported for the same system in previous studies. In those studies the decay was simply checked to be independent of velocity and measured at a low approach rate, rather than explicitly taking account of viscous effects: we explain why this gives indistinguishable outcomes for the screening length by noting that the viscous force is linear to very good approximation over a wide range of distances.

Figure 1

(a) Schematic showing the principle of SFB measurements. (b) Force profile measured during approach of two mica surfaces separated by $\left[{\mathrm{C}}_2{\mathrm{C}}_1\mathrm{Im}\right]\left[{\mathrm{NTf}}_2\right]$ at $v=0.075\pm 0.010\mathrm{nm}/\mathrm{s}$. At short distances, below ca. 7–8 nm, one observes a structural force with characteristic steps due to successive squeeze-out of ordered layers (also called solvation force). The configuration with $i=6$ cations along the gap is indicated by the green dashed line and is illustrated in the inset. At larger distances a long-range force is found, and the aim of this paper is to determine the static and dynamic contributions to it. (c) Chemical structure of $\left[{\mathrm{C}}_2{\mathrm{C}}_1\mathrm{Im}\right]\left[{\mathrm{NTf}}_2\right]$. (d) Location of the layers in the structural force region: the slope gives the average layer thickness.

Figure 2

(a) Trajectory recorded when the top lens is moving at $v=95\pm 5\mathrm{nm}/\mathrm{s}$ (piezo motion indicated by the dashed line in orange). The black curve is the experimental signal; the orange (resp. green) curve is the fit by the model with $b=4.2\pm 0.7\mathrm{nm}$ (resp. $b=0)$, ${\lambda }_{\mathrm{s}}=8.0\mathrm{nm}$, and $A=0.22\mathrm{mN}/\mathrm{m}$. These values of ${\lambda }_{\mathrm{s}}$ and $A$ are justified by fits at lower velocity as described later; as noted in the text a fit with $A=0$ would be indistinguishable at this velocity. (b) Same measurements, but with data presented in terms of force, in semi-log scales. The black curve is obtained from the difference between the black curve and the orange dashed line in (a). The total force (in orange), viscous contribution (in blue), and electrostatic contribution (in red) are deduced from the fit. In both graphs, the vertical line in gray indicates the time and distance at which the piezo reaches its maximum deformation and the top lens stops.

Figure 3

(a) Trajectory recorded when the top lens is moving at $v=0.24\pm 0.01\mathrm{nm}/\mathrm{s}$ (piezo motion indicated by the dashed line in orange). The black curve is the experimental signal, the blue curve is an attempt to fit with a single, viscous force $\left(A=0\right)$, and the orange curve is the fit by the model with ${\lambda }_{\mathrm{s}}=8.0\pm 0.5\mathrm{nm}$ and $A=0.22\pm 0.02\mathrm{mN}/\mathrm{m}$ and three different values $b\in \{3.5\mathrm{nm};4.2\mathrm{nm};4.9\mathrm{nm}\}$ (zoom in inset). (b) Same measurements, but with data presented in terms of force, in semi-log scales. The black curve is obtained from the difference between the black curve and the orange dashed line in (a). The total force (in orange), viscous contribution (when fitting with nonzero A; in blue) and electrostatic contribution (in red) are deduced from the fit.

Figure 4

(a) Trajectory recorded when the top lens is moving at $v=1.13\pm 0.04\mathrm{nm}/\mathrm{s}$ (piezo motion indicated by the dashed line in orange). The black curve is the experimental signal, the blue curve is an attempt to fit with a single, viscous force $\left(A=0\right)$ with $b=4.2\mathrm{nm}$, and the orange curve is the fit by the model with ${\lambda }_{\mathrm{s}}=7.5\pm 0.5\mathrm{nm}$ and $A=0.18\pm 0.02\mathrm{mN}/\mathrm{m}$. The gray dashed line is a linear fit a large distances, which doesn't take into account the strong curvature observed initially (zoom in inset) and provides the velocity $0.99\pm 0.04\mathrm{nm}/\mathrm{s}$. (b) Same measurements, but with data presented in terms of force, in semi-log scales. The black curve is obtained from the difference between the black curve and the orange dashed line in (a). The total force (in orange), viscous contribution (in blue), and electrostatic contribution (in red) are deduced from the fit with nonzero $A$, whereas the force profile in gray line is computed from the gray dashed line in (a). Inset: same data plotted in linear scales. The vertical dashed line in gray indicates the intersection between the viscous and electrostatic contributions, and the second gray dashed line is a guide for the eyes to illustrate the linearity of the viscous component at large distances.

Figure 5

(a) Electrostatic and viscous forces calculated numerically for four approaches at $v\in \{0.1\mathrm{nm}/\mathrm{s};1\mathrm{nm}/\mathrm{s};10\mathrm{nm}/\mathrm{s};100\mathrm{nm}/\mathrm{s}\}$, with $b=4.2\mathrm{nm},{\lambda }_{\mathrm{s}}=7.5\pm 0.5\mathrm{nm},$ and $A=0.18\pm 0.02\mathrm{mN}/\mathrm{m}$. (b) Ratio of the viscous force to the approach velocity as a function of the distance between the surfaces, for the same trajectories.