Electrohydrodynamics of an ionic liquid meniscus during evaporation of ions in a regime of high electric field

Numerical investigations are presented for an ionic liquid meniscus undergoing evaporation of ions in a regime of high electric field. A detailed model is developed to simulate the behavior of a stationary meniscus attached to a liquid feed system. The latter serves as a proxy for commonly utilized electrospray emitters such as needles and capillary tubes. Two solution families are identified for prototype liquid analogous to the ionic liquid 1-ethyl-3-methylimidazolium tetrafluoroborate (EMI-BF4). The first belongs to a regime of low electric field in which the meniscus is blunt and does not emit charge. The second belongs to a regime of high electric field in which a conelike meniscus produces charge from a sharp tip. Electrohydrodynamic features of the meniscus in this regime are presented. These reveal that the meniscus is Stokesian and hydrostatic and governed by conduction. The applied electric field influences both the shape of the meniscus and the current that it produces while the impedance of the feed system—which must be above a threshold value in order to ensure that the current, and therefore the flow, remains below a maximum value—influences the meniscus current but not the macroscopic shape. In general, this shape deviates from Taylor's idealized cone.

Figure 1

Evolution of a cone-jet as the flow rate is reduced. Top: At high flow rates $Q$ the meniscus resembles a classical Taylor cone (angle ${\theta }_T\approx 49.2^{\circ }$) and exhibits a characteristic protuberance at the tip. The tip supplies a train of electrosprayed droplets that propagate under the influence of the external field $E$. It may eventually shed ions and other charges of low solvation as its diameter $d_j$ decreases in response to waning flow. Bottom: Under certain conditions, very low flow rates lead to complete extinction of the jet, such that only ions emanate from a sharp tip through electrically induced evaporation.

Figure 2

Modeling configuration for an evaporating ionic liquid meniscus. An axisymmetric meniscus is attached to a conductive plate under vacuo with a contact radius $r_0$ (arbitrary contact angle) and stressed by an applied electric field $E_0$ that is asymptotically uniform at distances much greater than $r_0$ from the plate. Fresh liquid enters the meniscus during steady evaporation at a pressure $\overline{p}$ by a reservoir that is charged to a prescribed pressure $p_r$ and a feeding line with characteristic impedance $R_h$.

Figure 3

Comparison of evaporation results for ionic liquid droplets of constant volume. The findings of Higuera [31] for volumes $\mathcal{V}=$ 0.5, 1, 1.5, and 2 (dashed curves) are shown next to those computed with the present model. (a) Elongation of the droplets, $Z_e=h(r=0)$, as a function of applied electric field. The droplet volume is increasing in the direction of increasing elongation. (b) Surface charge $\sigma$ at the tip of the meniscus. The droplet volume increases from right to left.

Figure 4

Select characteristics of solutions in the low-field regime for $p_r=-0.50$, $-0.25$, 0, 0.25, 0.50, 0.75, and 0.90. (a) Limiting meniscus shapes at the end of the stable solution branch (turning point) in the low-field regime with electric field $E_0=0.72$, 0.61, 0.52, 0.42, 0.33, 0.23, and 0.14, respectively. The reservoir pressure increases in the direction of increasing elongation. (b) Electric field $E_n^v$ at the tip of the meniscus. The maximum field corresponds to the turning point (maximum curvature) but remains limited to $E_n^v=\mathcal{O}\left(1\right)$, precluding emission from a large meniscus. The reservoir pressure decreases from left to right.

Figure 5

Select properties of the steadily evaporating meniscus residing at the point $E_0=1$, $1/B=20$, $P_r=0$, and $C_R={10}^3$ in parameter space. (a) The meniscus shape (solid curve) plotted alongside a classical Taylor cone (dashed curve) for comparison. (b) The distribution of stresses across the liquid-vacuum interface. The viscous stress accumulated in the meniscus itself, ${(-\mathit{n}·\mathit{\tau }·\mathit{n})}_m$, is separated from the pressure drop in the feeding line, $-I{C}_R$, for clarity. (c) The distribution of electric potential across the liquid-vacuum interface, normalized by the characteristic ${\mathrm{\Phi }}^{*}=E^{*}{r}^{*}/{\epsilon }_r$. (d) The distribution of surface charge across the liquid-vacuum interface, normalized by the normal component of the displacement field in vacuum. (e) Normal (solid curve) and tangential (dashed curve) components of the fluid flow at the liquid-vacuum interface, normalized by the characteristic $u^{*}$. (f) The temperature of the liquid-vacuum interface normalized by $\mathrm{\Delta }{T}^{*}$.

Figure 6

Select parametric characteristics in the high-field regime. Effects of the meniscus size [(a) and (d)], electric field [(b) and (e)], and impedance [(c) and (f)] are represented. (a) Meniscus shapes for $E_0=0.7$ and $C_R={10}^3$. The meniscus size $1/B=$ 3.19, 4.26, 6.38, 17.0, 100, increasing in the direction of increasing elongation. A classical Taylor cone (dashed) provides spatial reference. (b) Meniscus shapes for $1/B=106$ and $C_R={10}^3$. The electric field $E_0=$ 0.585, 0.605, 0.625, 0.645, 0.665, 0.685, increasing in the direction of decreasing elongation. A classical Taylor cone (dashed) provides spatial reference. (c) Electrical current for $E_0=0.62$ and $1/B=106$ over a range of impedance. (d) Radius of curvature at the meniscus tip for $E_0=0.7$ and $C_R={10}^3$ over a range of $B$. (e) Electrical current for $1/B=106$ and $C_R={10}^3$ over a range of field. (f) Radius of curvature at the meniscus tip for $E_0=0.62$ and $1/B=106$ over a range of impedance.