An understanding of the processes enabling field-assisted evaporation of ions from leaky dielectric liquids, i.e., liquids that are substantially less conductive than liquid metals, has historically been elusive in comparison to those of conventional electrohydrodynamic emission modes such as that of the cone-jet. While select ionic liquids have been shown to yield nearly monodisperse beams of molecular ions under certain conditions, the dearth of direct observation (visualization) and theoretical insight has precluded a fundamental appreciation for the inherent mechanics. In this paper, we present a family of equilibrium meniscus structures that shed measurable charge when the meniscus is large in relation to a characteristic emission scale. Such structures reside in a region of parameter space where empirical evidence suggests that steady emission may occur and also where stationary interfaces have not been reported before. In this regime, we show (i) that the macroscopic shape of the meniscus may vary only with the applied electric field; (ii) that the feeding flow is very germane to the emission characteristics, unlike liquid metal ion sources; and (iii) that while the balance of stresses governing the interface shape may in some cases be very similar to that of the classical Taylor cone, the widespread notion of a ubiquitous 49° half-angle is unfounded. Further study of this family may be helpful in elucidating a number of outstanding questions surrounding the pure ion mode.

Field-induced evaporation of charged species from the menisci of liquids with modest electrical conductivity, e.g., ionic liquids, is a kinetic phenomenon that occurs under the influence of strong electric fields, similar to those experienced by the cone-jets that typify many electrospray sources.^{1} Although ion evaporation may sometimes proceed alongside steady droplet evolution, i.e., in connection with a jet producing a train of electrosprayed droplets,^{2} time-of-flight mass spectrometry has confirmed the existence of a so-called pure ion mode corresponding to an exclusive evaporation of solvated ions at low liquid flow rates.^{3} While such a mode is already technologically meaningful (see, e.g., Refs. 4, 5, and 6 for focused ion beam and micropropulsion applications, respectively), the continued elusiveness of direct observations and theoretical traction has enabled a number of important questions to persist. The purpose of this paper is to introduce a family of meniscus structures that may be useful in elucidating some of the more salient of these, including (i) why purely evaporating menisci have never been visualized and (ii) why specific emitter architectures are advantageous.

Ion emission from the meniscus of a leaky dielectric liquid formed in vacuo is taken to obey the kinetic law^{7,8}

where *j _{e}* is the current emitted per unit area of the meniscus surface,

*σ*is the density of surface charge,

*k*is Boltzmann's constant,

_{B}*T*is the liquid temperature,

*h*is Planck's constant, Δ

*G*is the activation energy barrier for solvated species, $Env$ is the magnitude of the normal component of the electric field acting on the meniscus exterior, and $G(Env)$ is the reduction of Δ

*G*due to this field. The latter is assumed to take the form $G(Env)=(q3Env/4\pi \u03f50)1/2$, based on the Schottky hump for polar media, when

*q*is the charge state of evolving particles and

*ϵ*

_{0}is the permittivity of vacuum. Since Δ

*G*/

*k*is large in situations of interest, meaningful evaporation can only begin when $\Delta G\u2212G(Env)=O(kBT)\u226a\Delta G$, as permitted by fields near the characteristic value $E*=4\pi \u03f50\Delta G2/q3$ arising from the condition that $\Delta G=G(E*)$. The length scale of the evaporation region at the meniscus tip is $r*=4\gamma /\u03f50E*2$ owing to a balance of electric (order $\u03f50E*2/2$ (Ref. 9)) and surface tension (order 2

_{B}T*γ*/

*r**) stresses for a liquid with surface tension

*γ*(Ref. 10). Steady evolution of charge from this region reduces

*σ*from its equilibrium value $\u03f50Env$ (Ref. 9) and permits electric fields of order

*E**/

*ϵ*on the liquid side of the surface, where

*ϵ*is the dielectric constant of the liquid. These drive the characteristic conduction current density

*j*

^{* }=

*KE**/

*ϵ*and total evaporation current $I*=\pi r*2j*$. For many leaky dielectrics of interest—namely ionic liquids—the relevant intrinsic properties may include the solvation energy Δ

*G*∼ 1–2 eV; the electrical conductivity

*K*∼ 1 S/m; the surface tension $\gamma \u223c10\u22122\u221210\u22121\u2009N/m$; the dielectric constant

*ϵ*∼ 10–100; the specific charge

*q*/

*m*∼ 10

^{6 }C/kg, where

*q*and

*m*are the charge state and mass of each (solvated) ion, respectively; the liquid density

*ρ*∼ 10

^{3 }kg/m

^{3}; and the viscosity $\mu \u223c10\u22122\u221210\u22121\u2009Pa\u2009s$. Under these conditions,

*E*

^{* }∼ 0.1–1 V/nm,

*r*

^{* }∼ 10–100 nm, and

*I*

^{* }∼ 10

^{2}–10

^{3 }nA. Strong activating fields accompany very sharp liquid tips that are typically small in comparison to traditional electrospray emitter structures, e.g., solid needles or capillary tubes, but precipitate modest currents on account of tenuous conductivity.

In contrast to the liquid metal ion sources (LMIS), for which high currents and related space charge effects are preponderant, the modest emission of leaky dielectric ion sources precludes evaporating charge from providing significant feedback to the meniscus. Consider that the space charge density due to evaporating ions is $\rho sc=j*/vsc$ around the activated region, where $vsc\u223c(2qE*r*/m)1/2$ is an order-of-magnitude ion speed based on the exchange of electrical and kinetic energy. From the Poisson equation, $\u2207\xb7E=\rho sc/\u03f50$, the electric field induced by this charge is $Esc\u223c\rho scr*/\u03f50$ and thus $Esc/E*\u223c(m\gamma K2/q\u03f5\u03f503E*3)1/2$. This ratio is typically small, $O(10\u22121)$, or less for conditions of interest, ensuring that the meniscus is effectively insensitive to the cloud of emitted ions.

For the liquid around the tip, $u*=j*/\rho (q/m)$ is the characteristic speed of the flow induced by ion evaporation, residing in the range from 10^{−1} m/s to 10^{−2} m/s for many leaky dielectrics. The Reynolds number is $Re=\rho u*r*/\mu =O(10\u22122)$, while the attendant residence time of the liquid *r**/*u** is long on the scales of both the electrical relaxation time *ϵϵ*_{0}/*K* (the ratio of these is defined as $\Lambda =\u03f5\u03f50u*/Kr*$) and the characteristic evaporation time *h*/*k _{B}T* from the kinetic law, Eq. (1), with $r*/u*\u226b\u03f5\u03f50/K\u226bh/kBT$. These indicate that the effects of the liquid inertia are small and that the surface charge cannot reach its equilibrium value.

Figure 1 delineates the domain for a meniscus model in axisymmetric cylindrical space, where the *z* = 0 plane is coincident with the surface of a flat conducting plate supporting a mass of fluid on its vacuum side. The liquid interface maintains a fixed contact radius *r*_{0} but is otherwise beholden to a mechanical balance that is influenced in part by an imposed electric field *E _{∞}*. The latter becomes asymptotically uniform at distances from the meniscus that are large as compared to

*r*

_{0}. A feeding system supplies the meniscus with fresh fluid during steady evaporation and captures lumped hydraulic effects that may owe to the flow through conventional electrospray emitter architectures, e.g., capillaries or needles, in real sources. It includes an infinite reservoir charged to a pressure

*p*and a feeding line with characteristic impedance

_{f}*R*, such that $pf\u2212p\xaf=RhQ$, where $p\xaf$ is the mean pressure of the liquid on the plate and $Q=I/\rho (q/m)$ is the evaporation flow rate. Under these conditions, the dimensionless equations and boundary conditions governing the steady pure ion evaporation problem in the meniscus are as follows. Inside the liquid

_{h}where $v$ and *p* are the velocity and pressure of the liquid, respectively, $\tau \u2032=\u03f5CaB1/2[\u2207v+(\u2207v)T]$ is the viscous stress tensor, and $\varphi l$ is the electric potential in the liquid, with $El=\u2212\u2207\varphi l$. Outside the liquid

At the free liquid surface,

where ** n** and

**are unit vectors normal and tangent, respectively, to the liquid surface, such that $Env=Ev\xb7n,\u2009Enl=El\xb7n$ and $Et=Ev\xb7t=El\xb7t$. The terms $\tau ne=(Env)2\u2212\u03f5(Enl)2+(\u03f5\u22121)Et2$ and $\tau te=2\sigma Et$ are the dimensionless electric stress components normal and tangent, respectively, to the surface and $je=(\sigma /\chi )\u2009exp(\u2212\Gamma {1\u2212B1/4Env})$. At the plate (**

*t**z*= 0, modeled as a porous plug),

where ** i** is a unit vector normal to the plate and

*I*is the integral of

*j*over the surface of the meniscus. Finally, $\u2212\u2207\varphi v\u2192E0i$ far from the meniscus.

_{e}The equation set has been nondimensionalized in the following way: lengths have been scaled by the radius of the contact line, *r*_{0}; stresses by the capillary pressure *p _{c}* = 2

*γ*/

*r*

_{0}; electric fields by a value

*E*satisfying $\u03f50Ec2/2=pc$; electric potentials, surface charge densities, and current densities by

_{c}*r*

_{0}

*E*,

_{c}*ϵ*

_{0}

*E*, and

_{c}*KE*, respectively; and the velocity of the liquid by $KEc/\rho (q/m)$. The problem now depends on the nine dimensionless parameters

_{c}where *r** and *u** are the characteristic radius of curvature and the evaporation-induced liquid velocity at the meniscus tip, respectively, defined above. The parameter *B* is the ratio of the size of the evaporation region to the size of the meniscus, *p _{r}* is a dimensionless feed (reservoir) pressure, and

*C*is a dimensionless feeding impedance. Stationary solutions to the governing set of equations are computed numerically using the finite elements and iterative methods. Thermal effects, which are not expected to qualitatively affect the problem,

_{R}^{11,12}are omitted. Additional details on these steps and numerical validation can be found in Ref. 12.

Studies are performed after electing the properties *K* = 1 S/m, Δ*G* = 1 eV, *γ* = 0.05 N/m, *ϵ* = 10, *μ* = 0.037 Pa s, *T* = 300 K, *ρ* = 10^{3 }kg/m^{3}, and *q*/*m* = 10^{6 }C/kg, such that the test liquid nominally resembles the widely used ionic liquid 1-ethyl-3-methylimidazolium tetrafluoroborate (EMI-BF4)^{3,13} with characteristic values *E*^{* }= 0.7 V/nm, *r*^{* }= 47 nm, and *u*^{* }= 7 cm/s. A reservoir pressure *p _{r}* = 0 is taken to match the upstream condition of common leaky dielectric ion sources, particularly those of the externally-wetted variety, although positive

^{3}and negative

^{14}values are technically feasible. The corresponding dimensionless variables include

*Ca*= 0.0257, Λ = 1.32 × 10

^{−4},

*χ*= 0.0181, Γ = 38.6, and

*p*= 0, while those that remain (i.e.,

_{r}*E*

_{0},

*B*,

*C*) form a set of free parameters. Starting from the null-field configuration, in Figure 2(a), we show the evolution of the meniscus with respect to increasing

_{R}*E*

_{0}. The liquid surface gradually elongates with growing field but always remains regular, preserving a tip curvature of the order $\kappa \u223cr0\u22121$. As a result, measurable evaporation is precluded when the meniscus is of practical size, i.e.,

*r*

_{0}≫

*r**, such that its behavior in this regime is independent of specific

*B-C*combinations. Rounded structures of this type have been examined by a number of different authors and are known to exist for relatively low electric fields below a special value corresponding to a turning point, where the incremental elongation of the meniscus (defined by the tip location

_{R}*z*) with respect to the field,

_{t}*dz*/

_{t}*dE*

_{0}, becomes unbounded. Taylor

^{10}famously considered the elongation of a perfectly conducting droplet in free space, while Wohlhuter and Basaran

^{15}have studied the shapes of sessile dielectric droplets. Although the results of the latter suggest that regular structures may begin to develop sharp, cone-like apices at the turning-point (see also Ref. 16), no stationary solutions for conductive menisci have ever been identified beyond the corresponding field. This includes the findings of Higuera,

^{11}where the model admitted the physics of charge emission.

Here, however, we uncover a family of menisci in a high-field region to the right of the turning point. In this regime, the solutions are structurally disparate from their low-field counterparts, exhibiting the elevated tip curvature *κ* ∼ 1/*r** even when the menisci are rather large (*B* ≪ 1). This, in turn, enables the liquid to shed meaningful charge. Similar to the finding of Higuera^{11} for a meniscus with $B=O(1)$, this emission, once activated, is ballasted primarily by conduction in the fluid. Such ballasting implies a relative insensitivity to the kinetic law, Eq. (1), which is convenient in that the kinetics may sometimes be called into question (see Ref. 17). Consider that the normal electric field at the surface of the meniscus is observed to vary smoothly on the scale of *r**. It is equal to *E** at a closed curve $C$ encircling the tip and higher than *E** between this curve and the tip. Owing to the large value of Δ*G*/*k _{B}T*, the exponential factor in Eq. (1) changes from very small to very large in a narrow band around this curve. Since conduction in the liquid keeps

*j*bounded, the surface charge density decreases from the equilibrium value $\u03f50Env$ to nearly zero on crossing this narrow band. The ion current is essentially the integral of $KEnv/\u03f5$ over the region enclosed by the curve $C$, meaning that the solvation energy Δ

_{e}*G*enters only insofar as it determines this curve. This is particularly intriguing for two reasons: (i) it suggests that normal electric stresses acting on the tip, which are responsible for destabilizing the hydrostatic Taylor cone,

^{15}may be tempered by surface charge depletion and (ii) it admits electrical shear in the same vicinity. Although the numerical method does not inherently ensure stability, such effects are known to stabilize cone-jets

^{18}and could influence at least a subset of this high-field family of structures in a similar way.

Additional properties of interest may be enumerated as follows. First, in Figure 2(b), we show the evolution of the meniscus with the applied field for *B* = 10^{−2} and *C _{R}* = 10

^{4}, a proxy for the full family. Notice that the meniscus is now sharp rather than “egg-like” with a tip size

*r** (not fully resolved on the macroscopic scale of the figure,

*r*

_{0}) that allows it to shed charge. The height and volume of the meniscus decrease when the electric field

*E*

_{0}is increased, in contrast to the low-field solutions. Hence, the turning point coincides with the onset of a rapid morphological transformation, which thereafter enables the meniscus to be slowly “sucked” into the feeding source, similar to what is observed in cone-jets.

^{19}

Second, when *B* ≪ 1, the shape of the meniscus on the scale of *r*_{0} is effectively independent of *B* and *C _{R}* and depends only on the applied field

*E*

_{0}. This may be understood by noticing that, owing to the relatively small capillary number

*Ca*, the meniscus is very nearly hydrostatic outside of a small region encircling the tip and only one hydrostatic shape with a sharp tip exists for each value of

*E*

_{0}(Ref. 16). In particular, $p\xaf$ and

*E*

_{0}, which are independent parameters for smooth menisci, satisfy a certain relation $\u2212p\xaf=\zeta (E0)$ for a sharp tip meniscus. The feeding condition in Eq. (5) with

*p*= 0 determines then the electric current as $I=\zeta (E0)/CR$, and the small evaporation region around the tip of the meniscus must take care of this current. The electric field acting on this region is intensified by the sharp, nearly equipotential surface of the hydrostatic meniscus, which apparently provides a mechanism to effectively control the total evaporation rate with only small changes of the meniscus shape.

_{r}With this in mind, one might now observe that the feeding impedance must play an integral role in emission, in contrast to LMIS where strong space charge effects typically modulate the current independent of any details of the flow.^{20} Consider the case where $I0=\zeta (E0,s)/CR$ is the initial current for a generic source with “starting” field *E*_{0,}_{s}. After minor algebraic manipulation

where we have made use of the fact that the pressure curve *ζ* is highly linear in our findings to introduce the truncated Taylor expansion $\zeta (E0)=\zeta (E0,s)+d\zeta /dE0(\Delta E0)$. Recognizing that both *E*_{0,}_{s} and *dζ*/*dE*_{0} are effectively universal, Eq. (6) suggests that the slope of the I–V traces inherent to the high-field family should be inversely proportional to the impedance of the feeding system. Empirically, Fig. 3 collects the I–V data generated using several types of electrospray emitters and the liquid EMI-BF4. The results indicate a clear disparity in slope, with the shallowest curves corresponding to solid needles and the steepest to porous tips, congruent with attending impedance estimations.^{21–23}

Third, when parameter space is traversed at fixed field *E*_{0}, e.g., changing impedance *C _{R}*, the hydrostatic pressure of the bulk meniscus and its macroscopic shape are preserved by subtle modifications to the morphology of the tip. This is typically seen as either a sharpening of the tip (increasing field) or rounding of the tip (decreasing field) depending on the current that is to be drawn. There are, however, limits to how much the tip may be distorted, with our findings pointing to the absence of stationary solutions when currents greater than $O(I*)$ are required. This is reinforced in Fig. 4, which depicts a map of the

*E*

_{0}–

*B*parameter plane for

*C*= 10

_{R}^{3}and

*C*= 10

_{R}^{4}. Based on the associated upper bounds, higher impedances afford access to more of the parameter space since the corresponding currents ($I\u2009\u2272\u2009I*$) and tip curvatures ($\kappa \u2009\u2272\u20091/r*$) remain sufficiently modest.

Finally, Taylor's idealized cone is predicated upon a conducting liquid that obeys a mechanical interplay between electrical traction and capillary pressure only. The type of generic field distribution that this admits, $Env\u221dr\u22121/2$, is typically only permissible in the neighborhood of a boundary singularity, e.g., a perfectly sharp tip, and in this case requires a simple 49.29° half-angle to be defined by the liquid surface.^{10} Hydrostatic pressure values for high-field solutions of the present type are of order *γ*/*r*_{0} or less, leading one to expect the recovery of a very Taylor-like structure in the limit as *B* → 0, particularly in an intermediate region *r**≪ *r* ≪ *r*_{0}, where electrical and surface tension stresses dominate. Curiously, the menisci reported in Fig. 2(b) and their siblings deviate from the classical cone to a large degree yet still manage to exhibit global field distributions that are remarkably consistent with Taylor's (Fig. 5).

In conclusion, we have uncovered a family of menisci that support electrically-induced evaporation from the leaky dielectric liquids and have reported on several of their essential qualities. These include the basic findings that feeding hydraulics may play an important role in evaporation and that real menisci may not necessarily mirror the classical Taylor cone archetype, in contrast to the widespread conceptions. Such results begin to explain why purely evaporating menisci are difficult to visualize (e.g., very small menisci with $r0<O(10\u22125m)$ may be favored unless the impedance is appropriately large) and why emitters like solid needles are advantageous (higher inherent impedance). Further investigation of this family, particularly with respect to dynamic stability and characteristics for very low values of *B*, may prove extremely useful in the continued elucidation of these and many other related questions.

The Air Force Office of Scientific Research (AFOSR) and NASA Space Technology Research Fellowships Program are acknowledged for the financial support. F.J.H. acknowledges the support of the Spanish MINECO through Project Nos. DPI2013-47372-C02-02 and CSD2010-00011.