The domain modeler is a web-based modeling utility for the design of microfluidic components for eventual placement in an electrospray system network. The user designs a specific domain for which microfluidic properties of the domain are computed and displayed. The properties are computed with direct access to the properties database. There are two types of domains: feed system domains and emitter domains. Feed system domains include various flow media including cylindrical (capillary) or rectangular open channels, or porous media of similar shapes. Emitter domains incorporate the liquid Taylor cone charge emission physics, and the effects of the substrate on the Taylor cone base.
3.2.2. Emitter Domains
In order to set up an emitter component, the user selects an emitter type consisting of a Taylor cone substrate base design.
Table 4 lists the currently supported emitter types and associated references used to develop the utility and component models. ESPET utilities compute emitter properties for a specified volume flow rate at the onset voltage assuming a cone-jet mode of operation. The default flow rate is given by the minimum flow rate,
Qmin, at which a Taylor cone-jet can be sustained [
12]:
where
γ is the liquid surface tension,
ε is the relative permittivity,
ε0 is the vacuum permittivity, and
K is the liquid conductivity.
A number of outputs is produced by the emitter utility: onset voltage, hydraulic resistance of the emitter, the minimum flow rate, the droplet and ion currents, the charge to mass ratio of the spray current, thrust,
Isp, the maximum surface electric field, the mass flow, and the polydispersive efficiency. A screenshot of the capillary Taylor cone emitter domain modeler is shown in
Figure 5. It illustrates the required user inputs for estimating individual emitter performance and for developing a SPICE component. Note that for this model involving an IL propellant, the user needs to define the polarity of the emitter (parameter F in
Figure 5).
For dielectric propellants (e.g., ILs) the droplet current,
Idroplet, is computed from the empirical expression derived by Gañán-Calvo et al. [
47]:
The maximum cone-jet surface field is obtained from [
11]:
We then follow the ion field evaporation theory of Higuera [
39] and Coffman [
17] to compute the ion field evaporation current from:
where
σ (
Emax) is the surface charge density at the region of maximum field,
and
h are the Boltzmann and Planck constants,
T is the temperature, Δ
G is the ion solvation free energy, and
G is the free energy associated with the external field. The effective field evaporation area,
A(
Emax), is estimated from:
Equation (5) assumes field evaporation occurs at the neck of the cone-jet. Coffman provides justification to neglect the convection current in the field evaporation region near the Taylor cone tip. In this case the field evaporation current is equal to the conduction current given by:
and the internal electric field,
, can be computed assuming a steady-state surface charge density from:
This approach neglects ion emission at the jet tip. Measurements by Gamero-Castaño [
48] observed a substantial ion current fraction at high flow rates of a EMI-TFSI system, and concluded that ion emission occurs primarily at the jet tip at these conditions. At high flow rates,
Emax is too low for measurable field evaporation. Currently, the ESPET cone-jet model does not include jet tip ion emission (high flow rate mixed ion-droplet regime).
Component models require that the microfluidic network provides input flow and pressure as a function of extraction field strength. This is straightforward for actively pressurized systems where the flow rate is given by the ratio between the pressure drop across the feed system and the feed system hydraulic resistance. For passively pressurized emitters, the electric field drives the flow. In a cone-jet mode, we compute the field-induced pressure to produce flow as an excess pressure beyond the onset voltage,
, as proposed by Perez-Martinez [
31]:
where
is the extraction voltage,
rc is the tip curvature for an externally wetted or porous emitter, or the capillary inner radius for an internally wetted or capillary emitter, and
D is the tip to extractor distance.
pscale is an empirical adjustable parameter that determines the fraction of the electric field-induced force that results in induced flow.
For emitters in a pure ionic regime (PIR), we follow the work of Coffman et al. [
18]. This is still an area of active research and there are no simple analytical formulae that predict that a Taylor cone operates in a PIR. For both dielectric and LM propellants, it is known that pure ion emission is more probable for feed systems with high hydraulic resistance [
13,
16,
18]. Coffman and coworkers [
18] determined that the ion evaporation current for dielectric propellants with significant conductivities is inversely proportional to a dimensionless feed system hydraulic resistance parameter,
CR:
where
rbase is the Taylor cone base radius and
B is a slope parameter that Coffman et al. suggest is universal in that it applies to all dielectric propellants that operate in a conduction limited regime (liquid metals are known to operate in a space charge limited regime [
10]). The group of Lozano recently conducted single emitter measurements in the PIR. The measurements include a newly tested substrate consisting of a xerogel with very narrow pore-size distribution [
31,
43]. We also include measurements by Guerra-Garcia et al. [
49] using borosilicate tips.
Figure 6 plots the experimental voltage–current (VI) slopes against 1/
CR to determine the value
B. The data are in polarity pairs and show that only the EMI-BF4 results can be reasonably subjected to a linear regression. The derived slope for EMI-BF4 measurements corresponds to a value of
B = 6.06 × 10
−9 Ω
−1. The time-of-flight measurements conducted by Perez-Martinez [
31] demonstrate that the EMI-TFSI system operated in a cone-jet mixed ion-droplet regime and the relations in Equation (10) for P are thus not applicable.
The results in
Figure 6 raise the question whether the
B slope value is indeed universal. The data suggest that EMI-BF4 can operate in a PIR at lower values of
CR than EMI-TFSI. Currently we propose that the user sets a propellant-dependent threshold value of
CR at which the emitter starts to operate in a PIR. Our data survey suggests
CR ≈ 3 for EMI-BF4, and at least 10 for EMI-TFSI. We also note that the numerical calculations by Coffman et al. [
18] focused on systems with
CR ≥ 1000, which is a regime normally applicable to externally wetted systems. Finally, using
CR as the switching parameter neglects the role of the surface electric field,
Emax, and does not do justice to the pioneering work of Coffman in navigating the extensive
CR,
Emax, and
rbase stability phase space [
17,
18]. Critical in the future development of ESPET is finding an algorithm that incorporates these results properly. Nevertheless, a switching value of
CR provides a means to turn on and off the PIR when analyzing array data, as demonstrated below.
For LM systems, ESPET incorporates the theory introduced by Mair who derived a current–voltage expression for “low-impedance” liquid metal capillaries or low-impedance, grooved externally wetted emitters or emitters with roughened surfaces [
13]:
where
M is the atomic mass of the liquid metal propellant,
e is the unit charge, and α
T is the Taylor cone half angle of 49.3°. For high-impedance LM Taylor cones (smooth externally wetted tips, or small capillaries), a flow-impedance (hydraulic resistance) factor is introduced:
where
Z is the electric impedance. For LMs we also estimate the mass efficiency based on an empirical expression developed by Tajmar and coworkers for liquid indium systems [
14]. The mass efficiency corresponds to the mass fraction of the output mass flow attributable to atomic ions, the remainder of the mass flow being due to droplets. The theory for droplet formation in LM systems is still underdeveloped.
Porous conical or edge emitters, and externally wetted emitters have the additional complication of multiple emission sites per emitter. Edge emitters can have hundreds of emission sites per centimeter. For typical IL contact angles, emission sites of porous substrate emitters are located at specific surface pores in areas of high surface curvature. Emission sites on externally wetted needle emitters are associated with surface protrusions that exceed the average roughness. The protrusions and pores form a base for a Taylor cone if sufficiently wetted by the propellant. Since the resulting Taylor cone bases and tip curvatures are subject to length scale distributions, each emission site is associated with a different onset voltage. In addition, the more displaced the pore is from the emitter tip or edge, the weaker the surface electric field, thus further broadening the distribution of onset voltages. As the voltage is raised, the number of emission sites exposed to field strengths exceeding the onset field increases. Unlike the single emission site VI curves, which are close to linear, those of multiple emission site emitters have a parabolic appearance [
2,
16].
Further complications arise when the Taylor cone base radii are greater than the pore size for highly wetted propellant/substrate combinations (small contact angles). A model for porous emitters must, therefore, also account for propellant pooling. In
Section 5, we discuss benchmarking a porous model with multiple emission sites using experimental data that spatially resolves emission currents from different emission sites of a porous conical emitter and an externally wetted emitter.