Inferring ion cluster lifetimes from energy-resolved mass spectra of an EMI-BF₄ electrospray plume

$BF 4 −$

EMI-BF₄ anion (negative ion)

$E$

Electric field strength, V/m

$EM I +$

EMI-BF₄ cation (positive ion)

$f n i$

Fraction of the total plume current carried by species $n i$

$h$

Planck constant, $J H z − 1$

$k B$

Boltzmann constant, $J K − 1$

$n 0$

A “monomer”—consists of a single $EM I +$ cation

$n 1$

A “dimer” cluster—consists of one $EM I +$ cation and one $[ EMI][ B F 4]$ neutral pair

$n 2$

A “trimer” cluster—consists of one $EM I +$ cation and two $[ EMI][ B F 4]$ neutral pairs

$r c$

Emitter tip radius of curvature, $m$

$T$

Ion cluster temperature, $K$

$t$

Elapsed time since species emission, s

$x$

Axial distance from the emitter tip, m

$Δ G S , n i 0$

Ion cluster binding energy, $eV$

$μ$

Ratio of mass after fragmentation to mass before fragmentation

$ϕ$

Electrical potential, V

$ϕ E m$

Emitter potential, V

$ϕ E x$

Extractor potential, V

$ϕ frag$

Potential at which an ion cluster fragments, V

$ϕ S P$

Stopping potential (kinetic energy per unit charge), V

$τ n i$

Mean lifetime of species $n i$

Ionic liquid ion sources (ILISs) are a type of electrospray ion source that emit charged particles by field-assisted evaporation of ions directly from an ionic liquid surface. ILISs have a variety of applications, including spacecraft propulsion and microfabrication.¹ Despite their simple design, ILISs typically emit plumes containing species with a large range of kinetic energies and mass-to-charge ratios. One mechanism that leads to these broad distributions is the fragmentation of metastable ion clusters in the plume, which can degrade performance and limit ILIS lifetime due to excess mass deposition on the extractor electrode.^2,3 Despite the importance of fragmentation in ILIS plumes, experimental data on fragmentation rates and their sensitivity to factors, such as ionic liquid (IL) chemistry, temperature, and electric field strength, are scarce because measurements are difficult due to the complex energy and mass-to-charge distributions in the plume. While significant progress has been made with theoretical investigations of IL cluster fragmentation, that progress is limited by the lack of experimental data for model validation.

Experimental measurements have been reported for IL ion clusters with lifetimes of the order of milliseconds. For example, Prince et al. used an ion trap and thermalizing gas to measure the fragmentation rate of Bmim-DCA (1-Butyl-3-methylimidazolium dicyanamide) ion clusters at known temperatures.⁴ They found that the mean lifetime of positive BMIM-DCA dimers was approximately $τ=62 ms$ from $T=422 K$ to $T=431 K$⁠. Those results compared reasonably well with the lifetimes estimated from molecular dynamics (MD) simulations, which ranged from $τ=32 ms$ to $τ=19 ms$ over the same temperature range. Another experimental method used to investigate IL cluster lifetimes is reported by de la Mora et al., who used a differential mobility analyzer (DMA) to measure the lifetimes of (1-Ethyl-3-Methylimidazolium Tris(Pentafluoroethyl)Trifluorophosphate) ion clusters ranging from $T=25 ° C$ to $T=95 ° C$⁠.⁵ DMAs determine cluster lifetime by measuring the mobility of ion species moving through a background gas under the influence of a weak electric field, which requires those species to survive long enough to drift significant distances (∼mm). They report mean lifetimes of the order of 1 ms and present a method of data analysis that can extend the DMA measurement range down to ∼1 μs. However, cluster lifetimes for the most popular electrospray propellants, such as EMI-Im and EMI-BF₄, are reported to range from ∼1 ns to ∼1 μs.^2,3,6–8 These lifetimes are too short to be directly measured using methods, such as ion traps and DMAs. Consequently, the ultra-short lifetimes typical of electrospray propellant ion clusters must instead be estimated from the energy and mass-to-charge distributions measured in the electrospray plume.

Most experimental investigations of ion cluster fragmentation for popular IL electrospray propellants have relied on measuring the energy distribution in ILIS plumes.^2,3,6,8 By analyzing characteristic features of those energy distributions, the fragmentation rates of ion clusters can be estimated. For example, Miller and Lozano used a retarding potential analyzer (RPA) to measure the mean lifetimes of ion clusters for four different ionic liquids.⁶ They used a spherical RPA geometry to minimize error due to angular expansion of the plume and measured the plume energy distribution at multiple distances from the ion source. Using the same method as Coles et al.,⁷ they identified a characteristic step in the RPA curve that corresponds to the fragmentation of the smallest ion clusters, known as dimers, in the region between the ion source and the RPA instrument. They modeled those data as an exponential decay and found that the mean lifetime ranged from 1 to 5 μs, with $τ=1.49μ s$ for the positive EMI-BF₄ dimer.⁶ 

The energy distribution measured in ILIS plumes indicates that a significant portion of fragmentation happens within the electric field of the ion source, in the so-called acceleration region. The residence time of ions in the acceleration region is of the order of 2–20 ns.³ Thus, the mean lifetimes in the acceleration region must be significantly shorter than the lifetimes measured by Miller et al. in the “drift” or “field-free” region, where the electric field is negligible. Furthermore, it is fragmentation within the acceleration region that is responsible for degrading ILIS performance and lifetime. Consequently, further investigation of fragmentation in the acceleration region is needed.

Singly charged ion clusters can be described by their solvation state, n, which denotes the number of neutral cation–anion pairs in the cluster. Single ions are called monomers and have a solvation state of $n=0$⁠. Similarly, ion clusters consisting of one ion bound to one neutral pair are called dimers and have a solvation state of $n=1$⁠. In this work, we use notation $n 0$ for monomers, $n 1$ for dimers, and $n 2$ for trimers. The fragmentation of each type of cluster in the acceleration region leaves a distinct signature in the plume energy distribution, which manifests as a sloped region in the RPA curve. However, those signatures overlap one another, making it difficult to distinguish between the fragmentation of different cluster types. In fact, only the dimer signature has a portion that is free from this overlap, making RPA data nearly useless for studying the fragmentation of trimers and other large clusters in the acceleration region. Furthermore, this overlap restricts analysis of dimer fragmentation to a small portion of the acceleration region.

Despite the limitations of using RPA data to study fragmentation in the acceleration region, several studies have used a combination of molecular dynamics and other computational methods to estimate ion cluster lifetimes and temperatures by comparing simulated and experimental RPA data. Although the energy distribution of individual plume species cannot be measured using a conventional RPA, these modeling approaches provide a means to further investigate ion cluster fragmentation. For example, Nuwal et al. used 3D-particle-in-cell (PIC) and molecular dynamics (MD) to implement a multi-scale model of an EMI-BF₄ ILIS plume.⁸ They used MD to estimate dimer fragmentation rates as a function of cluster temperature and electric field strength. For positive dimers at a temperature of 600 K, they found mean lifetimes of 549 ps and 6.76 μs for electric fields of 1.5 and 0.6 V/nm, respectively. At 300 K, they predicted a mean lifetime of 394 ns at 1.5 V/nm, while they predicted no fragmentation at all at 0.6 V/nm. They concluded that dimer fragmentation is determined almost entirely by the temperature of the cluster, with negligible contribution from the electric field at values below 0.6 V/nm. Next, they used a PIC method to simulate RPA data and compared the shape of the simulated RPA curve to experimental data. They found the closest agreement between simulated and measured data for a mean lifetime of 5 ns, which corresponds to a cluster temperature of 1300 K according to their MD results. While this approach did produce a reasonable match between simulated and measured RPA curve shapes, it implicitly assumes a constant lifetime throughout the acceleration and field-free regions, and it does not accurately predict the magnitude of fragmentation in the field-free region. Furthermore, the mean lifetime assumed in the PIC model was more than two orders of magnitude lower than the field-free lifetime measured by Miller et al. (5 ns vs 1.5 μs).^6,8 That is, they could not simultaneously explain the lifetimes in the acceleration and field-free regions under the assumption that all dimers are at a single temperature.

Petro et al. also implemented a multi-scale model of an EMI-BF₄ ILIS plume and compared simulated and measured RPA data.² They used molecular dynamics to estimate the lifetimes of positive dimers and trimers and used an n-body approach to simulate the macroscopic behavior of the plume. Using MD, Petro et al. found that positive dimer and trimer lifetimes are nearly constant below 0.01 V/nm. Comparing simulated and measured RPA data, they found the best agreement for a dimer temperature of 1000 K and a trimer temperature of 1350 K. Their MD results suggest that the cluster mean lifetimes for those temperatures are 12.3 ns for dimers and 1.75 ns for trimers. Again, the mean lifetimes required to reproduce the features of the measured RPA curve that correspond to the acceleration region are not compatible with the measured lifetime in the field-free region. Like Nuwal et al., Petro et al. report a mean lifetime for positive dimers in the field-free region that is more than two orders of magnitude lower than the value measured by Miller et al.

More recently, a similar approach was reported by Schroeder et al., who used molecular dynamics in conjunction with a simplified electric field model derived from previously reported electrohydrodynamics results.³ Like the previous two examples, their MD results suggest that the electric field has minimal effect on cluster lifetime at field strengths below 0.01 V/nm. They assume a single temperature and a mean lifetime for each type of ion clusters compare simulated RPA data to experimental measurements. For positive ion clusters, they found the closest agreement for dimer and trimer temperatures of ∼590 and ∼948 K, respectively. They report that the corresponding field-free mean lifetimes are 14.6 and 3.25 ns, respectively. Like previous MD results, the estimated mean lifetime for positive dimers in the field-free region is two orders of magnitude lower than the measurements reported by Miller et al. In order to match the simulated RPA curves to experimental measurements, Petro et al. and Schroeder et al. assume a mean lifetime of 1.49 μs for dimers that exit the computational domain despite the disagreement between the field-free MD results and experimental measurements.

Earlier work by Coles et al. suggested that ion clusters may not be emitted as a single temperature, as has been assumed in the studies conducted by Petro et al., Schroeder et al., and Nuwal et al.^2,3,8 Coles et al. used RPA measurements and MD simulations to investigate the effects of temperature and electric field strength on the fragmentation of EMI-BF₄ ion clusters.⁷ They performed MD simulations of EMI-BF₄ droplets subjected to an electric field of 1.5 V/nm. Those simulations predicted the emission of ion clusters with a distribution of temperatures. For cation emission, those temperatures ranged from roughly 250 K up to about 1100 K, with a most probable temperature around 450 K. They report a much wider distribution for anions, ranging from $T≈150 K$ to $T≈2250 K$⁠, with the most probable temperature around 550 K. The temperature distributions for both cations and anions appear to roughly follow a Maxwell–Boltzmann distribution. Their conclusion that clusters are emitted with a distribution of temperatures is further supported by their experimental measurements, which suggest that there are significant populations of species that fragment within the acceleration region, in the field-free region, and some that do not fragment prior to reaching their RPA. They propose that the observed phenomenon is best explained by a distribution of cluster temperatures, which allows for the fast fragmentation of high temperature clusters within the acceleration region and the slower fragmentation of moderate and low temperature clusters in the drift region.

One notable experimental study of ion cluster fragmentation in ILIS plumes is provided by Miller, who used time-of-flight measurements to infer the mean lifetimes of dimers in the acceleration region.⁹ They analyzed the time-of-flight signal in between the primary monomer and primary dimer signals. They identify this range of times as species, which were emitted as dimers but fragmented into monomers in the acceleration region. The fragmented dimers have a flight time that is faster than primary (non-fragmented) dimers, but slower than primary monomers. Thus, by analyzing the shape of the time-of-flight curve within this range, the dimer fraction within the acceleration region can be found. Miller paired those measurements with a simple electrostatic model of the electric field within the acceleration region and used numerical methods to simulate the flight of ion clusters from the emission site through the acceleration region. The flight time simulations were used to calculate the dimer fraction as a function of flight time from the dimer fraction as a function of potential in the acceleration region. In this study, we use a similar methodology where species fractions are estimated as a function of potential within the acceleration region and then used to calculate species fractions as a function of time using trajectory simulation results. A key difference, however, between the study conducted by Miller and the one presented here is that this study relies on tandem retarding potential/time-of-flight mass spectrometry (RP/ToF-MS) measurements rather than conventional ToF-MS data. The key benefit provided by using RP and ToF-MS in tandem is the ability to clearly distinguish monomer, dimer, and trimer species at all plume energies. Fragmentation pathways that produce low-energy monomers (e.g., $n 2→ n 0$⁠) will produce a signature in conventional ToF-MS measurements that overlaps the trimer-to-dimer fragment signal because those low-energy monomers can have flight times longer than primary dimers. Thus, analysis of trimer fragmentation using standalone ToF-MS data is more uncertain than with RP/ToF-MS because RP/ToF-MS does not suffer from the same signal convolution problem. Nonetheless, Miller's work uses many of the same methods employed here and is a commendable step toward developing experimental methods to investigate ion cluster fragmentation in the acceleration region.

Though significant progress has been made by using molecular dynamics to supplement RPA measurements, conventional RPA data are fundamentally insufficient to fully investigate the fragmentation of multiple species (⁠ $n 1$⁠, $n 2$⁠, etc.) in the acceleration region. Instead, the energy distributions of each individual species must be known. Lyne et al. recently addressed this problem by using an RPA in tandem with time-of-flight mass spectrometry.¹⁰ They reported separate energy distributions for positive monomers and dimers in an EMI-BF₄ ILIS plume. In addition, they estimated the current fraction of monomers, dimers, and trimers within the acceleration region of the ILIS as a function of electric potential. However, their analysis does not provide a way to estimate the mean lifetime of ion clusters within the acceleration region. Here, we present a model of the electric field in the ILIS acceleration region based on the geometry of the ILIS used by Lyne et al.¹⁰ We then simulate the trajectory of ion clusters through that field to relate electric potential to a physical position and elapsed time. Using those results, we analyze the monomer and dimer energy distributions reported by Lyne et al. to estimate the mean lifetimes of dimers and trimers within the acceleration region of the ion source. Unlike previous work (e.g., Refs. 2, 3, and 8), our analysis does not rely on molecular dynamics to estimate ion cluster lifetimes from conventional RPA data. Instead, for the first time, we infer those lifetimes from the separate species energy distributions provided by RPA in tandem with mass spectrometry. We show that tandem RP/ToF-MS data, when analyzed with the methods presented here, can be used to estimate ion cluster lifetimes down to ∼1 ns without using MD or other computational methods.

This work describes a method of analyzing energy-resolved mass spectra of an EMI-BF₄ electrospray plume and reports the results calculated using that method, such as cluster lifetime and plume composition as a function of distance from the emitter tip. That analysis relies on measurements published by Lyne et al., who reported the energy distributions of individual species in the plume of an ILIS with 576 emitters in parallel.¹⁰ In contrast, several previous fragmentation studies have relied on measurements published by Miller, who reported the energy distribution averaged over all species in the plume of a single electrospray emitter.^2,3,6,11 This section describes the ion source, experimental methods, and experimental data that are analyzed in this study. Differences between the source and the experimental approach used in this study and those used by Miller are further discussed in Sec. 4 of the  Appendix.

Ionic liquid ion sources are a type of electrospray source that produces charged particle beams by using strong electric fields to evaporate ions from the surface of an ionic liquid. When a single ion or ion cluster has evaporated from the liquid surface, it accelerates through a potential drop to reach velocities typically on the order of 10 km/s. The ILIS used by Lyne et al. is an AFET-2 ion source described by Adduci.¹² The emitter substrate is a porous borosilicate glass with a pore size of 1.0–1.6 μm (ROBU Glasfilter-Geraete GmbH, P5 porosity). The emitter substrate has 576 individual emitter tips machined into a square grid with ∼546 μm separating each emitter tip from its nearest neighbor. The geometry of an individual emitter and a single extractor aperture is shown in Fig. 1. The mean emitter height was 247 ± 2 μm and the mean radius of curvature of the emitter tips was 32 ± 2 μm, measured by optical profilometry.¹² For the data reported by Lyne et al., the distance between the emitter tip and the front face of the extractor was approximately 200 μm, the emitter was operated by a ±1800 V square wave with a period of 10 s, and the extractor was grounded.

It is important to note that, although the ILIS emitters were manufactured with relatively high precision (±2 μm), differences between individual emitter geometries and porosities may lead to variations between the plumes produced by each emitter. The measurements analyzed in this work represent the aggregate result of 576 individual emitter plumes, each containing monomers, dimers, and trimers. That is, each emitter produces its own plume, characterized by a unique species population distribution that depends on the individual emitter properties, and the measurements analyzed here represent an average over all of these emitter plumes.

The data set analyzed in this work was collected over a period of roughly 60 s using a signal averaging method described by Lyne et al.¹⁰ During that period, the source was held in positive polarity at an emitter voltage of +1800 V. Emitter current initially peaked at 113 μA and steadily fell to 65 μA by the end of the data acquisition. Thus, the results reported in this work (e.g., mean lifetimes) reflect the time-averaged plume composition and species temperatures. The average source current during the measurement was 74.6 μA, which corresponds to an average current of 129.5 nA per emitter. Section 5 of the  Appendix further discusses the ion source current during the measurement interval.

Figure 2 is a diagram of the instrument used by Lyne et al. This instrument has been described in detail previously;^10,13,14 therefore, only a brief description is provided here. The instrument uses a retarding potential analyzer (RPA) in tandem with a time-of-flight mass spectrometer (ToF-MS). The RPA is used to measure species’ energy distributions, while the mass spectrometer is used to distinguish between different plume species according to their mass-to-charge ratios. Lyne et al. used the tandem RP/ToF-MS instrument to measure separate energy distributions for monomers and dimers in an EMI-BF₄ ILIS plume, which represent the plume conditions at the RPA plane of the instrument, 52 mm from the ion source exit. The ion detector used in their study was a 19 mm copper disk, connected directly to a transimpedance amplifier, which acts as a fast electrometer (1 V/μA, ∼5 MHz bandwidth). The RPA and ToF gate were constructed from five stainless steel plates spaced 2.5 mm apart, each with 15.9 mm apertures covered by a nickel mesh with 88% transparency and a center-to-center wire spacing of 268 μm. A stainless steel tube with an inner diameter of 22.9 mm separates the gate from the ion detector, providing a region free from electric fields that allows ions to drift and their velocities to be measured. The total drift distance between the gate and detector is 130 mm. High voltage is applied to the RPA and gate plates as needed, while the other plates, the drift tube, and the ion detector are kept at ground potential. Lyne et al. used this instrument to measure a composite mass spectrum and found no evidence of droplets or species larger than trimers in the ILIS plume.

Based on the instrument dimensions, ions up to 2.9° off-axis can reach the detector. As described in the Appendix of Ref. 10, the uncertainty in the stopping potential and the mass-to-charge ratio depends on several factors, including the true values of $ϕ S P$ and $m/q$ of the measured species. For example, primary monomers (⁠ $ϕ S P=1800 V$⁠, $m/q=111 amu/ q$⁠) have associated confidence intervals of $1604 V< ϕ S P<1868 V$ and $98.4 amu/ q<m/q$ $<116.4 amu/ q$⁠.¹⁰ 

The energy spread observed in ILIS plumes can be attributed to the fragmentation of ion clusters within the acceleration region of the ion source.¹⁰ Consider an ion cluster with solvation state n that evaporates from the emitter tip and begins to accelerate through the electric field of the ion source. While accelerating, that ion cluster may fragment, typically producing an $n−1$ ion cluster and a neutral cation–anion pair. Figure 3 depicts a dimer $( n 1)$⁠, which fragments into a single EMI⁺ ion $( n 0)$ and an [EMI][BF₄] neutral pair.

The relationship between measured RPA data (i.e., $ϕ s p$⁠) and fragmentation, as illustrated by Eq. (3), is the basis for previous studies of fragmentation in the ILIS acceleration region (e.g., Refs. 2, 3, and 8). However, Eq. (3) also explains why conventional RPA data are insufficient to study fragmentation in plumes containing trimers and larger ion clusters. Consider the example of a positive EMI-BF₄ plume with an emitter voltage of $ϕ E m=1800 V$⁠. The mass ratio for dimers is $μ n 1 → n 0=0.36$ and for trimers $μ n 2 → n 1=0.61$⁠. Thus, the stopping potential of products formed by dimer fragmentation in the acceleration region $0< ϕ frag<1800 V$ will fall in the range $647 V< ϕ s p<1800 V$⁠. Similarly, the range of stopping potentials for trimer products is $1097 V< ϕ s p<1800 V$⁠. Consequently, only fragment products with stopping potentials in the range $647 V< ϕ s p<1097 V$ can be attributed to dimer fragmentation alone. Species with stopping potentials $1097 V< ϕ s p<1800 V$ may be from fragmentation of dimers or trimers. Thus, conventional RPA data only provide a clear measure of dimer fragmentation in a portion of the acceleration region and provides no clear measure of the fragmentation of trimers or larger clusters.

Figure 4 shows the stopping potential distributions reported by Lyne et al., who used an RPA in tandem with time-of-flight mass spectrometry.¹⁰ Those data represent the energy distributions of monomers and dimers in the ILIS plume at the RPA plane of their instrument, 52 mm from the exit of the ion source. They obtained these data by using the RPA to limit the range of stopping potentials allowed within each mass spectrum measurement and then fitting each spectrum within a range of mass-to-charge values corresponding to monomers (Fig. 4, left) and dimers (Fig. 4, right). The amplitude of the each fitted curve is proportional to the total number (or “count”) of that species within the associated energy range. The height of the bars in Fig. 4 shows the amplitudes of those fits after normalizing by the total measured current, where the range of stopping potentials included in each measurement is represented by the “width” of each bar. Lyne et al. estimated the uncertainty in the species fraction by reporting the 95% confidence interval calculated from each curve fit.

In Sec. III, we present the methods used in this work to analyze the stopping potential distributions shown in Fig. 4 to estimate the mean lifetime of dimers and trimers in the ILIS acceleration region, which were not reported by Lyne et al.¹⁰ 

The energy distributions of major species in the ILIS plume (Fig. 4) can be used to estimate the mean lifetimes of ion clusters in the acceleration region of the ion source. The mean lifetime is presumed to be a function of cluster temperature and the magnitude of the accelerating electric field, $[ τ n i ] − 1=f( T n i , E →)$⁠. In this section, we present a model of the electric field in the ion source and trajectory simulations that relate elapsed time to other state variables, such as position, electric potential, and electric field strength.

In order to relate potential $(ϕ)$ to the time since ion cluster emission $(t)$⁠, a model of the electric field in the ion source is needed. Then, ion trajectories can be simulated to estimate the potential at an ion's location based on the time since the emission of that ion, $ϕ(t)$⁠. The model used in this study is based on a numerical solution of the Laplace equation in 3D for a static emitter geometry. The geometry of a single emitter was modeled, with symmetry boundaries applied at the perimeters to simulate other emitters in the array.

Results from the numerical model provide a good estimate of the electric field structure far from the emitter tip. However, the ionic liquid at the emitter tip can deform and sharpen, further concentrating the electric field. An electric field strength of $∼ 10 9 V / m$ is required for field evaporation of ions from the ionic liquid surface.¹⁵ However, the maximum field strength in the numerical electrostatic solution is of the order of 10⁷ V/m. To provide a more accurate model for the electric field close to the tip, a hybrid model was developed that transitions to the classical Taylor solution close to the emitter tip. This hybrid model is not meant to replace sophisticated physics-based models, such as Electrohydrodynamics (EHD), in providing the most accurate simulation of the electric field in the acceleration region. Rather, it is meant to provide a simple extension to the numerical results obtained from a Laplace solution based on the emitter geometry, providing field strengths at the emitter tip that are in-line with the $∼ 10 9 V / m$ field strength required for ion emission. Furthermore, we note that the methodology presented here for inferring ion cluster lifetimes from RP/ToF-MS measurements is agnostic to the choice of an electric field model. That is, the same methodology can be used with other field models, including EHD and other advanced models of electrospray emission.

The numerical solution for the electric field was found by solving the Laplace equation in 3D using a MATLAB's PDE solver. A single emitter was simulated, as shown in Fig. 1, with dimensions closely matching the experimental conditions of Lyne et al.^10,12 The simulated emitter is 250 μm tall, has a half angle of $α= 29 °$⁠, and the tip has a radius of curvature of 30 μm. A gap of 200 μm separates the emitter tip from the upstream plane of the extractor, which is 100 μm thick and has an aperture diameter of 465 μm.

A constant potential boundary condition was applied to the surface of the emitter and the surface surrounding the emitter base, $ϕ= ϕ E m$⁠. Constant potential boundary conditions were also applied to the extractor surfaces, $ϕ= ϕ E x=0$⁠. Symmetry conditions were applied along the half-plane of the emitter (bisecting the geometry shown in Fig. 1) and along the surfaces on the outer boundary of the geometry in order to model the effect of the surrounding emitters on the electric field. Finally, a constant potential boundary condition was applied to a plane far from the source, $ϕ=0$⁠, at $x=5000μ m$⁠.

The numerical solution for the electric field provides a good estimate of the electric field at distances far from the emitter tip. However, at distances closer than the emitter tip radius of curvature, $r c$⁠, the numerical solution becomes inaccurate because it neglects the deformation of the liquid meniscus and the resulting concentration of the electric field. The classical Taylor solution gives an analytical expression for the electric field as a function of position for an infinitely sharp cone, $r c→0$⁠.^11,16 Here, we develop an electric field model that uses the numerical solution in the far-field and a scaled Taylor solution in the near-field.

The constant k equation (6) is found by setting the potential at the emission site to the emitter potential, $ϕ( x 0)= ϕ E m$⁠. That is, they were re-normalized so that the total potential drop between the point of ion emission $( x 0)$ and ground $( ϕ ∞)$ equals the emitter potential: $ϕ( x 0)− ϕ ∞= ϕ E m$⁠. Note that both potential and electric field are scaled by the constant k. The start point for emitted ions $( x 0)$ was estimated by finding the distance at which the scaled Taylor model [Eq. (6a)] predicts a field strength of 10⁹ V/m. Figure 5 shows the calculated electric field and potential after normalization.

Note that the scaling and stitching of these numerical and analytical models into the hybrid model described in this section are only meant to be a simple, reasonably accurate way to find $ϕ(t)$⁠. In scaling the analytical solution, the physical consistency of the model is not maintained. That is, the model is no longer rigorously accurate, and it only provides an estimate of the field strength enhancement provided by the deformation of the liquid meniscus.

The electric field model used in this work is a relatively simple approximation compared to some others presented in the literature. Here, the field close to the emission site is modeled using Taylor's analytical solution for an infinitely sharp liquid meniscus, which concentrates the field to sufficiently high values to evaporate ions from the liquid meniscus $( E ≈ 10 9 V / m)$⁠. More sophisticated models of electrospray emission typically use electrohydrodynamics simulations to solve for the shape of the liquid meniscus, which accounts for physical phenomena, such as joule heating of the meniscus and fluid flow rate.^2,3,17 For example, Schroeder et al. used EHD to model a Laplacian electric field for the experimental conditions reported by Miller and then simulated the trajectory of ion clusters through that field.^3,6,11 Coupling those trajectory results with the cluster lifetimes predicted by MD simulations, they produced simulated RPA curves and compared them to the RPA curves measured by Miller. The approach of modeling a Laplacian electric field and simulating ion cluster trajectories through it is also used in this study, though the electric field is approximated here by the simpler hybrid field model described in Sec. III A 2. Figure 6 shows the trajectory results obtained in this study (solid lines) and those reported by Schroeder et al. (dashed lines) in terms of the normalized potential. Note that in addition to the difference in modeling approaches, the emitter geometries are different between these two studies. The point at which the field model transfers from the near-field model to the far-field model can be seen by the sharp change in the slope at $ϕ/ ϕ E m=0.53$⁠. Figure 7 shows the corresponding electric field for this work and reported by Schroeder et al. For both dimers and trimers, the electric fields predicted by the two field models match within a factor of 2 over the majority of the flight trajectories.

The estimated species fractions within the ILIS acceleration region were previously reported by Lyne et al. as a function of potential.¹⁰ However, they did not report those fractions as functions of distance from the emitter tip. Thus, their analysis of the plume composition within the ion source is incomplete and cannot be used to estimate cluster lifetimes. Here, we extend their work by using the electric field model and trajectory results described in Sec. III to express the species fractions as a function of physical position. Our analysis implicitly makes the same assumptions as the original method used by Lyne et al., namely, (1) the plume is assumed to only contain monomers, dimers, and trimers; (2) ion clusters can only undergo $n→n−1$ fragmentation; and (3) fragment products cannot undergo subsequent fragmentations. The first assumption is well supported by time-of-flight data of the ILIS plume, included in the  Appendix of this work. The second assumption relies on previous findings, which generally show that ion clusters based on the EMI+ cation are most likely to fragment by shedding a single neutral pair.⁴ The third assumption is supported by the methodology used in previous studies (e.g., Ref. 8) and is further supported by the tendency for thermal fragmentation tends to reduce the total internal energy of the molecules in the cluster.¹⁸ We should note, however, that some previous studies have found that fragmentation may increase the internal energy of large clusters under certain conditions.⁵ 

Figure 8 shows the species current fractions in the ILIS plume as a function of distance from the emitter tip, estimated by combining the electric field and trajectory models presented here with the experimental data previously published by Lyne et al.¹⁰ The hollow markers at $x=52 mm$ represent the species fractions at the RPA plane of the tandem RP/ToF-MS instrument. Lyne et al. calculated species fractions within the ion source by starting with the species fractions measured in the field-free region (at the RPA plane), then accounting for ion clusters that fragmented at the lowest range of energies $( 0 < ϕ frag < 105 V)$⁠, determined by the data in Fig. 4. They proceeded in this manner to estimate the species fractions throughout the acceleration region, ending with species that fragmented at the highest potentials, closest to the emitter tip. Uncertainty in the species fraction within the ion source was estimated using the error bars shown in Fig. 4, and uncertainty was compounded for each subsequent calculation. Thus, uncertainty in each species fraction increases monotonically from the RPA plane to the emitter tip (right to left in Fig. 8). The trimer fraction at the RPA plane was below the detection limit of the RP/ToF-MS instrument (<1% of the total current fraction). Consequently, the calculations shown in Fig. 8 assume zero trimer fraction at the RPA plane with zero associated uncertainty. In reality, the trimer fraction and associated uncertainty at this point is nonzero, but we have no rigorous means of estimating those quantities and have assumed that they are negligible for this analysis.

The position for each data point in Fig. 8 was calculated from the potential, $ϕ frag$⁠, that Lyne et al. estimated from their RPA measurements. Thus, uncertainty in position was derived from their reported uncertainty in fragmentation potential, which they estimated through a detailed uncertainty analysis.¹⁰ The sources of uncertainty they considered were (1) off-axis entry angles into the instrument, (2) uncertainty in the voltage applied to the RPA grid, (3) potential variation at the RPA plane due to the size of the mesh used for the RPA grid and the geometry of the RPA, and (4) the stopping potential “bin width” or a change in the applied RPA voltage between subsequent mass spectra. The uncertainty in position shown in Fig. 8 was calculated directly from the uncertainty in potential, without accounting for any additional uncertainty associated with the electric field model used in this work.

In Sec. IV A, the energy distributions of ILIS plume species were used in combination with electric field and ion trajectory models to estimate the current fractions of major species in the plume. The trajectory results were used to transform the x axis from potential, which was previously published by Lyne et al., to a physical position. Similarly, those trajectory results can be used to relate position to time-since-emission for each ion species. This section describes how those time-series data can be used to estimate the mean lifetimes of ion clusters within the ion source. Note that the cluster lifetimes estimated with this method, and all other results presented in this article, were calculated from the species’ energy distributions shown in Fig. 4 and knowledge of the source geometry and operating conditions.

The x axis of Fig. 8 can be transformed from position to time-since-emission using the trajectory results for each primary ion cluster, $t p , n i=f(x)$⁠. From those data, the quantities $Δ f p , n i$⁠, $f ¯ p , n i$⁠, and $Δt$ can be directly computed, as shown in Fig. 9. Note that all trimers are assumed to be primary species $( f p , n 2 = f n 2)$ because the fragmentation of larger clusters is not considered in this analysis. Mean species fractions, $f ¯ p , n i$⁠, were calculated using linear interpolation. Uncertainty in time-since-emission was calculated by transforming the uncertainty in position (shown in Fig. 8) to an equivalent uncertainty in time using trajectory simulation results. No additional uncertainty was added to account for uncertainty in the flight time of ion clusters obtained from trajectory simulations. Note that all data points in Fig. 9 have error bars showing the uncertainty in species fraction. However, some of those error bars are eclipsed by the data marker. Refer to Table IV in the  Appendix for the same data in a tabular form (including uncertainties).

Theoretical studies have shown that ion cluster lifetime is decreased by the presence of an external electric field.^2,3,8 In this section, we consider the possibility that electric field strength is the primary cause of decreased cluster lifetime inside the acceleration region. Using the electric field model and trajectory results, the average electric field strength was calculated for each lifetime data point. The mean electric field was computed using a time-weighted average, and the uncertainty in the electric field was computed from the uncertainty in position shown in Fig. 8. Figure 10 shows the mean lifetimes vs electric field strength computed in this work, along with previously published results based on experimental (Exp.) or molecular dynamics (MD) studies.

The results from this work and from Miller et al. were calculated from experimental measurements, while the results of Nuwal, Petro, and Schroeder were estimated from molecular dynamics simulations optimized using experimental data.^2,3,8,9 Miller's data include both the field-free measurement of a dimer lifetime (triangular data marker) as well as the lifetimes they reported based on conventional time-of-flight measurements. The methodology used to obtain those estimates was substantially similar to the methods used here. Like this study, they also used a simple Laplacian electric field model and simulated ion trajectories through the acceleration region and then used those results to relate their experimentally derived estimates of a dimer fraction as a function of potential into a dimer fraction as a function of time [i.e., $f n 2(ϕ)→ f n 2(t)$]. Miller's data, shown in the top left of Fig. 10, agree with the trends found in this work relating electric field strength to mean lifetime. Unlike previous studies, the analysis presented in this work relies on experimental data that provide separate energy distributions for each plume species. This prevents the trimer-to-dimer $( n 2 → n 1)$ fragmentation signal from being convoluted with the low-energy monomer signal resulting from alternative fragmentation pathways (e.g., $n 2→ n 0$⁠). Thus, RP/ToF-MS can be used to reliably calculate the lifetimes of larger clusters, such as trimers.

The mean lifetimes calculated here show a strong inverse correlation with electric field strength, which is well supported by molecular dynamics results. Additionally, we find that dimer lifetimes exceed trimer lifetimes for all field strengths, in agreement with prior theoretical findings. However, the MD studies suggest that only very strong electric fields are capable of significantly affecting cluster lifetime. Petro et al. (dashed–dotted lines) and Schroeder et al. (dashed lines) found no significant effect on lifetime for fields below 10⁷ V/m. Similarly, Nuwal et al. found that field strengths below 6 × 10⁸ V/m do not significantly decrease dimer lifetime. While these studies each use state-of-the-art computational methods, none of them is able to explain the field-free lifetime of dimers reported by Miller and Lozano.⁶ Using the data at 10⁶ V/m to approximate field-free conditions, the results published by Petro et al. and Schroeder et al. imply field-free dimer lifetimes of 12.3 and 14.7 ns, respectively. These values are both more than two orders of magnitude lower than the value of $1.49μ s$ measured by Miller and Lozano.⁶ This discrepancy is discussed further in Sec. V A.

Another approach to understanding how mean cluster lifetime changes as the ions travel through the acceleration region and into the field-free region is to consider how the mean lifetimes change as a function of time since emission. This approach allows our results to be easily compared to simulation results, regardless of the fragmentation model used in those simulations. For example, it may be necessary to assume a distribution of temperatures for each cluster type to reproduce the observed fragmentation in the acceleration region and in the field-free region. The results of that simulation could be easily compared to the results presented here if mean lifetime is plotted as a function of time since emission, rather than electric field strength. Furthermore, if thermal fragmentation is the primary mechanism that explains the discrepancy between lifetimes in the acceleration region and the field-free region, then presenting mean lifetime as a function of time since emission more accurately captures the underlying physics.

Figure 11 shows the primary cluster lifetimes as a function of time since emission. Figures 10 and 11 show the same estimated lifetime data, but the x axis in Fig. 11 is given as time rather than electric field strength. The mean lifetime generally increases over time for both dimers and trimers. This trend is a necessary consequence of short lifetimes in the acceleration region and longer lifetimes in the field-free region reported by Miller et al. and others.^2,3,6,8 The field-free dimer lifetime reported by Miller et al. is shown in the top right corner of the plot, at $τ=1.49μ s$⁠.

The tendency for mean lifetime to increase with time can be explained by two possible mechanisms, which are not mutually exclusive. First, the ion clusters may be emitted with a distribution of temperatures, rather than at a single temperature. For example, Coles et al. used MD to simulate an EMI-BF₄ droplet in a strong electric field and found that clusters were emitted at a range of temperatures spanning hundreds of kelvin.⁷ The second possible explanation is that the electric field in the acceleration region, which decreases over cluster flight time, is initially strong enough to drastically decrease the mean cluster lifetime. As the clusters travel away from the emitter tip, the electric field becomes weaker, and the lifetime increases. In Sec. V, we consider the latter possibility and fit our estimated lifetime data to a fragmentation model that allows the cluster temperature and binding energy to be calculated. We then compare those values to published results from molecular dynamics simulations.

We propose that there are two plausible mechanisms to explain why theoretical methods have, thus far, been unable to simultaneously explain short lifetimes (∼1–10 ns) in the acceleration region and ∼1 μs lifetimes in the field-free region. The first possibility, as noted previously by several authors, is that ion clusters are emitted with a wide range of temperatures, rather than at a single temperature for each species.^3,7 Generally, numerical simulations have assumed a single temperature for each cluster type.^2,3,8 A second possibility is that molecular dynamics simulations have not yet been able to accurately capture the fragmentation process, perhaps due to the structural complexity of the molecules involved. Here, we introduce a model that relates mean lifetime to the ion cluster temperature and its binding energy, which is the energetic barrier to fragmentation.

Figure 12 shows the estimated cluster lifetimes and the results of fitting those data to Eq. (17). Those fits allow the binding energy and temperature to be estimated for primary dimers and trimers. Table I reports those values calculated using the image potential and dipole potential models. Using the image potential model, the binding energy and temperature for dimers are estimated as $Δ G S , n 1 0=0.57±0.24 eV$ and $T n 1=437±202 K$⁠, respectively. For trimer clusters, the binding energy and cluster temperature estimated from the fitted data are $Δ G S , n 2 0=0.49±0.22 eV$ and $T n 2=416±209 K$⁠. If the temperature and binding energies are calculated using the dipole potential model instead of the image potential model, similar results are obtained. When performing the dipole potential calculations, a separation distance and an angle of $d=1$ nm and $θ= 45 °$ were assumed, and x was chosen to minimize $G(E)$⁠.¹⁹ Using the dipole model and fitting the data to Eq. (17), the binding energy and temperature for dimers are estimated as $Δ G S , n 1 0=0.49±0.22 eV$ and $T n 1=394±214 K$⁠, respectively. For trimers, the binding energy and the cluster temperature estimated from the fitted data are $Δ G S , n 2 0=0.40±0.18 eV$ and $T n 2=365±182 K$⁠. The uncertainty reported for binding energy and temperature reported here represents the 95% confidence interval $( 2 σ)$ on the fit coefficients for $Δ G S 0$ and T. The field-free lifetime reported by Miller and Lozano is included in the dimer lifetime plot (hollow circle) and the associated fit.⁶ 

Fitting the lifetime vs electric field data, which were estimated from the measured energy distributions for monomers and dimers in the plume, does not make assumptions about the molecular structure or chemistry of the ion clusters. Instead, fragmentation was assumed to follow a kinetic model describing the field-assisted evaporation of charge. Fitting lifetime data to that kinetic model allowed the binding energy and temperature to be estimated without resorting to complex theoretical methods, such as molecular dynamics.

The binding energies calculated by fitting the data from Fig. 12 to the image potential or dipole potential models are considerably lower than binding energies reported elsewhere in the literature. The fact that the calculated binding energies are considerably below those predicted by molecular dynamics suggests that the observed fragmentation cannot be fully explained by the model presented in Eq. (16). Specifically, the assumption that each cluster population can be characterized by a single temperature is not well supported. As with the previously reported computational studies, we find that the lifetimes estimated from experimental data are inconsistent with the assumption of a single cluster temperature.^2,3,8

Studies of other ionic liquids further support the conclusion that the binding energies and temperatures estimated using Eq. (17) are lower than is realistic. Prince et al. used MD, validated by measurements using an ion trap and thermalizing gas, to estimate the binding energy of positive BMIM-DCA dimers as 1.27 eV.⁴ In another study, de la Mora et al. used a differential mobility analyzer to measure lifetimes of positive trimers of EMI-FAP, concluding that the activation energy for those clusters is 0.927 eV.⁵ Both studies controlled for ion cluster temperature and directly measured ion cluster lifetime, enabling binding energies to be calculated with relatively low uncertainty. In both cases, they found the IL cluster binding energies to be of the order of ∼1 eV. Later, Miller and Lozano used the published binding energies of EMI-FAP and EMI-Im to estimate the temperature positive dimers in an ILIS plume, which they found to be $T=557 K$ and $T=602 K$⁠, respectively.⁶ Their findings support the conclusion that the electrospray emission process tends to increase the internal energy of ion clusters, as proposed by Coles et al. and others.⁷ However, we should note that others have found that fragmentation may increase cluster temperatures under certain conditions.⁵ Taken together, the theoretical and experimental evidence suggest that ILIS plumes cannot be accurately modeled by assuming a single temperature for each type of ion cluster. Instead, future modeling efforts should investigate the possibility that the electrospray emission process produces clusters with a distribution of temperatures.

To date, studies of ion cluster fragmentation in EMI-BF₄ ILIS plumes have relied on measuring the plume energy distribution using retarding potential analyzers (RPAs) or species velocities using time-of-flight and comparing those measurements to simulated data obtained from numerical models. However, RPAs measure the plume energy distribution averaged over all plume species, making it difficult to accurately determine the lifetime of each individual species. This is especially true of species that fragment within the acceleration region of the ion source, which have the greatest effect on the performance and lifetime of the ion source. Furthermore, standalone ToF cannot accurately determine species mass if its energy is not well known. Estimating cluster lifetimes based on ToF measurements is complicated by the wide energy distributions found in ILIS plumes, convoluting the signals of interest. Here, for the first time, we estimate ion cluster lifetimes based on tandem RPA and time-of-flight measurements. Those tandem RP/ToF-MS measurements, previously published by Lyne et al.,¹⁰ provide separate energy distributions for each major species in the plume. In this work, we demonstrate how those species’ energy distributions can be analyzed to estimate ion cluster lifetimes within the acceleration region of the ion source. Using that analysis method, we estimate ion cluster lifetimes down to ∼1 ns, which is considerably shorter than the minimum lifetime that can be measured using differential mobility analyzers (∼1 μs) or ion traps.

We find that the mean lifetimes for positive EMI-BF₄ ion clusters in the acceleration region range from $3.7$ to $124 ns$ for dimers and $1.5$ to $23 ns$ for trimers. For each data point, we report the estimated electric field strength and time since ion cluster emission. Those results are shown in Figs. 10 and 11 and in Table II in the  Appendix. We introduce a simple theoretical model that assumes a single, constant temperature for each species and accounts for the effect of an external electric field. Those data were fit using two different models for the effect of the electric field on the fragmentation rate, the image potential model, and the dipole potential model. Using the IPM to fit the lifetime vs electric field data, we estimate that the binding energy and temperature for positive EMI-BF₄ dimers are $Δ G S , n 1 0=0.57±0.24 eV$ and $T n 1=437±202 K$ and for positive trimers is $Δ G S , n 2 0=0.49±0.22 eV$ and $T n 2=416±209 K$⁠, where the uncertainty bounds represent a 95% confidence interval on the fit coefficients. Similarly, fitting the data using the dipole potential model, we estimate that the binding energy and temperature for positive EMI-BF₄ dimers are $Δ G S , n 1 0=0.49±0.22 eV$ and $T n 1=394±214 K$ and for positive trimers is $Δ G S , n 2 0=0.40±0.18 eV$ and $T n 2=365±182 K$⁠. However, previous experimental and theoretical studies suggest that realistic values for binding energy and temperature are $Δ G S 0≈1 eV$ and $T≥600 K$⁠. The large discrepancy between the binding energies estimated through other means and those found here suggests that the apparent reduction of cluster lifetime may be more closely related to time since emission $(t)$ than it is to an electric field. This behavior would be expected if the clusters had a wide temperature range that resulted in a mixture of prompt and delayed fragmentation (e.g., see Ref. 7). Consequently, we suggest that future modeling efforts more thoroughly investigate the possibility that clusters are emitted with a distribution of internal energies, which may or may not be well represented by a bulk temperature following a Maxwellian energy distribution.

Last, we reiterate that tandem RP/ToF-MS has considerable advantages over conventional RPA or ToF measurements for investigating cluster lifetimes in multi-species plumes. We also note that the data analysis method presented here is independent of the choice of the electric field model. In future work, this method could be used to analyze energy-resolved mass spectra, such as those obtained from RP/ToF measurements, with more accurate electric field models by using EHD to simulate a liquid meniscus shape and accounting for the effects of space charge.

The data and MATLAB code that support the findings of this study are available in the supplementary material.

This work was supported by a NASA Space Technology Research Fellowship (Grant No. 80NSSC19K1165). The authors would like to thank Anthony Adduci for his assistance in setting up and operating the ionic liquid ion source. We would also like to thank Miron Liu for his extensive assistance in building and troubleshooting the RP/ToF-MS instrument.

The authors have no conflicts to disclose.

Christopher T. Lyne: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Funding acquisition (equal); Investigation (lead); Methodology (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Joshua L. Rovey: Conceptualization (supporting); Funding acquisition (equal); Project administration (lead); Supervision (lead); Writing – review & editing (equal).

The data that support the findings of this study are available within the article and its supplementary material.

This appendix presents the numerical results presented in this work in tabular form. Table II summarizes the estimated ion cluster lifetimes. Tables III and IV summarize the estimated monomer, dimer, and trimer species fractions within the acceleration region. Finally, Table V summarizes the monomer and dimer species fractions previously reported by Lyne et al., which were used to calculate all other results in this work.

Table I includes the dimer lifetimes $τ n 1$ and trimer lifetimes $τ n 1$ calculated within the acceleration region of the ion sources, as well as the mean values of fragmentation potential $ϕ frag$⁠, time since emission t, and electric field strength E. Each quantity is reported in terms of its nominal value, as well as the upper and lower uncertainty bounds, UB and LB, respectively. Those uncertainty bounds were calculated using the methods described in the main text of this work. In short, uncertainty bounds on $ϕ frag$ were determined from the uncertainty the stopping potential of fragment products measured at the RPA plane of the RP/ToF-MS instrument. The uncertainty bounds for time and electric field were calculated directly from the uncertainty in $ϕ frag$ using an electric field model and trajectory simulation results. Uncertainty in cluster lifetime $τ$ was estimated as described in Sec. IV B.

Table II presents the species fractions calculated for the acceleration region of the ion source. Those results are the distance from the emission site x along with its upper and lower confidence bounds, calculated from uncertainty in $ϕ frag$⁠. The monomer and dimer species fractions (⁠ $f n 0$ and $f n 1$⁠) and their uncertainties (⁠ $u f n 0$ and $u f n 1$⁠) are also given, along with the species fraction of dimer primaries $( f n 1 , p)$ and its uncertainty $( u f n 1 , p)$⁠. Table III presents those same values for trimer species. Note that the first row of both Tables II and III represent species fractions at the RPA plane of the RP/ToF-MS instrument.

Table IV summarizes the monomer and dimer energy distributions, which were previously reported by Lyne et al., calculated from tandem RP/ToF-MS measurements.¹⁰ These data were analyzed in this work to obtain all numerical results presented here, including estimates of ion cluster lifetimes and species fractions within the acceleration region of the ion source. The energy distributions reported in Table IV represent the plume conditions at the RPA plane of the RP/ToF-MS instrument, located 52 mm from the ion source exit.

Figure 13 shows the mass spectra reported by Lyne et al.¹⁰ and by Miller¹¹ for EMI-BF₄ ion sources. The measurements reported by Lyne et al. and the species’ energy distributions computed from those measurements (reproduced in Fig. 4) were analyzed in this work to obtain all numerical results presented in this article. The vertical dashed lines in Fig. 13 correspond to the known mass-to-charge values of positive monomers (111 amu/q), dimers (309 amu/q), and trimers (507 amu/q). Miller used a carbon xerogel ion source, which produced a plume consisting of 64% monomers and 21% dimers at an emitter current of 462 nA. The remaining 15% of plume current was not attributed to any particular species, though they estimate ∼0% fractions for both trimers and droplets (Table 6.9 in Ref. 11). Based on an SEM image of their emitter included in their work [Figs. 4–8(b) in Ref. 11], their source had a tip radius of curvature of ∼3 μm.

The ToF spectrum reported by Lyne et al., which represents the data analyzed in this work, suggests that the plume composition is 38% monomers, 56% dimers, 5% trimers, and ∼1% larger species. The significant difference in monomer and dimer fractions compared to Miller may result from the larger tip radius used in Lyne et al.'s work (∼30 vs ∼3 μm), as well as the larger pore size (1.0–1.4 vs ∼0.45 μm).^10,11 Schroeder et al. and Petro et al. both used the measurements reported by Miller in their studies, minimizing the difference between their simulated RPA curves and Miller's experimental RPA curve to estimate ion cluster temperatures.^2,3 Another significant difference is that the source studied here is a thruster, consisting of 576 emitters in parallel, whereas Miller used a single-emitter source. In addition, Miller used a large spherical RPA to capture the majority of the emitted current, while the data in this study was measured by Lyne et al. using a flat RPA that sampled the center of the plume, capturing species up to $± 2.9 °$ from the center axis. Although the flat RPA measurements are subject to error due to beam spreading that is not present in the spherical RPA measurements, the small sampled angle of $± 2.9 °$ limits the magnitude of that error to a maximum of 0.3%.¹⁰ Nonetheless, Miller's measurements represent the energy distribution for the full plume, while Lyne et al.'s measurements represent the center of the plume. Any differences in plume composition with angle will also lead to differences between their measurements.

The small fraction of large species suggested by Lyne et al.'s ToF data supports the analysis presented in Sec. IV A, which assumes that the plume is composed exclusively of monomers, dimers, and trimers, with no significant droplet fraction. Lyne et al. also report a mass spectrum calculated from their tandem RP/ToF data, which corrects for the error in calculated mass-to-charge resulting from low-energy species formed by fragmentation in the acceleration region. That RP/ToF mass spectrum represents the plume conditions at the RPA plane of the RP/ToF instrument, approximately 52 mm from the ion source exit. That spectrum suggests that only monomers and dimers still remain in the plume by that point, with no evidence of larger species including trimers or droplets. This finding suggests that the ∼1% current fraction seen in the ToF spectrum above 600 amu/q is likely due to low-energy fragment products, rather than true evidence of droplets in the plume. That is, the mass-to-charge ratio calculated for those low-energy fragments is higher than the true value, creating the appearance of larger species in the ToF spectrum.

The data analyzed in this study were reported by Lyne et al., who measured the energy-resolved mass spectra in the plume of a porous glass electrospray source with 576 emitters operating in parallel.¹⁰ The data analyzed in this study were measured using retarding potential analysis and time-of-flight mass spectrometry in tandem (RP/ToF-MS). Lyne et al. averaged together 320 ToF spectra at each retarding potential in order to improve the signal-to-noise ratio of the measurements. This process took approximately 60 s, during which the ILIS emitter voltage was maintained at +1800 V. Figures 14 and 15 show the current emitted by the ILIS source (sampled at a frequency of 100 Hz) during the period when the RP/ToF-MS data were measured. The emitter voltage was switched from −1800 V to +1800 V at $t=0$⁠. The source current initially peaks at 113.6 μA, then slowly falls as the source is held in positive polarity, reaching 81.6 μA by $t=10 s$ and 65.6 μA by $t=60 s$⁠. The average current per emitter initially peaks at 197 nA, falling to 141.6 nA by $t=10 s$ and 113.9 nA by $t=60 s$. The average source current over the duration of the RP/ToF-MS measurement was 74.6 μA, which corresponds to an average current of 129.5 nA per emitter. The relatively low current per emitter further suggests that the ion source operated in pure ion mode, with minimal droplet emission.