Introduction

Recently, nanosatellite operations have been rapidly increasing because of their lower costs and shorter development periods compared to conventional, large satellites [1, 2]. For advanced missions, such as satellite constellations and deep space exploration, nanosatellites need propulsion systems for orbital maneuvers and attitude control [3]. To this end, nanosatellites require propulsion systems that can achieve tight positioning control, low thrust/high Isp, and low power consumption within an allocated mass and size budget [4]. Among the various electric propulsion systems used today, such as Hall thrusters, gridded ion thrusters (GIT), or pulsed plasma thrusters (PPTs), ionic liquid electrospray thrusters (ILESTs) could further increase nanosatellite payload capabilities because of their small size and flexible operating modes [5, 6].

Figure 1 shows a schematic of the electrospray thruster, which mainly consists of an emitter array and an extractor electrode. Applying a high voltage between the emitter and extractor enables the ionic liquid to reach the emitter tip by the electric force, forming a Taylor cone [7]. When the local electrostatic pressure at the emitter tip exceeds an internal pressure by the surface tension of the ionic liquid, ions or droplets are extracted and accelerated, and then the thrust is provided to a spacecraft [8]. ILESTs can operate as either ionic liquid ion sources (ILIS) [9], where propellant is passively fed through an externally wetted emitter or porous material [10, 11] to emit purely monomer ions, or a droplet-dominated emission mode, where propellant is actively fed through a capillary emitter [12].

Fig. 1
figure 1

Schematic of the electrospray thruster

Ionic liquids contain only anions and cations without solvent. The vapor pressure of ionic liquids is almost zero because of the strong Coulomb force between cations and anions [13, 14], indicating that ionic liquids can stay as a liquid phase even in space. As opposed to gridded ion and Hall thrusters, during bi-polar pulse high voltage electrospray operation, both anions and cations can be extracted such that no neutralizers are required [15]. High-pressure gas storage and feeding systems are also unnecessary for ILESTs. These features allow the entire propulsion system to be miniaturized significantly compared to other electric propulsion systems, thus enabling concepts like staging of entire thruster modules or multiplexing of ILEST emitters for improved redundancy and scalability [3, 16]. Moreover, droplet-mode ILEST technologies, such as the Colloid Micronewton Thrusters (CMNTs) on-board the LISA Pathfinder technology demonstration, exhibit key mission capabilities such as micronewton thrust precision and low thrust-noise. [17, 18]

Previous studies indicated that some of the divergent ion beams collided with the extractor, which could erode the extractor [19,20,21]. To prevent extractor erosion, we have investigated trajectories of ion beams extracted from the emitter tip using particle-in-cell (PIC) simulations [22]. However, we found that the initial conditions at the emitter tip, such as the kinetic energy and the extracted angles, strongly affected ion trajectories, where the kinetic energy and the extracted angles had to be assumed in the particle simulations.

This observation, that charged particle trajectories are most sensitive to initial ion kinetic energies and extraction angles, is corroborated by similar electrospray plume models and experiments. For instance, Nuwal et al. [23] conducted 3D-PIC simulations using an unstructured adaptive automatic mesh refinement/octree grid to show the large electric field gradients near the emitter, spanning multiple orders of magnitude in length scale. It follows that the electric potential energy is equal to the kinetic energy of a charged particle at the site of emission. Furthermore, previous retarding potential analyzer (RPA) experiments have shown that variations in initial ion emission energy can influence the spacing between the cation-anion pairs. Past a certain energy threshold, the cation and anion molecules may overcome their attractive Coulomb forces, resulting in ion cluster dissociation or fragmentation of downstream solvated ion clusters [24, 25]. Furthermore, Gamero-Castaño [26] implemented an Eulerian electrospray beam model based on the line-of-point-charges approach and demonstrated a strong dependence of charged droplet trajectories on their initial velocities. Parmar et al. [27] conducted a variance-based, global sensitivity analysis on the inputs for a reduced-order, electrospray plume model and concluded that a particle’s final position is most sensitive to other conditions such as its initial emission angle by orders of magnitude relative to its initial position and the electrode voltage. Thus, the primary objective of this work is to investigate the kinetic energy and the extracted angles at the emitter tip. To this end, we have conducted molecular dynamics (MD) simulations, which can analyze the physical motions of atoms and molecules.

MD simulations are widely used for many advanced applications [28,29,30,31,32,33] or for obtaining basic physical parameters, such as electrical conductivity [34], surface tension [35], viscosity [36], diffusion coefficient [37], and wettability [38]. They are also employed for ion extraction of electrospray thrusters because electrospray phenomena occur on an atomic scale [35, 39,40,41,42,43]. However, because the scale is a few orders of magnitude smaller than the real scale electrospray thrusters, MD simulations are time-consuming. To reduce calculation time, coarse-grained (CG) models have also been investigated by representing an ionic liquid molecule by only treating groups of atoms instead of each individual atom [34, 41, 42]. While CG models can achieve length-scales of up to a few 10s of nanometers, it is extremely small compared with a typical capillary scale of a few micrometers.

Our approach is similar to the work by Prince et al. [44], where a droplet of 4 nm in diameter is simulated by an all-atom model, not by coarse-grained approaches. As mentioned in their paper, the droplet simulations do not probe the interaction between ionic liquid and electrodes, but can estimate the jet’s tip, which leads to droplet fission or evaporation. Moreover, nanodroplets often exit in the acceleration region, where a high electric field is applied between the emitter and extractor electrodes. In the following study, we investigate the kinetic energy and extracted angle of anions and cations that are extracted from a 4–5 nm droplet under a uni-directional uniform electric field condition. The following section describes our numerical model. In “Results & discussion” section, numerical results are shown, where we employ two types of ionic liquids, EMI–BF4 (1-ethyl-3-methylimidazolium tetrafluoroborate) and EMI–Im (1-ethyl-3-methylimidazolium bis((trifluoromethyl)sulfonyl) imide), and focus on the position, angle, and energy of extracted ions. Finally, conclusions are drawn in “Conclusion” section.

Simulation methods

Molecular dynamics overview

Molecular dynamics can simulate atomic-scale phenomena, which consider intramolecular and intermolecular forces. In MD simulations, we calculate three-dimensional equations of motion to obtain the positions and velocities of each atom. Forces used in the equation of motion are derived from the differential equations of the potential:

$$F_{i} = -\frac{\partial\phi(r_{i})}{\partial r_{i}},$$
(1)

where ϕ is the interaction potential of the molecular system and Fi and ri are the force and position of the ith atom, respectively. In this study, all the MD simulations were carried out using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) package [45]. A Package for Building Initial Configurations for Molecular Dynamics Simulations (Packmol) is also used for the molecular arrangement during the setting of the input files [46]. Packmol implements an optimal packing algorithm that can speed up system equilibration and thermalization [47].

The interaction potential, which is necessary for Eq. 1, consists of intramolecular and intermolecular potentials that can be described by the following functional form:

$$\begin{aligned} \phi & = \sum\limits_{{bonds}}^{{N}}k_{r} (r_{ij} - r_{0})^{2} + \sum\limits_{{angles}}^{{N}}k_{\theta} (\theta_{ijk} - \theta_{0})^{2} \\ & + \sum\limits_{{dihedrals}}^{{N}} k_{{\phi}} \left[1+ \cos {(n}{\phi}_{ijkl} {-\delta)} \right ] \\ & + \sum\limits^{{N-1}}_{{i=1}}\sum\limits^{{N}}_{{j>i}} s_{ij} \left(4\epsilon_{ij}\left[ \left(\frac{\sigma_{ij}}{r_{ij}}\right)^{12} - \left(\frac{\sigma_{ij}}{r_{ij}}\right)^{6} \right] + \frac{1}{4\pi\epsilon_{0}}\frac{q_{i}q_{j}}{r_{ij}}\right). \end{aligned}$$
(2)

The intramolecular potential models the molecule’s structure, where kr, kθ, and kϕ, are the corresponding bond, angle, dihedral angle force constants, r is the distance between atoms attached by covalent bonds, θ is the angle within three atoms connected by covalent bonds, and the torsional angles are formed on two planes that include more than three molecules, where ϕ is the dihedral angle based on four atoms connected in a sequence. The dihedral angle potential also includes an integer value representing the multiplicity or number of local minima in a complete rotation about bond jk, n, and the associated phase factor, δ. Figure 2 depicts the physical significance of each term as it relates to the cations and anions used in this work [48].

Fig. 2
figure 2

Visual aid of terms in Eq. 1, where representative bonds, angles, and dihedral angles are shown in the (a) EMI + cation, (b) BF\(_{4}^{-}\) anion, and (c) EMI–Im anion often denoted as Tf2N, respectively

The intermolecular potential determines the structure and the movement of the overall population of molecules in the system and accounts for four intermolecular interactions: electrostatic, induced, dispersion force, and repulsive. In MD simulations, the four interactions are approximately divided into long-range and short-range interactions. As given by Eq. 2, the Coulomb potential is used for the long-range interaction, and the 12-6 Lennard-Jones (LJ) potential is used for the short-range interaction, where εij is the potential depth, σij is the distance between atoms i and j at which the potential reaches zero, and sij is an empirically-determined scaling factor. Because the long- and short-range interactions indicate that the potential vanishes only at infinity, we need to define a certain distance within which the potential affects the atoms, i.e., the cutoff distance. The values of the cutoff distance are shown in “Nanodroplet simulation procedure and equilibration” section. To reduce the computational cost of intermolecular interactions, only two-body interactions are calculated in this study. The potential between N atoms is calculated by adding the potential between two atoms, thus reducing the number of interactions to nC2=N(N−1)/2. Additionally, intermolecular interactions were neglected (i.e., sij=0) for covalently bound atoms separated by up to two bonds, scaled down by 50% (i.e., sij=0.5) for atoms separated by three bonds, and fully considered for atoms separated by four or more bonds (i.e., sij=1.0).

In order to accurately calculate all the relevant forces acting on molecules and atoms in a system, selecting the proper force field, which refers to the functional form of the interaction potential and the empirical parameters therein, is one of the most critical factors in MD simulations. In previous MD investigations of ionic liquids for electrospray propulsion applications, a wide range of force fields have been tested. For example, the effective-force coarse-graining (EFCG) model [49] has achieved significant reductions in computing time for mesoscale EMI–BF4 simulations [34, 37, 39] by replacing the cations and anions to their most essential functional groups (i.e., the imidazolium ring, the CH2 and CH3 in the alkyl chains, or the entire BF4). Despite achieved computational savings, CG models have shown difficulties in reproducing basic ionic liquid properties. To this end, all-atom force fields have been used instead and follow the most common Class I force fields: AMBER [34, 50, 51], CHARMM [44, 52], and OPLS [39]. Class I force fields typically follow the functional form given in Eq. 2 and can model a wide range of condensed-phase, organic systems [48]. Since the initial use of MD for imadazolium-based ionic liquids, significant research has been dedicated towards obtaining the most accurate set of Class I force field parameters, either by experimental techniques or ab initio or density functional theory (DFT) calculations [53]. At the cost of additional runtime, higher-fidelity polarizable [54] and reactive force [55] fields are capable of accounting for more complex physical interactions, such as locally induced dipole moments or thermal decomposition, respectively. In this study, we used the Optimized Potentials for Liquid Simulations-All Atom (OPLS-AA) force field, one of the general potentials developed for organic liquids [56]. The scaling factor sij for the long-range interactions in Eq. 2 is defined by results from Lopes et al. [57].

Nanodroplet simulation procedure and equilibration

We employed 125 EMI–BF4 and 125 EMI–Im molecules as a droplet of approximately 4–5 nm in diameter, which was assumed to be the diameter of the jet’s tip or meniscus [58]. We analyzed ion extraction by applying a uniform electric field of 1.3–1.7 V nm−1, which is enough to extract ions, assuming EMI–BF4 and EMI–Im droplets were placed in a vacuum without gravity.

Before applying the electric field, we equilibrated the droplet for 300–400 ps from the initial condition using the canonical (NVT) ensemble at T= 300 K. The initial velocity of each atom was set randomly to satisfy the Gaussian distribution, and a Nose-Hoover thermostat was used. We set six different initial conditions using different random seeds. Although the system reached an equilibrium state at 300 ps, we kept the calculation and obtained three more different equilibrium states at 325, 350, and 400 ps; a total of 24 different equilibrium conditions were prepared. Then, we applied the electric fields of 1.3–1.7 V nm−1 to each of the prepared 24 droplets for 30–50 ps, using the microcanonical (NVE) ensemble. We also used shrink-wrapped boundary conditions, where the boundary expands depending on the molecular motions.

The cutoff distance of the LJ potential was conventionally set to be approximately three times the longest distance of σij, where the LJ potential is almost zero. On the other hand, the cutoff distance of the Coulomb potential is set to be 100 Å, which is long enough to simulate 4-nm droplets.

Evaluation of ion extraction

In this study, the time step was set to 0.5 fs, and the positions and velocities of each atom were output at every 50 fs to analyze ion extractions. The kinetic energy and the angle of extracted ions were analyzed based on the output images, the position, and the energy of the extracted ions. Here, we assumed that ions were extracted when the ions left the droplet by approximately one or more molecular sizes and when the kinetic energy rose above the equilibrium state. It should be noted that this study focuses only on monomers because more than 90% of extracted ions were monomers. A few percent of dimers and trimers did not break up in a small simulation domain considered. Figure 3 shows the definition of the extracted angle. The extracted angles are defined as the angle between the velocity vector of extracted ions and the direction of the applied electric field, which is in the x-direction. Therefore, the extracted angles are described as

$$\alpha = \cos^{-1}\left(\frac{v_{x}}{\sqrt{v_{x}^{2} + v_{y}^{2} + v_{z}^{2}}}\right).$$
(3)
Fig. 3
figure 3

Definition of the extracted angle α

We also employ a solid angle to evaluate ion emission. The solid angle is obtained by assuming that an ionic liquid droplet is a sphere, and the colatitude is defined as the inclination from the line passing through the center of mass and parallel to the x-direction. When only a cation or anion is extracted, the droplet starts to move in the opposite direction. Since the nanodroplet’s motion has a non-negligible impact on the following ion emissions, we analyzed ion extraction until the droplet began to move significantly.

Results & discussion

Firstly, we set the cutoff distance of the LJ potential to 12 Å (>3σij) because this is the distance at which the potential is almost zero. Figure 4 shows simulated molecular images of the EMI–BF4 in an equilibrium state and the state after the electric field has been applied for 15.0 ps after the equilibrium, where the red molecules are cations, and the blue ones are anions. To verify the droplet reached equilibrium, various system properties were investigated. Using the open-source software package TRAVIS [59], the number density profile was calculated for each unique molecule as a function of z distance above and below the x-y plane. As shown in Fig. 5, the observed distribution of cations and anions indicates a well-mixed system and nearly isotropic behavior of molecules within the droplet. Furthermore, the spherical nature of a nanodroplet can be assessed based on elements of the moments of inertia tensor [44, 60]. As shown in Fig. 6, after ∼50 ps, the components fluctuate within 5% of an average value and indicate a quasi-spherical state. Lastly, the average radial mass density is shown in Fig. 7 [61]. The interior of the nanodroplet results in bulk-phase densities that are in excellent agreement with the experimental values for EMI–BF4, 1.28×103 kg m−3 [62]. The bulk-like behavior drops significantly at the droplet-vacuum interface due to a droplet diameter of ∼4.0 nm. We applied a uniform electric field along the x-direction (the right direction of the image). As shown in Fig. 4, a cation was extracted from the right side of the droplet due to the applied electric field. The distance between the extracted cation and the droplet was approximately 14 Å, which is equal to the molecular size of EMI–BF4 and EMI–Im.

Fig. 4
figure 4

EMI–BF4 (a) equilibrated droplet and (b) a state after electric field has been applied in the positive x-direction for ∼ 15.0 ps. The cations and anions are visualized in red and blue, respectively. Extracted ions are on the order of ∼14 Å

Fig. 5
figure 5

EMI–BF4 number density as a function of z

Fig. 6
figure 6

EMI–BF4 droplet moments of inertia as a function of time

Fig. 7
figure 7

EMI–BF4 droplet density as a function of radial distance

In the following analysis, only monomers and dimers are considered as extracted ions. The population fraction of species extracted from the droplet for EMI+-[EMI–Im]n is shown in Fig. 8 and compared with previous modeling and experimental results. The present study is in excellent agreement with similar MD simulation results provided by Prince et al. [44] and the slight discrepancy can be accounted for by the difference in electric field strength, where the stronger electric field of 1.5 V nm−1 results in a greater fraction of n=0 ions, or monomers. Both literature and present modeling results show greater fractions of monomers than the experimental results, which are taken from mass spectrometry and retarding potential analyzer (RPA) measurements from Ticknor et al. [63] and Lozano [64]. The same trend is given in Fig. 9 for EMI+-[EMI–BF4] based on experimental results by Natisin et al. [65] and Chen et al. [66].

Fig. 8
figure 8

Emitted species population fraction compared with literature-produced experimental and modeling values for EMI-Im

Fig. 9
figure 9

Emitted species population fraction compared with literature-produced experimental and modeling values for EMI-BF4

A few hypotheses arise to explain the increased number of dimer and trimers shown experimentally that are not accounted for in the MD simulations. First, Prince et al. note that emission behavior and the angle-resolved species may be especially sensitive to the emitter tip aspect ratio [44]. Moreover, it is noted that both MD simulations from Fig. 8 are using nonpolarizable force fields. In contrast, polarizable simulations capture local electric field effects that may strengthen Coulombic interactions and screen repulsive forces between cation-anion pairs; these effects have shown to be crucial in order to accurately model ionic liquid thermophysical properties [67]. In a comparative study by Yan et al. [68], nonpolarizable simulations of an imadazolium-based ionic liquid over-predicted surface tension by nearly 20 mN m−1. With higher surface tensions, nonpolarizable simulations may artificially suppress the evaporation of large solvated ion clusters and neutrals from the nanodroplet surface. Therefore, it is likely that polarizable simulations may result in higher numbers of dimers and trimers that would compare more favorably with experimental data. Regarding neutral species, while measurements are limited in the literature and are not a part of this effort, future simulations could be used to investigate the reported anomalous mass loss [10] that may hinder electrospray performance.

Figure 10 shows the energy of each ion constituting the droplet of ionic liquid when one of the cations is extracted. The horizontal axis indicates the ion number assigned to each ion in the MD simulation, and the vertical axis indicates the energy of each ion. The red marker within the green circle of Fig. 10 represents the cation extracted and leaving the droplet. Although some cations and anions have energies over 0.2 eV, those ions do not have enough energy to be extracted and remain in the droplet.

Fig. 10
figure 10

The energy state of each droplet

Figure 11 shows the time variation of the kinetic energy of the extracted cation of EMI–BF4 at the electric field of 1.5 V nm−1. Since the kinetic energy continuously changed, we could not determine the kinetic energy threshold when the ion was extracted. In this study, we judged whether the ion was extracted by comparing the output image with the value of the kinetic energy. As mentioned earlier, the ion was assumed to be extracted when the distance between the ion and the droplet was more than one or more molecular sizes. In the present case, the intermolecular distance was approximately 14 Å (Fig. 4), and this value exceeded the cutoff distance of the LJ potential (12 Å), which determined the short-range interaction between molecules. Beyond the cutoff distance of the LJ potential, only the uniform electric field and the Coulomb potential are applied to ions except for the intramolecular potential, and the uniform electric field of 1.3–1.7 V nm−1 is larger than the Coulomb force at 14 Å. Therefore, ions did not return to the droplet when the distance between the extracted ion and the droplet exceeded the cutoff distance of the LJ potential. As mentioned in the “Nanodroplet simulation procedure and equilibration” section, the data was output every 50 fs for the extraction process. Thus, we have uncertainty depending on this time frame. The uncertainty in energy is 0.1–0.2 eV and that in angle is 1–2, depending on the output data at 50 fs earlier or later.

Fig. 11
figure 11

Time variation of the kinetic energy of an EMI–BF4 cation at the electric field of 1.5 V nm−1

The ion kinetic energies are consistent with previously developed theory on ion evaporation and dissociation. Originally postulated by Iribarne and Thomson [69], the evaporative flux of charged particles from a droplet surface is defined by

$$j = \sigma \frac{kT}{h} \exp{\left[-\frac{1}{kT}\left(\Delta G_{0} - \sqrt{\frac{q^{3}E}{4\pi\epsilon_{0}}}\right)\right]},$$
(4)

where σ is the surface charge density, k is Boltzmann’s constant, T is the system temperature, h is Planck’s constant, ΔG0 is the energy barrier for the particle of charge q to be emitted from the surface, and E is the electric field strength. For common ionic liquids, ΔG0∼ 1–2 eV [70], thus requiring E∼ 1–2 V nm−1 for ion emission, as observed in the presented results.

We counted the number of ions extracted every 5 ps after applying the uniform electric field and investigated how easily cations and anions were extracted from EMI–BF4 and EMI–Im droplets to validate our MD simulations. The time until the first emission event was recorded for each case and compared with results from Prince et al. in Fig. 12. It is noted that the Prince et al. simulations used a detection plane located 10 nm away from the droplet surface, thus accounting for the difference in timescale magnitudes. All simulations show that with increased electric field strength, the time for an emission event to occur decreases monotonically. The comparison between EMI–Im and EMI–BF4 can be explained by their fundamental difference in electrical conductivity as well as molecular level theory. The EMI–Im anion, often denoted as [TF2N], is a relatively bulky molecule compared to the BF\(_{4}^{-}\) anion, resulting in EMI–Im cation-anion configurations that experience stronger, longer lasting interactions between the imadazolium-ring hydrogens and the [TF2N] charged atoms [71]. In effect, the EMI–BF4 molecules can experience faster rotational and translational motion, especially in response to an external electric field. Figure 12 also shows that these differences between EMI–Im and EMI–BF4 become less apparent at stronger electric field strengths.

Fig. 12
figure 12

Comparison of initial emission event as a function of electric field strength

Since the droplet could not keep the shape as time passed after ion extraction, the measurement time was set at up to 50 ps. Figure 13 shows the number of ions extracted for every 5 ps for different applied electric fields of 1.3–1.7 V nm−1. Note that the calculation time was set to 30 ps for EMI–BF4 at the electric fields of 1.5 and 1.7 V nm−1 because the EMI–BF4 droplet could not keep the shape and was broken up into smaller pieces under the strong electric field after approximately 30 ps. As shown in Fig. 13(a)-(c), the cation of EMI–BF4 started to be extracted in a few picoseconds after the uniform electric field was applied, while EMI–Im cations tended to be extracted slowly. When the applied electric field increased, the time of the ion extraction decreased for both EMI–BF4 and EMI–Im. Compared with EMI–Im, a large number of EMI–BF4 cations were extracted within 30 ps, although the cation of EMI–BF4 is the same as that of EMI–Im. Regarding the anions (Fig. 13(d)–(f)), EMI–Im anions are much less extracted than EMI–BF4 anions. Since the electrical conductivity of EMI–BF4 (1.57 S m−1) is higher than that of EMI–Im (0.91 S m−1), the extracted current of EMI–BF4 is higher than that of EMI–Im in experiments. The tendency of the MD simulations is in reasonable agreement with the experimental one.

Fig. 13
figure 13

Number of ions extracted every 5 ps after the electric field is applied

Figure 14 shows the number of ions extracted per steradian as a function of solid angle for both EMI–BF4 and EMI–Im at the applied electric fields of 1.3, 1.5, and 1.7 V nm−1. As shown in the figure, most ions are extracted around 0 sr, and the solid angle increases when increasing the applied electric field. Compared with EMI–Im cations and anions, more ions are extracted around 0 sr for EMI–BF4. This tendency is summarized in Table 1. Approximately 60% of both EMI+ and BF\(_{4}^{-}\) ions are extracted within 0.2 sr (14.5) at 1.3 V nm−1 for EMI–BF4, while 92% of EMI+ ions and 73% of BF4 ion are extracted within 0.4 sr (20.6). The percentage decreases when increasing the applied electric field. These values for EMI–Im cations and anions tend to be lower than those for EMI–BF4. Since the number of EMI–Im ions extracted is smaller than that of EMI–BF4 ions as shown in Fig. 13, the percentage varies more widely for EMI–Im. Nonetheless, the percentage also tends to decrease when increasing the applied electric field, and the values of EMI–Im are lower than those of EMI–BF4. This tendency is probably because EMI–Im ions take longer to be extracted than EMI–BF4 as shown in Fig. 13, indicating that the droplet shape tends to be deformed by the applied electric field before ions are extracted.

Fig. 14
figure 14

Number of ions extracted every 0.2 sr

Table 1 The ratio of ions extracted within a solid angle

Furthermore, the extracted angle of EMI–BF4 tends to increase when increasing the applied electric field, indicating the active motion of ionic liquid droplets. The resultant high-energy ions are produced; then, the off-axis velocity components also increase. However, the extracted angle of EMI–Im tends to decrease when increasing the applied electric field. As shown in Fig. 13, EMI–Im ions take longer to be extracted than EMI–BF4, and therefore, extracted ions have enough time to face the same direction of the applied electric field before extraction while the ionic liquid droplet has been affected by the electric field. Although most ions are extracted within the solid angle of 0.4 sr (20.6), such ions can also obtain large extracted angles over 20, which affects the ion beam trajectory and leads to collisions against the extractor electrode.

The influence of electric field strength on the off-axis velocity component can also be illustrated in the polar plots shown in Fig. 15. For the cation cases in Fig. 15(b)(d), the average angle decreases when the electric field strength increases from 1.3 V nm−1 to 1.5 V nm−1 and then increases at 1.7 V nm−1. Past the critical electric field strength of 1.5 V nm−1, the nanodroplet begins to fracture and breakup into multiple, satellite droplets, thus creating a subsequent increase in the velocity angle histogram. This phenomenon was also observed by Prince et al. Overall, Fig. 15 indicates that ion extraction dynamics under the explored electric field strengths is more directional than a Lambertian cosine emission (that would peak at 45 degrees).

Fig. 15
figure 15

Normalized angular distribution of anions, (a) and (c), and cations, (b) and (d), for EMI–Im and EMI–BF4

Figure 16(a)–(c) show the distributions of the extracted angle and the kinetic energy of cations for both EMI–BF4 and EMI–Im. The average and standard deviation of the kinetic energy are summarized in Table 2. Here, the standard deviation of EMI–Im anion is not shown because only three anions were extracted during the time examined. The averaged kinetic energy of the extracted cation is in the range 0.77–1.20 eV for EMI–BF4 and 0.67–0.91 eV for EMI–Im, with a slight increase as the electric field increases. Since the surface tension of the EMI–BF4 and EMI–Im is 50.3 mN m−1 and 33.2 mN m−1, respectively, and electrospray phenomena occur when electrostatic pressure is higher than the internal pressure by the surface tension, the difference in kinetic energy is derived from the difference in surface tension. Regarding the anions, EMI–BF4 indicates a similar tendency compared with its cations, showing a slight increase in kinetic energy when increasing the applied electric field. Because the anions of EMI–Im are hard to extract as shown in Fig. 13(d)–(f), they need much more time for the extraction after the electric field is applied. This long-time electric field application causes an increase in the internal energy of EMI–Im droplet and the resultant high kinetic energy of extracted anions of EMI–Im, which is comparable to that of EMI–BF4 anions. It should be noted again that the droplet was charged due to ion extraction, and then it started to move in the opposite direction of the electric field. It is desirable to fix the droplet or set the wall and prevent the droplet from moving; this work will be performed in the future.

Fig. 16
figure 16

Distributions of the kinetic energy versus extracted angle

Table 2 Average and standard deviation of kinetic energy for extracted ions

Conclusion

We have conducted molecular dynamics simulations of the ionic liquid electrospray thruster and investigated extracted energies and angles from an approximately 4–5 nm droplet under the uniform electric field condition of 1.3–1.7 V nm−1. In this study, we employed EMI–BF4 and EMI–Im as the propellant. When we counted the number of ions extracted every 5 ps, we found that EMI–BF4 was easier to extract than EMI–Im for both anions and cations. This tendency is in reasonable agreement with experimental results, where EMI–BF4 has a higher electrical conductivity, leading to a higher current than EMI–Im. The kinetic energy at ion extraction seems to increase slightly with the applied electric field, and EMI–Im with lower surface tension leads to lower initial kinetic energy at the onset of cation extraction than EMI–BF4. Although most ions were extracted at a small solid angle with respect to the x-direction, which is the same direction of the applied electric field, such ions could also have a relatively large extracted angle because of the active motion in the droplet imparted by the electric field, which probably results in collisions against the extractor electrode.

The primary conclusions can be further summarized as follows:

  1. 1.

    The primary species emitted from the nanodroplet surface are monomers. While this result is in agreement with atomistic simulations that use a similar force field [44], the reason for discrepancy between mass spectrometry and RPA measurements requires further investigation.

  2. 2.

    Energies of extracted ions were on the order of 1–2 eV and matched theoretical expectations.

  3. 3.

    At lower electric field strengths, the differences in EMI–Im and EMI–BF4 emission behavior can be attributed to the attractive forces between the cation-anion pairs. Namely, EMI–Im exhibited slower initial emission, likely due to the greater hydrogen bond strength between its cations and anions. Moreover, the present study showed higher extracted current profiles for EMI–BF4 than EMI–Im.

  4. 4.

    Electric field strengths past the critical value of 1.5 V nm−1 resulted in an increase in the overall angular velocity distribution of cations for both EMI–Im and EMI–BF4, indicative of nanodroplet fracturing and breakup.