4.1. Ionic liquid physical properties and model inputs

The results presented in this section follow the same characteristic non-dimensional numbers based on the properties of standard ionic liquids as defined in Coffman et al. (Reference Coffman, Martínez-Sánchez and Lozano2019). These properties are similar to those of $\text {EMI}-\text {BF}_4$, which is a widely used ionic liquid in the literature of pure-ion evaporation (Romero-Sanz et al. Reference Romero-Sanz, Bocanegra, Fernández De La Mora and Gamero-Castaño2003; Legge & Lozano Reference Legge and Lozano2011).

The physical properties are $\kappa _0 = 1$ S m$^{-1}$, $\kappa ' = 0.04$ S m$^{-1}$ K$^{-1}$, ${q}/{m} = 10^6$ C kg$^{-1}$, $\mu _0 = 0.037$ Pa s, $\kappa _T = 0.2$ W m$^{-1}$ K$^{-1}$, $c_p = 1500$ J kg$^{-1}$ K$^{-1}$, $\gamma = 0.05$ N m$^{-1}$, ${\rm \Delta} G = 1$ eV, $\rho = 10^3$ kg m$^{-3}$ and $\varepsilon _r = 10$. These properties determine most of the non- dimensional parameters shown in tables 1 and 2, namely $\varLambda = 12$, $\psi = 38.6$, $\chi = 1.81 \times 10^{-3}$, $We = 2.26 \times 10^{-6}$, $Ca = 0.026$, $Gz = 0.024$, $K_c = 1.32 \times 10^{-4}$ and $H = 0.176$.

The reported results contain variations of parameters that are mostly external to the physical properties of the working ionic liquid. The space of independent variables that are numerically explored are $\hat {E}_0$, $\hat {R}$ and $\hat {Z}$. The reservoir pressure is taken to be $\hat {p}_r = 0$, since this is the most common case for operation of passively fed emitters.

4.2. Diagram of the regions of static stability

A more detailed version of the stability diagram presented in Coffman (Reference Coffman2016) is presented in this section. In particular, this analysis extends the range of exploration of solutions from an interval of non-dimensional contact line radius $\hat {R} \in [10,110]$ in Coffman (Reference Coffman2016) to $\hat {R} \in [6,210]$.

Figure 3 shows the combinations of non-dimensional external electric field $\hat {E}_0$ and contact line radius $\hat {R}$ that yield statically stable menisci. Static equilibrium solutions are found at a given $\hat {R}$ for combinations of electric fields outside the black stripped region above $\hat {E}_{max}$ and below the solid grey lines at their correspondent value of non-dimensional hydraulic impedance coefficient. According to the characteristics of the equilibrium solutions, the stability diagram is divided in four regions.

Figure 3. Map of the static stability boundaries as a function of the non-dimensional external electric field $\hat {E}_0$ and non-dimensional contact line radius $\hat {R}$ for seven hydraulic impedance coefficients. Static solutions exist at a given $\hat {R}$ for external electric fields smaller than the limit boundary for the aforementioned impedance. Dashed lines show the regions of the stability diagram that share the same contact line angle $\theta$ with the electrode for $\hat {Z} = 0.0839$. Contact angle values can be extrapolated to the other hydraulic impedance coefficients.

Region I spans the set of non-dimensional contact line radii above the critical $\hat {R}_{crit} \approx 16$ and external fields below $\hat {E}_0 \approx 0.513$. The region I is characterized by a lack of meaningful current output. This family of hyperboloid-like equilibrium solutions is well known in the literature (Basaran & Scriven Reference Basaran and Scriven1989a) and out of the scope of discussion in this paper. These non-emitting equilibrium shapes experience turning solutions when going past the field $\hat {E}_0 = 0.513$. As mentioned in § 3.2.2, solutions turn back on themselves as a symptom of imminent instability at turning points.

The existence of a critical radius below which no turning point exists ($\hat {R}_{crit}$) suggests that the disparity between $r*$ and $r_0$ is important for stability. On the limit where $r_0 \gt \gt r^*$ (high $\hat {R}$) the non-dimensional critical electric field scales as ${E^*}/{E_c} = \hat {E}^* \sim \hat {R}^{{1}/{2}}$ (see the non-dimensional kinetic law for charge evaporation in table 2). The invariance of the turning point at $\hat {E}_0 = 0.513$ at high $\hat {R}$ confirms that the associated instability is not driven by the activated emission process, but by standard Rayleigh instability. In other words, if the evaporation process were significant in this loss of stability, the maximum local electric field in the vicinity of the meniscus tip would be of the order of the critical field. Instead, equilibrium surfaces on the verge of the turning point instability ( $\hat {E}_0 < 0.513$) are observed to be mostly independent of $\hat {R}$ and $\hat {Z}$, and the local electric fields at the menisci tip are more than one order of magnitude smaller than $\hat {E}^*$.

The lack of ion emission precludes any ion transport and ${\varepsilon _r \hat {j}_{conv}}/{\hat {K}}$ can also be neglected. The surface charge expression in (3.2) can therefore be reduced to $\hat {\sigma } = -\hat {\boldsymbol {\nabla }}\hat {\phi }^{v} \boldsymbol {\cdot } \boldsymbol {n}$. The latter expression indicates the surface charge can be considered to be fully relaxed, and the meniscus behaves like a conductor.

Beroz, Hart & Bush (Reference Beroz, Hart and Bush2019) showed that the stability of a conducting axisymmetric droplet exposed to an external electric field and pinned on a conducting surface follows a scaling law of the form

(4.1)\begin{equation} \frac{r^3_0}{V} > \frac{{\rm \pi} \varepsilon_0 E_0^2}{\dfrac{2 \gamma}{r_0}}, \end{equation}

where $r_0$ is the pinning radius and $V$ is the volume of the droplet. This scaling law predicts the stability limits obtained numerically by Basaran & Scriven (Reference Basaran and Scriven1990) for the cases of negligible hydrostatic pressure inside the droplet.

Using the reference magnitudes, the non-dimensional form of (4.1) becomes

(4.2)\begin{equation} \frac{1}{\hat{V}} > 2 {\rm \pi}\hat{E}^2_0. \end{equation}

The non-dimensional volume in the region of non-dimensional electric fields close to the lower turning point is shown in figure 4. It is observed that increasing the electric field yields equilibrium shapes of higher volume. The convergence criteria (3.5) was reached for non-dimensional electric fields up to $\hat {E}_0=0.513$. As seen in figure 4 for electric fields slightly higher than this limit, and contact line radii higher than $\hat {R}_{crit}$, the volume of the shapes along the successive iterations approaches the Basaran–Beroz stability boundary until the volume is large enough to trigger the Rayleigh instability. It is worth mentioning that the derivative of the volume with respect to the external field becomes singular at the instability, as expected by its turning point nature.

Figure 4. Non-dimensional volume of the equilibrium shapes in region I of the stability diagram. Comparison with the Basaran–Beroz limit (Beroz et al. Reference Beroz, Hart and Bush2019) in green. Solutions for $\hat {R}$ greater than $\hat {R}_{crit}$ are shown in red, whereas solutions at smaller $\hat {R}$ are shown in dashed blue. The volume of the shapes at selected iterations for the first unstable $\hat {E}_0$ are shown in the black markers, where the volume can be seen to grow exponentially before breaking the numerical procedure.

Region II spans non-dimensional contact line radii greater than $\hat {R}_{crit} \approx 16$ and fields greater than $\hat {E}_0 \approx 0.485$. These high electric field solutions are characterized by menisci with substantial charge evaporation.

Figure 3 shows the combination of electric fields and contact radius $\hat {R}$ where statically stable emitting solutions were found in region II for seven different non-dimensional hydraulic impedance coefficients ($\hat {Z}$). Upper limits for increasing values of $\hat {Z}$ are shown in brighter grey-shaded hard lines. As shown in figure 3, for a given $\hat {R}$, the range of electric fields where static solutions were found increases for higher hydraulic impedance coefficients until a maximum range ending at $\hat {E}_{max} \approx 1.414 \sim \sqrt {2}$. The upper limit of stability corresponding to $\hat {Z} >= 0.0305$ collapses at $\hat {E}_0 = \hat {E}_{max}$ for $\hat {R} > \hat {R}_{crit}$.

Figure 3 also shows the meniscus contact angle isolines with the downside electrode $\varGamma ^v_D$, $\theta$, for the different combinations of $\hat {R}$ and $\hat {E}_0$. Simulations show that $\theta$ is very weakly dependent on the $\hat {Z}$ and $\hat {R}$ in this region. Contact angle isolines in figure 3 correspond to $\hat {Z} = 0.0839$ and they could be extrapolated to other values of $\hat {Z}$ within either region of static stability. The dependence of the contact angle on the external electric field $\hat {E}_0$ is distinct enough that solutions in region II can be classified further in two subregions.

Subregion II.a is limited to electric fields below $\hat {E}_0 \approx 1.1$ and characterized by equilibrium shapes that increment their contact line angle $\theta$ and decrease their volume for increasing values of the electric field. Solutions within this moderate field range were explored by Coffman et al. (Reference Coffman, Martínez-Sánchez, Higuera and Lozano2016) and showed a sharper interface than the hyperboloidal menisci in region I. Prototypical interface geometries can be seen in figure 5(b). These static menisci have a characteristic emission region of non-dimensional size ${r^*}/{r_0} = \hat {R}^{-1}$, where the non-dimensional electric fields are of the order of the critical field $\hat {E}^* \sim \hat {R}^{{1}/{2}}$. The surface charge on these menisci is not relaxed and the temperature is around 3 %–5 % higher than in the bulk ionic liquid due to heating by ohmic dissipation (Coffman et al. Reference Coffman, Martínez-Sánchez and Lozano2019). Figure 6 includes the flow structure of a prototypical equilibrium interface in subregion II.a. Streamlines show the recirculation cells occupying a large volume of the meniscus. This could be related to the low characteristic flow rates of menisci in the pure-ion mode (Herrada et al. Reference Herrada, López-Herrera, Gañán-Calvo, Vega, Montanero and Popinet2012). The emission region is amplified on the top of figure 6, where electric fields of the order of the $E^*$ are found.

Figure 5. Characteristic equilibrium shapes of representative regions identified in the stability diagram. Equilibrium shapes in region I are depicted in (a) with $\hat {Z} = 0.0839$ and $\hat {R} = 43$. Region II.a characteristic equilibrium shapes are in (b) with $\hat {Z} = 0.0147$ in solid and $\hat {Z} = 0.147$ in dotted lines for $\hat {R} = 54$. Region II.b contains shapes depicted in (c) for $\hat {Z} = 0.1586$, $\hat {R} = 54$. Shapes along iterations for a combination of $\hat {E}_0 = 1.43$, $\hat {R} = 32$ and $\hat {Z} = 0.1586$ in region III are shown in (d). Equilibrium was not reached in the latter simulation.

Figure 6. Prototypical pure-ion menisci internal flow structure. Operational space parameters used in this figure correspond to $\hat {R} = 43$, $\hat {E}_0 = 0.7$, $\hat {Z} = 0.0839$, $\hat {p}_r = 0$. The non-dimensional magnitude of the electric field is shown on the left. Field intensity is of the order of $E^*$ near the tip, where the evaporating fluid velocity streamlines end. The effect of ohmic heating transport near the tip is represented in the temperature plot on the right side subfigure.

The balance of stresses in the normal direction of a prototypical equilibrium shape in subregion II.a are shown in figure 7. Equilibrium shapes in this region look similar to a flattened Taylor cone, with a closed small region at the apex, where the meniscus is emitting. Near the emitting region, the curvature is high enough to sustain the majority of the electric stress needed for pure-ion evaporation. Near the contact line region, the meniscus does not emit. In this regard, the velocity field is negligible and the pressure is mostly that from the boundary conditions in (2.36), or the one originated due to friction of the fluid with the walls upstream. In this region near the contact line, the meniscus tends to a planar geometry, therefore the electric stress is compensated mostly by the hydrostatic pressure.

Figure 7. Plot (a) shows the distribution of dimensionless normal stresses for a prototypical equilibrium shape in region II.a ($\hat {E}_0 = 0.71$, $\hat {R} = 64.2$, $\hat {Z} = 0.0305$). Values at $\hat {r} = 0$ correspond to the stresses onto the meniscus axis of symmetry. Values at $\hat {r} = 1$ correspond to stresses onto the meniscus contact line with the electrode. Electric stresses in red, surface tension in green, hydrodynamic fluid stresses in blue. The corresponding equilibrium shape is shown in (b). The relative residual used as a criterion of convergence is shown in (c). The absolute residual is shown in (d).

Regions I and II.a overlap in a narrow range of electric fields between $\hat {E}_0 \sim 0.485$ and the turning point in $\hat {E}_0 \sim 0.513$ (green zone in figure 3). Whether the solver converges to an emitting equilibrium shape of subregion II.a or non-emitting equilibrium shape in region I depends on the initial guess provided to the solver. Figure 8 shows emitting (II.a) and non-emitting (I) solutions existing for the same external field $\hat {E}_0 = 0.49$. The current diminishes when the electric field is decreased with a starting solution from the emitting subregion II.a. The current being very small at these field magnitudes undermines the relative importance of the hydrodynamic stress with respect to the surface tension and the electric stress. In this sense, the equilibrium shapes tend to resemble the canonical Taylor solution with negligible static pressure. The exact Taylor conical shape cannot be recovered with this setting due to the planar electrode geometry sustaining the meniscus and the hydrostatic suction pressure originated by the small but non-zero current flow. This hysteresis behaviour is well documented experimentally for LMIS (Forbes Reference Forbes1997), where the extinction voltage is typically smaller than the one needed for the onset of pure-ion emission.

Figure 8. Equilibrium shapes in the hysteresis region for the emitting case (solid red) and non-emitting case (dotted red). Taylor cone geometry and characteristic emitting meniscus at higher stable fields are shown for cross-reference.

The turning point nature of the instability when approaching subregion II.a from non-emitting interfaces in region I, suggests the existence of a dynamic mechanism with mass ejection that cannot be described by the time-independent meniscus model with a closed interface presented in this paper. It is difficult to speculate what the emission outcome would be in this transition. An option for this could be droplet breakup that might be preceded by both cone-jet formation and ion evaporation. If such a cone jet were to exist in this region, it would be reasonable to infer a substantial deviation of its interface shape from the Taylor solution due to the high hydraulic impedance of capillaries feeding pure-ion menisci. This shape would change rapidly, resembling a ‘suctioned’ Taylor cone with a volume that would decrease at higher values of the electric field until the field was high enough to sustain steady ion emission.

The reduction of meniscus volume in subregion II.a due to the increase of the external field is accompanied with a rise in the contact angle $\theta$ with the downside electrode. It is known that electric fields could exhibit unbounded singular behaviours near sharp corners when these corners are greater than $180^{\circ }$ (Li & Lu Reference Li and Lu2000). The corner sharpens as the values of $\theta$ reach approximately $185^\circ \text {--}186^\circ$ and the equilibrium geometric shapes augment their curvature to compensate for the stronger electric stress that appears near the singularity. This curvature increase manifests as a small bump appearing near the contact line for external fields higher than $\hat {E}_0 \approx 1.1$. This point marks the beginning of subregion II.b.

Subregion II.b is only accessible when sufficient hydraulic impedance is provided. Equilibrium shapes contain this cylindrical bump near the contact line as seen in figure 5(c). The shapes also reduce their contact line $\theta$ and raise their bump amplitude for increasing values of the external electric field $\hat {E}_0$. The cylindrical bump does not emit any charge for the span of electric fields simulated in this region.

It should be emphasized that the model presented in this paper is axisymmetric and static. This prevents a determination of the effects of possible three-dimensional disturbances on the surface of this cylindrical bump that resembles a toroid. Disturbances like this originate capillary pinch-off instabilities and the eventual breakup of similar toroidal interfaces into smaller menisci (Mehrabian & Feng Reference Mehrabian and Feng2013; Fragkopoulos & Fernández-Nieves Reference Fragkopoulos and Fernández-Nieves2017). The determination of the dynamic stability of the equilibrium shapes in this subregion is beyond the scope of this study. However, it is certainly relevant to fully understand the structure and behaviour of these menisci and should be studied in detail.

Region II terminates at external electric fields $\hat {E}_0$ higher than $\hat {E}_{max} \approx \sqrt {2}$, when sufficient hydraulic impedance is provided. In dimensional form, the previous statement can be recast as a function of a reference electric pressure. It is helpful to define such pressure as a function of the electric field downstream from the emission region. In the case of a planar electrode such as the one studied in this paper, this reference field is taken as the external field $E_0$,

(4.3)\begin{equation} \frac{1}{2}\varepsilon_0 E^2_0 > 2 \left(\frac{2 \gamma}{r_0}\right). \end{equation}

It is then seen that the pure-ion emission cannot be sustained by a meniscus of pinning radius $r_0$ when the reference electric pressure is higher than approximately two times the surface tension stress of a liquid sphere of the same radius.

When $\hat {E}_0 > \hat {E}_{max}$, menisci in region III exhibit a sharp transition towards instability depicted in figure 5(d): the cylindrical contact line bump grows to such an extent that the electric field on its crest becomes of the order of the critical field, while the central emission region protuberance shrinks progressively until it disappears. At this point, the cylindrical bump transforms into an emitting corona with a significantly larger emission area, thus producing a dramatic increase in the current output that, in turn, produces a large pressure drop through the feeding channel. This pressure drop induces a sudden suction on the meniscus interface near the axis of symmetry, quickly terminating the simulation as the numerical procedure cannot track these changes.

At $\hat {E}_0 = \hat {E}_{max}$ point, the equilibrium interfaces turn on themselves when increasing the values of the electric field in a similar way described in Basaran & Wohlhuter (Reference Basaran and Wohlhuter1992) for the electrified menisci in region I. This can be seen in figure 9, where the aspect ratio of the equilibrium shapes obtained exhibits this singularity.

Figure 9. Aspect ratio of equilibrium shapes in region II at different hydraulic impedances.

The scaling in (4.3) appears to be independent of all parameters of the operational space considered in this study, namely $\hat {p}_r$, $\hat {Z}$ and $\hat {R}$ (when $\hat {R} > \hat {R}_{crit}$) and cannot be described in detail with the axisymmetric and static models implemented for the same dynamic instability reasons mentioned previously.

Regardless, reporting the existence of this sharp transition could be informative for future investigations of menisci bifurcation phenomena that are known to exist in the operation of pure-ion emission sources. Bifurcation is observed when the applied voltage increases over a critical value that depends on source geometry and liquid properties (Pérez-Martínez & Lozano Reference Pérez-Martínez and Lozano2015). Such critical voltage would correspond to a non-dimensional field that, according to the results presented here, cannot exceed the upper bound field value of the stability range. This is an important empirical validation point that requires more in-depth work with versions of this model based on source geometries and domains similar to those used in experiments.

Figure 10(a) shows the stress distributions along the meniscus interface for $\hat {E}_0 = 1.41$, thus, very close to the instability boundary (4.3). Solutions for three different reservoir pressures $\hat {p}_r = -1,0$ and 1 are shown in dotted, solid and dashed lines, respectively. The non-dimensional currents emitted are $\hat {I} = 0.920 \times 10^{-4}, 1.971 \times 10^{-4}$ and $2.978 \times 10^{-4}$, respectively (if non-dimensionalized by the characteristic emitted current, ${I}/{I^*} = 0.0814, 0.175$ and $0.264$, respectively). Differences in the stress distributions are concentrated in the vicinity of the emission region, where electric fields need to increase to accommodate higher current outputs at higher reservoir pressures. When emission is irrelevant, such as in the vicinity of the contact line where the bump forms (figure 10b) and $\hat {\sigma }$ is relaxed, stress distributions are a function of the external electric field only and directly independent from any parameter resultant from the emission. At this location, the only stress that would contain direct information from the emission region is the fluid hydrodynamic stress, where the local pressure equals that from the drop in the channel, thus, proportional to the total emitted current. However, simulations show that this pressure near the contact line $\hat {p} = \hat {p}_r-\hat {I} \hat {R}^{{5}/{2}} \hat {Z}$ is mostly invariant from $\hat{p}_r$, $\hat {Z}$, $\hat {T}$ on $\varGamma ^l_D$, $\psi$ and $\varepsilon _r$, therefore, mostly a function of $\hat {E}_0$. This results in a set of equations that locally resemble the equilibrium of a perfect non-emitting conductor subject to an upstream suction stress, but with a sole degree of freedom or $\hat {E}_0$. This fact confers the limit observed in (4.3) some sense of universality and independence from ionic liquid physical properties, other than $\gamma$.

Figure 10. Plot (a) shows non-dimensional normal stresses and equilibrium shapes for solutions at $\hat {E}_0 = 1.41$, $\hat {R} = 43$, $\hat {Z} = 0.8394$ as a function of the non-dimensional radial coordinate $\hat {r}$. Stress solutions with three different reservoir pressures are shown in dashed, solid and dotted lines corresponding to $\hat {p}_r = 1,0$ and $-1$, respectively. Electric stress distribution in red, surface tension stress in green and fluid hydrodynamic stress in blue. Plot (b) shows corresponding equilibrium shapes. The relative and absolute residuals are shown in (c,d), respectively.

In cases where the hydraulic impedance is not sufficiently high, statically unstable solutions appear at values below $\hat {E}_{max}$. This can be seen in figure 11. The diagram is similar to the one shown in figure 3, but instead of using the nominal non-dimensionalization used in this paper, results in this analysis are presented with reference values of the field relating to the emission region ($E^*$). Recall that the critical electric field depends exclusively on the ionic liquid properties and not on the source geometry, whereas nominal field $E_c$ is a function of the non-dimensional contact line radius $r_0$. For this reason, this alternative non-dimensionalization is more useful for relating simulation results to experimental data. In this non-dimensionalization the maximum electric pressure limit decays with the field (green line), instead of being a vertical line.

Figure 11. Boundaries of static stability as a function of the external field non-dimensionalized by the critical field. Boundaries are shown for different dimensionless hydraulic impedance values $\hat {Z}$. The minimum non-dimensional impedance for the existence of emitting solutions in the range of $\hat {R}$ displayed in the figure is shown in black. Limits for increasing values of $\hat {Z}$ are shown in grey. Extrapolated values are shown in dotted lines. The hypothetical bifurcation point is shown in green. For the analysed impedance values greater than $\hat {Z} = 0.0096$, the maximum current limit crosses the presumable bifurcation limit at $\hat {R}_{cross} \approx 180,75,40$ and $25$ for $\hat {Z} = 0.0147,0.0302,0.0514$ and $0.0833$, respectively.

First, the need of a minimum hydraulic impedance of $\hat {Z} \approx 0.0031$ for static solutions to exist can be noticed for any of the $\hat {R}$ in the simulated range. The corresponding dimensional impedance is approximately $Z = 4.32 \times 10^{18}$ Pa m$^{-3}$ s$^{-1}$ for the ionic liquid EMI-BF$_4$. This impedance is very close to that observed by Romero-Sanz et al. (Reference Romero-Sanz, Bocanegra, Fernández De La Mora and Gamero-Castaño2003) for achieving the pure-ion regime in capillary tubes of similar diameter as those reported here. The value of this impedance was predicted to be $Z \approx 4 \times 10^{18}$ Pa m$^{-3}$ s$^{-1}$ by Pérez-Martínez (Reference Pérez-Martínez2016). Second, it can be seen that the stability ranges are widened in figure 11 for increasing values of $\hat {Z}$.

Figure 12 shows the isocurrent lines at three values of $\hat {Z}$. The limits of static stability for each $\hat {Z}$ are also shown with bolder lines. Note how the increase of the stability boundaries is at the expense of a lower current output at fixed ${E_0}/{E^*}$ and $\hat {R}$. This trade-off between current output and meniscus stability is well known in the experimental pure-ion evaporation literature (Castro et al. Reference Castro, Larriba, Fernandez De La Mora, Lozano and Sumer2006; Castro & Fernández De La Mora Reference Castro and Fernández De La Mora2009; Hill et al. Reference Hill, Heubel, Ponce De Leon and Fernando Velásquez-García2014).

Figure 12. Dimensionless isocurrent maps as a function of the contact line radius and external field referenced to $r^*$ and $E^*$, respectively. Results shown for three different values of hydraulic impedance. Hypothetical bifurcation point is shown in green. Hypothetical maximum current limit is shown in hard black for each of the $\hat {Z}$ displayed. Extrapolations are shown dotted.

Figure 5(b) shows how equilibrium shapes adapt to this current reduction when changing the hydraulic impedance at fixed $\hat {R}$ and $\hat {E}_0$. For the higher impedance case (dotted line), equilibrium shapes are smoother in the neighbourhood of the emission region. In this case, local electric fields are less intense because of the lower current throughput demand. Therefore, surface tension can balance the electric stress with larger radii of curvature. Equilibrium shapes near the region close to the contact line are practically invariant with the increase of $\hat {Z}$.

Third, the limit of stability for every hydraulic impedance shown in figure 12 resembles an isocurrent line of about ${I}/{I^*} \sim 2.2$ for all the $\hat {Z}$ shown in figure 12. This suggests that the static stability of a meniscus in the pure-ion mode is linked to a limit in current throughput, when $\hat {E}_0 < \hat {E}_{max}$.

The existence of a maximum current appears to be related to a reduction in the area of emission at the apex of the meniscus. The contraction of the emission area is linked to a decrease in the radius of curvature that is needed to compensate for the higher electric stress. This trade-off between the reduction of the emission area and growth of the current density appears to limit the current that can be extracted from the meniscus for increasing values of $\hat {E}_0$ (see Appendix D). This phenomenon was predicted to exist also for viscousless liquid metals (Forbes et al. Reference Forbes, Ganetsos, Mair and Suvorov2004).

From the data shown in figures 11 and 12 at a given value of $\hat {Z}$, the two competing instability phenomena will occur at different ranges of $\hat {R}$. Menisci would lose their stability by a presumably bifurcation phenomena if their size $\hat {R} > \hat {R}_{cross}$, and will be limited by a maximum current throughput when $\hat {R} < \hat {R}_{cross}$. Interestingly, $\hat {R}_{cross}$ provides the largest span of stable electric fields. As seen in figures 11 and 12, this $\hat {R}_{cross}$ decreases when more hydraulic impedance is provided, and the range of fields widens. For representative values of $r^*$ in ionic liquids (${\sim }50$ nm) and impedances greater than $Z = 10^{19}$ Pa m$^{-3}$ s$^{-1}$, $r_{0_{cross}} = \hat {R}_{cross} \cdot r^*$ is found to be below 3 $\mathrm {\mu }$m in dimensional form $(\hat {R}_{cross} \sim 100)$.

If the range of stable fields was a measure of the probability of finding the meniscus at any $\hat {R}$, then $\hat {R}_{cross}$ would be a good estimation of this value. As mentioned previously, the scale of $\hat {R}_{cross}$ is close to the diffraction limit of standard optical observation systems, thus explaining in part the reason why non-invasive direct observation of pure-ion emitting menisci has not been reported by the scientific community.

The characteristic small meniscus sizes where the static stability ranges are maximum ($\hat {R}_{cross}$) are not in contradiction with the findings of Castro et al. (Reference Castro, Larriba, Fernandez De La Mora, Lozano and Sumer2006), Garoz et al. (Reference Garoz, Bueno, Larriba, Castro, Romero-Sanz, De La Mora, Yoshida and Saito2007) or Romero-Sanz et al. (Reference Romero-Sanz, Aguirre De Carcer and Fernández De La Mora2005), where the pure-ion regime is achieved for substantially larger diameter capillaries between $40$ and $200\,\mathrm {\mu }$m. The results in figures 3, 11 and 12 only show the predicted static stability ranges for menisci of non-dimensional radius between $\hat {R} = 4$ and $\hat {R} = 210$. For the $r^*$ of EMI-BF$_4$, these ranges correspond to radii in between $0.1$ to $10\,\mathrm {\mu }$m. If the maximum field limit (4.3) is extrapolated to these radii, stable menisci are still found, yet at a lower range of electric fields. It is worth mentioning that having direct observation of these menisci could be very valuable, particularly to discard any emission process governed by smaller ill-anchored menisci at the rim of the capillary channel.

The effect of the two mechanisms that lead to static instability on the current is shown in figure 13. Figure 13 shows the current–field curves for different pairs of $\hat {R}$ and $\hat {Z}$. The curves with smaller radii and higher hydraulic impedance are shown in grey. The maximum current achieved in these cases corresponds to an external field $\hat {E}_0 = \hat {E}_{max}$, therefore losing stability by the presumed bifurcation of the meniscus. These results show how the maximum currents achieved for such bifurcating menisci are typically smaller than the current limit of ${I}/{I^*}_{max} \approx 2.4$ obtained for the cases of lower $\hat {Z}$ and higher $\hat {R}$. In these latter cases shown in black, stability is lost when reaching that current. Note how in curves of such lower impedances, the current emitted per unit field is higher. This effect is well known in the literature (Krpoun et al. Reference Krpoun, Smith, Stark and Shea2009).

Figure 13. Current emitted as a function of the external field for menisci in region II. From left to right, the non-dimensional radius of the curves correspond to $\hat {R} = 100, 75, 64, 54, 43$ and $21$. In that order, the non-dimensional hydraulic impedance coefficients correspond to $\hat {Z} = 0.0096, 0.021, 0.030, 0.045, 0.083$ and $0.47$. For the radius depicted in grey, the presumably bifurcation point was reached before ${I}/{I^*}_{max}$.

The dimensionless flow parameter $\eta =\sqrt {{\rho \kappa _0 Q}/{\gamma \varepsilon _r \varepsilon _0}}$ defined by Fernández De La Mora & Loscertales (Reference Fernández De La Mora and Loscertales1994) is also shown on a right vertical axis in figure 13. Unlike electrosprays in the mixed droplet-ion regime, where decreasing values of $\eta$ are typically needed for achieving higher currents (Lozano & Martínez-Sánchez Reference Lozano and Martínez-Sánchez2002), electrosprays in the pure-ion mode exhibit larger current throughput at increasing values of $\eta$. It is also interesting to note that while conventional cone-jet electrosprays become unstable when approaching $\eta \sim 1$ from higher flow rates, the results in this work suggest that pure-ion electrosprays also become unstable near $\eta \sim 1$, but when approached from lower flow rates.

The current limit of stability appears to hold in a wide range of electric fields, radii and hydraulic impedances. Figure 14 shows the current emitted in the limit of stability for 21 different pairs of $\hat {Z}$ and $\hat {R}$ when the hydraulic impedance is not sufficient to trigger the bifurcation process. The range of maximum currents is between 2.1–2.4 times $I^*$ for all the simulated values.

Figure 14. Maximum values of the current reached for 21 different non-dimensional fields, radii and hydraulic impedances. Values of radius and external fields are chosen in region II. Values of the hydraulic impedance are chosen low enough for not triggering the bifurcation point at $\hat {E}_0 = \hat {E}_{max}$.

The effect of the meniscus geometry ($\hat {R}$) at fixed $\hat {Z} = 0.0096$ are shown in figure 15 for different values of $\hat {R}$. For all cases investigated in this figure, the values of $\hat {Z}$ and $\hat {R}$ are not sufficient to trigger the presumed bifurcation and static equilibrium solutions were found yielding current outputs below ${I}/{I^*} \approx 2.1$. Figure 15(a) shows the current output as a function of the non-dimensional external field ${E_0}/{E^*}$. Unlike electrospray cone-jets, where the liquid profile and emitted current is a function of the operational parameters and mostly independent from the electrode geometry (Fernández De La Mora & Loscertales Reference Fernández De La Mora and Loscertales1994; Gamero-Castaño & Magnani Reference Gamero-Castaño and Magnani2019), menisci in the pure-ion mode are typically smaller and more sensitive to changes in the electric field, as their emission region is comparatively closer to the electrodes and the space charge in the ion plume is negligible. The effect of this is seen in the higher steepness of the current–field slope for the smaller menisci.

Figure 15. Figure shows the non-dimensional current emitted by differently sized meniscus at $\hat {Z} = 0.0096$ as a function of ${E_0}/{E^*}$ (a) and an average of the ${E^v_n}/{E^*}$ fields in the neighbourhood of the meniscus tip, or $\hat {r} = 0$ (b). The average is performed as follows: $({1}/{A_0})\int _{A_0} ({E^v_n}/{E^*})\,\textrm {d} A_0$, where $A_0 = \int ^{{\rm \Delta} \hat {r}}_0 2{\rm \pi} \hat {r} \sqrt {1 + \hat {y}'^2} \,\textrm {d}\hat {r}$ for the portion of the meniscus ${\rm \Delta} \hat {r}$ that emits current (up to 1 % of the current density emitted at the apex).

It is worth mentioning that, when the current emitted is plotted against an average of the normal fields in the vacuum near the tip of the meniscus (${E^v_n}/{E^*}$), the results nearly collapse into a single curve (figure 15b). This reinforces the notion that current throughput could be regarded as a function of the local values of the electric fields, including the mechanism behind a possible limitation in current, such as the one described in Appendix D.

Region IV is defined for contact line radii below $\hat {R}_{crit}$ and it is characterized by the lack of a transition gap. Equilibrium menisci in this region evolve smoothly from a non-emitting configuration to an emitting configuration for increasing values of $\hat {E}_0$.

Simulations of equilibrium shapes have also been performed for contact line radii above $r_0 \approx 250$ nm ($\hat {R} \approx 6$). The continuum approach below this length scale is likely no longer valid due to the role that discrete molecular effects start to play.

Menisci in this region resemble those explored by Higuera (Reference Higuera2008). As discussed by Coffman et al. (Reference Coffman, Martínez-Sánchez and Lozano2019), the non-dimensional critical electric field $\hat {E}^* = \hat {R}^{{1}/{2}}$ is of the order of those found near the apex of the hyperboloidal shapes described by Basaran & Scriven (Reference Basaran and Scriven1990). The pressure drop created by the evaporation process compensates for the electric stress before the Rayleigh instability is triggered. This phenomenon can be seen in figure 4. For the cases where $\hat {R} < \hat {R}_{crit}$ (blue lines), the pressure drop reduces the volume increase due to the action of the electric field to shapes that lie within the Basaran–Beroz limit (Beroz et al. Reference Beroz, Hart and Bush2019). At this point, the meniscus is no longer hydrostatic, the surface charge is not fully depleted and the channel pressure drop is significant, making the Basaran–Beroz limit no longer valid.

If the electric field is increased further for emitting shapes with $\hat {R}< \hat {R}_{crit}$, then the hydraulic pressure drop becomes more relevant than the surface tension in compensating for the electric stress pull over the meniscus interface. It is observed that at very high hydraulic impedance coefficients ($\hat {Z} > 0.7136$), the instability described in region III is triggered at lower electric fields $\hat {E}_0 <\hat {E}_{max}$. Somewhat against intuition, for $\hat {R} < \hat {R}_{crit}$, this instability occurs at increasingly lower external electric fields when $\hat {Z}$ increases.

Unlike the distribution of stresses of the equilibrium solutions in region II, where most of the electric stress is balanced by the surface tension near the meniscus apex, solutions in region IV are somewhat planar when the electric field downstream approaches the limit of stability. Figure 16 shows the normal stress distributions for an equilibrium solution in region IV very close to the instability limit. The meniscus is practically hydrostatic in this region (fluid flow stress is negligible). The electric field stress is practically counteracted by the hydraulic pressure drop due to current evaporation. Without the surface tension playing a relevant role, the hydraulic impedance coefficient controls the sensitivity of the balance to the electric stress. It is observed that when the electric field remains close enough to the stability boundary, the suction pressure due to the hydrostatic drop grows beyond the value of the electric stress and turns the meniscus inside out, thus making it adopt a concave form which was considered to be unstable due to the aforementioned three-dimensional effects not captured in the axially symmetric formulation.

Figure 16. Distribution of the normal component of the stress to the meniscus interface for an equilibrium solution in region (IV) close to the electric field of instability ($\hat {E}_0 = 1.2$, $\hat {R} = 10.7$, $\hat {Z} = 71.4$); the meniscus axisymmetric interface profile $\hat {z}$ is shown in subfigure b). Normal electric and fluid stresses are shown in red and blue, respectively, surface tension stress in green. Relative and absolute residuals are shown in subfigures (c,d), respectively.

4.3. Influence of the liquid bulk temperature on emission and stability properties

The physical properties of ionic liquids depend on temperature, sometimes in a significant way. It is therefore expected that liquid bulk temperature variations will have an effect on the static stability of menisci investigated in this work.

Ohmic dissipation, as described by the energy transport equation (2.31) is the driving mechanism behind the temperature gradients in the liquid, specifically in the vicinity of the emission region where the current density is the highest.

The mechanical balance of stresses on the meniscus is affected through changes in electric conductivity $\kappa (T)$ and fluid viscosity $\mu (T)$, and through a modification of the activation law for ion evaporation (2.1). The global effect of a liquid bulk temperature increase on the emission characteristics can be seen in figures 17 and 18.

Figure 17. Plot (a) shows the non-dimensional emitted current density distribution along the meniscus interface in the vicinity of the emission region ($\hat {r}$ near $0$). Non-dimensional temperature distribution along the interface from the emission region to the contact line is shown in (b). The non-dimensional electric fields normal to the meniscus interface in the vicinity of the emission region are shown in (c,d) for the vacuum and liquid, respectively. Plot (c) also shows the non-dimensional surface charge distribution (dashed line). Results are shown for three different ionic liquid bulk temperatures and the isothermal case for comparison reasons. Simulation data corresponds to $\hat {R} = 64$, $\hat {E}_0 = 0.78$ and $\hat {Z} =0.0144$.

Figure 18. Current emitted scaled to $I^*$ as a function of the non-dimensional electric field. Results are shown for three different temperatures at $\hat {Z} = 0.0302$. From right to left, meniscus sizes are $\hat {R} = 107, 86, 64, 43$ and $21$.

Intuitively, the rise of the electric conductivity due to a temperature increase may incur more current throughput (for a meniscus with negligible convective charge transport, $\boldsymbol {j} = \kappa (T)\boldsymbol {E}$). However, while the current density distribution normal to meniscus interface increases near the meniscus apex (figure 17a), the area of emission is slightly reduced in such a way that the total current emitted (its integrated value along $\varGamma _M$) remains unexpectedly constant despite the temperature (and conductivity) increments. The total current emitted does not change even if an isothermal meniscus is considered at $\hat{T} = 1$. This total current emitted is $I/I^*=0.317$, for the parameters in figure 17.

Note that, for the linear conductivity model with temperature used in this paper and the values of the parameters simulated ($\chi = 1.81 \times 10^{-3}$ and $\varLambda = 12$), a higher conductivity also increases the ratio between the characteristic emission time ($\tau _e \sim {h}/{k_B T}$) and the charge relaxation time ($\tau _r \sim {\varepsilon _r\varepsilon _0}/{\kappa (T)}$). For moderate increases of temperature, namely $\hat {T} \approx 1.04$, the increase of the ratio ${\tau _e}/{\tau _r}$ is about $40\,\%$, where

(4.4)\begin{equation} \frac{\tau_e}{\tau_r} = \frac{\chi (1+\varLambda (\hat{T}-1))}{\hat{T}} \end{equation}

and $\chi = {h\kappa _0}/{k_BT_0\varepsilon _0\varepsilon _r}$ (see table 3). In this case, the meniscus is able to relax surface charge faster than the rise of emission time scale at higher bulk temperatures. This phenomenon can be seen in figure 17(c), where a more relaxed non-dimensional surface charge distribution ($\hat{\sigma} \sim \hat{E}^v_n$) is observed.

This over-relaxation of $\hat {\sigma }$ will tend to reduce the internal electric field, given the assumption that the external electric field $\hat{E}^v_n$ has a weaker dependence on the temperature (figure 17d). In dimensional form, this can be observed by using the interface field condition (2.17) to write the internal field as a function of $\sigma$,

(4.5)\begin{equation} E^l_n = \frac{\varepsilon_0 E^v_n - \sigma}{\varepsilon_0 \varepsilon_r}. \end{equation}

The validity of this assumption (see figure 17c) is supported by the fact that larger variations in the external electric field would affect exponentially the current output through (2.1).

The dependence of the emitted current density on the two phenomena can be better appreciated when writing it as an explicit function of the normal electric field acting on $\varGamma _M$, $E_n^v$,

(4.6)\begin{align} j^e_n = \kappa(T)E_n^l &= \kappa_0\left(1+\varLambda\left(\hat{T}-1\right)\right) \frac{\varepsilon_0 E_n^v-\sigma }{\varepsilon_0 \varepsilon_r} \nonumber\\ &= \frac{\kappa_0\left(1+\varLambda\left(\hat{T}-1\right)\right) \dfrac{E_n^v}{\varepsilon_r}}{1+\dfrac{\tau_e}{\tau_r}\exp{\dfrac{\psi}{\hat{T}} \left(1-\sqrt{\dfrac{E_n^v}{E^*}}\right)}}, \end{align}

where (2.1) and (2.17) have been used to relate $\sigma$ to $j^e_n$ and $E_n^v$.

Given these results, an anticipated increase of current due to a higher conductivity is cancelled out by a reduction of the electric field inside the meniscus due to charge relaxation. This effect can be seen in figure 18, which shows the negligible effect of the liquid temperature on the extracted total current for a given $\hat {Z}$ and $\hat {R}$ as a function of $\hat {E}_0$. These results support the hypothesis of Lozano & Martínez-Sánchez (Reference Lozano and Martínez-Sánchez2005), where the experimental increase of current at higher temperatures is associated to a decrease of the hydraulic impedance due to the lower viscosity of the ionic liquid.

Another effect linked to an increase of the bulk temperature of the liquid is shown in figure 18. It can be observed that the maximum current limit occurs at higher values for higher $\hat {T}$, as predicted by the lumped parameter model in Appendix D. Conversely, the effect of increasing the temperature is widening the range of electric fields where pure-ion emission is statically stable, irrespective of the meniscus radii $r_0$. The expansion is reflected in the increase of the maximum ${I}/{I^*}$ at higher bulk temperatures (figure 19). However, it is true that this range cannot increase without limit. According to the findings in this paper, the maximum range is determined by the upper limit electric field above which pure-ion emission cannot be sustained with a single axisymmetric meniscus (4.3). Regardless, menisci operating at electric fields below (4.3) that are not stable at a given impedance could stabilize if heated, while keeping the same impedance. This could give insight into explaining the temperature thresholds needed for achieving the pure ionic regime in capillary tubes of smaller impedance than porous tips (Romero-Sanz et al. Reference Romero-Sanz, Aguirre De Carcer and Fernández De La Mora2005; Garoz et al. Reference Garoz, Bueno, Larriba, Castro, Romero-Sanz, De La Mora, Yoshida and Saito2007).

Figure 19. Stability diagram obtained for different ionic liquid bulk temperatures at a constant hydraulic impedance coefficient of $\hat {Z} = 0.0302$. Stability boundary is also shown for a higher impedance $\hat {Z} = 0.0833$ in grey. Stability computed considering an isothermal meniscus is shown with a dashed line for reference. For the latter case, no energy equation was solved, and bulk temperature was set to $\hat {T} = 1$.

Figure 19 also shows that taking energy conservation into consideration is very relevant in describing the static stability boundaries. Dashed lines show how much narrower the stability field range would look like for $\hat {Z} = 0.0833$, when considering an isothermal meniscus (i.e.  without taking into account any heating effects). In fact, no statically stable solutions were found at $\hat {Z} = 0.0302$ for the isothermal case. This effect is consistently related to the fact that heated menisci are more accessible to higher maximum currents at similar values of $\hat {Z}$.

The energy transport results are shown in figures 20(a) and 20(b) as a function of the external electric field $\hat {E}_0$ for two different hydraulic impedances corresponding to $\hat {Z} = 0.0302$ and $\hat {Z} = 0.0833$, respectively. The non-dimensional contact line radius is $\hat {R} = 64.23$ (3 $\mathrm {\mu }$m for EMI-BF$_4$). Figure 20 shows $\hat {\dot {Q}}\hat {R}^2$, which is the non-dimensional power transported in and out of $\boldsymbol {\varOmega }_l$, normalized by the ionic liquid physical properties ($E^*, r^*, \kappa _0$). The first fact to note is that the enthalpy convected into $\boldsymbol {\varOmega }_l$ through $\varGamma _I$ (red solid line) is practically balanced by the enthalpy convected out of $\boldsymbol {\varOmega }_l$ through ion evaporation on the meniscus interface (red dashed line). The scale of the ohmic dissipation and conduction through the walls tends to dominate over the convected power at larger fields. It is shown also that viscous dissipation (in green) is negligible over ohmic heating (four orders of magnitude less).

Figure 20. Non-dimensional power transported by conduction through the channel walls (blue, solid) and the channel inlet (blue, dashed). Power transported by convection into the meniscus through the inlet (red, solid) and out from the meniscus through the meniscus interface (red, dashed). Ohmic heating and viscous power are shown in black and green, respectively.

Most of the steady-state ohmic heating is transported via conduction through the channel walls (blue solid line) and the channel inlet (blue dashed line). A rough first order of magnitude estimation of the impact of heat dissipation by conduction to a perfect thermally conducting emitter structure could be stated as

(4.7)\begin{equation} E^{*^2} r^{*^3} \kappa_0 \hat{\dot{Q}} \hat{R}^2 \approx \rho^e V_D c_p^e \frac{{\rm \Delta} T}{{\rm \Delta} t}, \end{equation}

where $\rho ^e$ is the density of the emitter material, $V^e_D$ is the dry volume of the emitter and $c^e_p$ is its specific heat. Using the values of $E^* \approx 6.95 \times 10^{8}$ V m$^{-1}$, $r^* \approx 46.7$ nm and $\kappa _0 \approx 1$ S m$^{-1}$ and a dry volume of $V^e_D = 0.5$ mm$^3$ per emitter, yields ${{\rm \Delta} T}/{{\rm \Delta} t} \approx 221\hat {\dot {Q}} \hat {R}^2$ K h$^{-1}$ for a carbon emitter ($c^e_p \approx 710$ J Kg$^{-1}$ K$^{-1}$, $\rho ^e \approx 2260$ Kg m$^{-3}$) and ${{\rm \Delta} T}/{{\rm \Delta} t} \approx 137 \hat {\dot {Q}} \hat {R}^2$ K h$^{-1}$ for a tungsten emitter ($c^e_p \approx 134$ J Kg$^{-1}$ K$^{-1}$, $\rho ^e \approx 19\,300$ Kg m$^{-3}$). For a moderately sized meniscus and electric field value in between the two shown in figure 20, $\hat {\dot {Q}} \hat {R}^2 \approx 5\times 10^{-3}$ and ${{\rm \Delta} T}/{{\rm \Delta} t} \approx 1.11$ and 0.69 K h$^{-1}$ for a carbon and tungsten emitter, respectively. The latter is a worst case estimation of the heating in a floating emitter. Generally speaking, the part of the emitter that captures the heat has substantially higher thermal diffusivity ($\alpha \sim 2.165 \times 10^{-4}$ m$^2$ s$^{-1}$ for carbon and $6.69 \times 10^{-5}$ m$^2$ s$^{-1}$ for tungsten) than the ionic liquid ($1.33 \times 10^{-7}$ m$^2$ s$^{-1}$), therefore, is able to dissipate heat with ease if connected to a thermal reservoir through a similar interface size. These scalings reinforce the notion that the emitter runs fundamentally cold in steady-state operation, and that stability of the source could be described with accuracy with the constant room temperature boundary condition at the channel walls $\varGamma ^l_ D$.

4.4. Other ionic liquids

The model presented in this paper is non-dimensional. Due to the similarities in scale for many non-dimensional numbers of ionic liquids numbers, these results are generalizable to other ionic liquids. In particular, from the results presented in this paper so far, it has been observed that the upper limits of stability appear to be dependent solely on $\gamma$, the meniscus size and the external field conditions $E_0$, and current emitted appears to be mostly determined by operational field conditions only ($\hat {E}_0,\hat {Z},\hat {p}_r$), when $r_0 \gt \gt r^*$, and very weakly dependent on temperature changes.

At the ${\rm \Delta} G = 1$ eV considered in this paper, hydrodynamic stresses play a minor role and the highest variability in the emission conditions and equilibrium configurations will mostly be given by parameters governing the electric problem, namely $\varepsilon _r$. Figure 21 shows the current density, normal electric fields and interfacial charge along the emission region for $\varepsilon _r = 10,15,20$, where most of the ionic liquids lie. Similar to what happens with the temperature increase, the effect of a higher charge relaxation time with $\varepsilon _r$, is balanced by higher electric fields in the liquid to yield almost equal currents. Note how interfacial charge departs from relaxation when the $\varepsilon _r$ increases. Recall the charge relaxation time $\tau _e = {\varepsilon _0 \varepsilon _r}/{\kappa _0}$. From the results shown, a maximum value of $\varepsilon _r$ is predicted beyond which charges cannot travel fast enough to the interface for the scale of the characteristic emission time $\tau _r$, and emission vanishes.

Figure 21. Plot (a) shows the non-dimensional current density along the emission region in the meniscus interface for different $\varepsilon _r$. Normal electric fields in the vacuum and interfacial charge in (b). Normal electric fields in the liquid in (c). Non-dimensional numbers dependent on $\varepsilon _r$ were updated as: $We = 10^{-6}$, $Ca =0.017$, $\chi = 1.21\times 10^{-3}$, $H = 0.078$, $Gz =0.016$ for $\varepsilon _r = 15$, and $We = 5.65 \times 10^{-7}$, $Ca =0.013$, $\chi = 9.05\times 10^{-4}$, $H = 0.044$, $Gz =0.012$ for $\varepsilon _r = 20$.

It is also worth mentioning that accurate values of ${\rm \Delta} G$ are not very well known for ionic liquids. Variations in ${\rm \Delta} G$ affect the critical field to the square power (2.3) and reduce the value of $r^*$ at a power four rate (2.6). The sensitivity of the results to increments of ${\rm \Delta} G$ is substantial and can be seen in figure 22, where the balance of normal stresses (a) and equilibrium shapes (b) are shown for two meniscus of equal radii and different ${\rm \Delta} G$ (1 and 1.3 eV in solid and dashed lines, respectively). Moderate variations of ${\rm \Delta} G$ originate equilibrium shapes with almost four times the magnitude of the normal stresses in the emission region. It is worth mentioning how hydrodynamic stresses start to become relevant in the emission region at higher values of ${\rm \Delta} G$, yet keeping the total current emitted constant, and invariant to changes in this property.

Figure 22. Plot (a) shows the balance of stresses in the normal direction for the cases of ${\rm \Delta} G = 1$ eV (solid) and ${\rm \Delta} G = 1.3$ eV (dashed). Results are shown up to $r=0.2$ to reinforce the values at the emission region. Electric stresses are shown in red, surface tension in green and hydrodynamic viscous stresses in blue. Equilibrium shapes are shown in (b). Relative and absolute residuals are shown in (c,d). The same dimensional contact line radius was used of 2 $\mathrm {\mu }$m, which corresponds to $\hat {R} = 42.8$ when using ${\rm \Delta} G = 1$ eV, and $\hat {R} =121$ when using ${\rm \Delta} G = 1.3$ eV. The non-dimensional electric field is $\hat {E}_0 = 0.77$. The non-dimensional hydraulic impedance is $\hat {Z} = 0.105$. Non-dimensional numbers dependent on ${\rm \Delta} G$ were updated for ${\rm \Delta} G = 1.3$ eV as: $K_c = 6.37\times 10^{-4}$, $\psi = 50.18$ $Ca =0.044$, $H = 0.062$, $Gz =0.041$.