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. 2025 May 24;147(22):18853–18863. doi: 10.1021/jacs.5c02845

Spontaneous Fission of Charged Water Nanodrops: Unveiling the Stochastic Nature of Fission Pathways and Dynamics

Veena S Avadhani 1, Conner C Harper 1, Zachary M Miller 1, Evan R Williams 1,*
PMCID: PMC12147151  PMID: 40411532

Abstract

Fission of highly charged micrometer-sized and larger droplets has been investigated using optical methods, but until recently, direct measurements of spontaneous fission of submicrometer droplets have not been possible. Charge detection mass spectrometry is used to track the mass, charge, and energy per charge of aqueous nanodrops that undergo evaporative water loss while they are trapped for up to 4 s. 154 of the 846 trapped nanodrops (18.2%) with charges ranging from 44 to 158% of the Rayleigh limit underwent fission. Although these spontaneous fission processes are highly heterogeneous, four distinct fission pathways that occur over times ranging from a few ms to 100s of ms with ejection of just a few to hundreds of progeny droplets were identified. One is a “continuous” pathway in which many small progeny droplets with progressively less charge are sequentially emitted over the course of ∼25 to 150 ms. Prompt and sequential prompt pathways in which one or a limited number of progeny droplets carry away a significant fraction of the precursor charge are the most common. “Prefission” events in which emission of just a few charges prior to a larger prompt fission event occur for some nanodrops charged above the Rayleigh limit, and these events appear to have similarities to “foreshocks” that often occur shortly prior to major earthquakes.

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Introduction

Charged water droplets are ubiquitous in the environment, where they are formed through a variety of natural processes, including ocean surf, waterfalls, and thunderstorms. Charged droplets are also electrostatically generated for use in a variety of industrial applications, including paint coatings, chemical analysis, and spacecraft propulsion. Charged droplets can undergo fission when the charge–charge repulsion exceeds the cohesive force associated with the surface energy of the droplet. Lord Rayleigh derived an equation that describes when fission is expected to occur for spherical droplets

zR=8π(ε0γR3)1/2 1

where z R is the maximum or Rayleigh charge limit, R is the droplet radius, γ is the droplet surface tension, and ε0 is the permittivity of the surrounding space. There have been many different experimental approaches aimed at gaining information about fission of micrometer-sized droplets that can be optically imaged. These include experiments with capacitor-type apparatus, Millikan condensers, , electrodynamic balances, − and light scattering techniques. − In all of these investigations, − ,− fission occurred for droplets that are charged between 70–120% of the Rayleigh limit, although the charge was not measured in some experiments, and fission was assumed to occur at the Rayleigh limit. A general outcome of this work is that fission of micrometer-sized droplets consisting of a variety of solvents occurs through a process or processes in which a relatively large loss of charge occurs (∼10 to 40% of the original charge on the droplet) with low or negligible loss of mass (<5%). For water, Beauchamp and co-workers reported that fission occurs near or slightly below the Rayleigh limit and results in a loss of 20–40% of the original droplet charge.

Information about the number of progeny droplets that are formed and what processes initiate fission events is more limited because they are seldomly measured directly. Lord Rayleigh predicted the ejection of a “fine jet” of very small droplets, and optical methods show the formation of Taylor cone-like distortions in large droplets that produce observable progeny droplets. − ,− The role of high external electric fields in many prior experiments is not always clear, but high electrical fields can induce droplet fission. Duft and co-workers showed droplet elongation and reproducible fission of ∼50 μm diameter ethylene glycol droplets where ∼100 progeny droplets with similar size were consistently formed via two fine jets in opposing directions. Field-induced droplet ionization in which charged or neutral droplets are held in a high electric field also induces a prolate elongation, leading to a jet of charged progeny droplets in the direction of the electric field. Submillimeter water droplets trapped in an electrodynamic balance underwent as many as 15 consecutive fission events with each event complete in 10s of microseconds. The charge on these droplets at the time of fission was assumed to correspond to the Rayleigh limit, and a similar fraction of charge was lost with each successive fission event. Photographs show fission by emission of a fine jet of droplets, although the number of progeny droplets was not reported. Fission occurred in the direction of the external applied electric field with a Taylor cone-like extrusion with a half angle of 28°. Various modeling and computational efforts have been undertaken to understand how charged droplet fission occurs. − Fernandez de la Mora proposed two modes of fission that depend on many factors, including the conductivity of the solvent. Fission can occur either by a jet-emitting Taylor cone (commonly observed) ,,, or a Coulomb explosion in which the droplet breaks up into large and equal fragments (for which there are few indirect observations).

In contrast to the micrometer-sized droplets investigated previously, significantly less is known about the spontaneous breakup of charged drops between 10 nm to 1 μm diameter. This size range is important for understanding a variety of processes, including electrospray thrusters for space propulsion − and understanding how charged analyte ions are generated in nanoelectrospray (nanoESI) ionization. , When nanoESI is combined with mass spectrometry, the masses and structures of large molecules and macromolecular complexes can be accurately measured, and this technique is used in thousands of laboratories for environmental chemical analysis, metabolomics and proteomics investigations as well as in many other applications. Molecular dynamics simulations of small ion-containing water nanodrops (<7 nm diameter) indicate that droplet breakup occurs between ∼80 and 100% of the Rayleigh limit through the formation of a thin jet, as has been observed for large droplets. − , Experimental data indicate that the range in the extent of charging of droplets from which ion emission occurs is much broader, occurring both well above and well below the Rayleigh limit for 15 to 30 nm diameter ion-containing aqueous nanodrops. Simulations also indicate surface fluctuations leading to the formation of a single protrusion or jet. ,, An external electric field can induce jet formation on opposite sides of the droplets, and multiple jets were computed to form from droplets that were initially charged well above the Rayleigh limit.

Direct experimental observation of aqueous nanodrop fission has been recently reported. Charge detection mass spectrometry (CDMS) was used to measure the masses, charges, and energy per charge of individual aqueous droplets. During these measurements, a small fraction of these droplets underwent fission. Fission events from seven nanodrops ranging in size from ∼40 to 120 nm in diameter led to the production of various numbers and sizes of progeny droplets that carried away 4–14% of the original droplet charge. The fission process occurred over time scales of <1 ms to 100s of ms. These results demonstrated that spontaneous fission of aqueous nanodroplets can be complex and occur via a number of different pathways and time scales. Here, over 800 individual charged nanodrops were investigated using CDMS to obtain a larger data set from which more robust statistics and categorization of fission pathways and dynamics can be made.

Experimental Section

Charge Detection Mass Spectrometry

Experiments were performed using a custom-built electrostatic ion trap charge detection mass spectrometer that is described in detail elsewhere. Positively charged aqueous nanodrops were generated by nanoESI from deionized water with a resistivity of 18.2 MΩ·cm at 25 °C (Milli-Q gradient ultrapure water purification system; Millipore, Billerica, MA). Borosilicate emitters (1.0 mm outer diameter, 0.78 mm inner diameter, with filament, Part no. BF100–78–10, Sutter Instrument, Novato, CA) pulled to an inner tip diameter of ∼18–20 μm using a P2000/G micropipette puller (Sutter Instrument) were positioned ∼3 mm from the CDMS instrument inlet. A voltage of 2.4–2.7 kV was applied to a platinum wire placed inside the emitter in contact with the water to initiate electrospray.

The resulting charged nanodrops enter the mass spectrometer through a heated inlet capillary (80 °C) and, subsequently, pass through an ion funnel and enter three sequential quadrupole ion guides. The frequencies and amplitudes of RF potentials applied on the funnel and quadrupole are similar to those used previously. The nanodrops that are trapped in the quadrupole ion guides are then pulsed into an acceleration region and subsequently enter an electrostatic cone trap for mass analysis for up to 4 s. The charged aqueous nanodrops induce a current in a cylindrical detector electrode located in the center of the cone trap. This signal is used to determine the charge (z), mass-to-charge ratio (m/z), energy per charge (eV/z), and mass (m) continuously throughout the entire time that a nanodrop is trapped. Short-time Fourier transform (STFT) analysis of time domain data containing individual ion signals was performed using a 25 ms window length, a 5 ms step size, and 5 zerofills except where otherwise indicated. The pressure of the vacuum chamber housing the cone trap was ∼1 × 10–8 Torr. Ion current was kept low to ensure that no more than one or two ions were trapped during any single trapping event and thereby avoid ion–ion interactions that can perturb ion frequencies of motion but are unrelated to droplet fission. Data for 846 nanodrops were acquired from 3882 individual measurements, corresponding to an average of one nanodrop in every 4 or 5 measurements. The instrument, including the internal charge-sensitive preamplifier, was operated at ambient temperature (20–25 °C).

Data Analysis

Automated tracing of the oscillation frequencies of analyte ions was done using in-house software. , A change in frequency of ion motion can be due to loss of energy owing to collisions with background gas, , loss of mass and charge as a result of fission, , and ion–ion interactions. A frequency drop during an ion–ion interaction corresponding to an increase in ion kinetic energy is always accompanied by a corresponding increase in frequency (lower kinetic energy) of the interacting partner as a result of the transfer of energy between the ions. Therefore, the possibility of misidentifying a frequency drop as a fission can be ruled out for the current study. All of the STFT frequency traces were manually inspected for the classification of fission pathways.

The diameters of the nanodrops were determined from the individually measured masses using a spherical model and a water density of 0.9998 g/cm3 (at 0 °C). The fraction of charge relative to the Rayleigh charge limit was determined from the measured charge and diameter values as a function of time. Loss of charge from a fission event was determined from the difference between measured charges prior to and after fission. Because the mass of a nanodrop continuously reduces due to evaporation, the mass loss for each fission event was obtained by averaging the mass over only 50 ms before and after a fission event. For determining the charge loss associated with each fission event, the charge measurement was averaged over as long of a period as possible before and after the fission event, terminating either at the beginning or end of the trapping period or at times when other fission events for the same nanodrop occurred. For fission events where only a few charges are lost, a frequency-based charge determination method that relates change in frequency to change in charge was used in order to obtain an estimate of the charge loss when the signal length was too short to obtain accurate amplitude-based charge loss measurements. An implicit assumption in this method is that there is a negligible mass loss, and uncertainties in the method obfuscate integer charge loss determination.

Results and Discussion

Fission Pathways

Electrospray ionization of pure water was used to produce positively charged nanodrops that were introduced into a charge detection mass spectrometer using soft source conditions. The CDMS instrument is capable of dynamic measurements that make it possible to obtain the mass, charge, and energy per charge (heretofore referred to as energy) of individual nanodrops throughout the entire time that they are stored in the electrostatic trap, where these measurements are made. The relationship between m/z and the fundamental frequency, f of ion motion within the trap, is given by eq

mz=C(E)f2 2

where C(E) is a function that depends on both the trap geometry (fixed) and ion energy (variable), m is the mass, and z is the charge on an individual nanodrop. Changes in the ion mass, charge, or energy can be determined from the changes in the frequencies of ion oscillation and the amplitudes of the signals. Measurements for 846 nanodrops ranging in size from 30 to 105 nm were made for trap times ranging from 0.5 to 4 s. 692 (81.8%) of these nanodrops underwent only neutral water loss with no charge emission. Water loss is indicated by a steady increase in the fundamental frequency of ion motion with time. The rate of water loss is determined by measuring the mass lost by nanodrops over time. An example of a nanodrop trapped for 1 s that underwent only neutral water losses is shown in Figure a. The initial mass and charge of this nanodrop are 173.74 ± 0.66 MDa (82 nm diameter) and 956.5e ± 1.2e, respectively. This nanodrop lost 14.09 ± 0.81 MDa of water over the 1 s trapping period, or ∼8.0% mass loss, corresponding to the loss of ∼783,000 water molecules per second.

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Evolution of the frequency of motion of charged aqueous nanodrops with time inside an electrostatic ion trap (1 s trapping duration) showing characteristic behaviors representative of 846 charged nanodrops. (a) Evaporation with no fission for a 82 nm diameter nanodrop with initial charge of 956.5e ± 1.2e, (b) prompt fission of a 49 nm diameter nanodrop with 660.2e ± 1.5e, (c) prompt fission preceded by small prefission emission events (circled) of a 38 nm diameter nanodrop with 432.2e ± 4.1e, (d) sequential fission comprising of three charge loss events for a 62 nm diameter nanodrop with 689.1e ± 1.6e, (e) continuous fission process for a 59 nm diameter nanodrop with 806.3e ± 2.6e, and (f) a complex fission processes for a 51 nm diameter nanodrop with 700.4e ± 5.8e.

For the remaining 18.2% of the nanodrops, one or more fission events occurred. This extent of fission is consistent with the value of 7.2% in prior experiments, in which shorter trapping times were used. With longer trapping times, more evaporation of water from the nanodrops occurs, leading to a higher probability of fission and a higher average charge relative to the Rayleigh limit for nanodrops that undergo only evaporation. Fission events for nanodrops are easily identified by the accompanying decrease in the frequency of ion motion that occurs as a consequence of charge loss (eq ). Prior results on seven charged nanodrops with diameters between 40 and 120 nm indicated that each nanodrop underwent one or more unique fission processes, with both the time frame and the number of charges lost varying significantly. The same is true for the 154 charged aqueous nanodrops investigated here, but characteristic “pathways” for fission events were identified from the larger number of nanodrops investigated and the higher precision of these measurements compared to the earlier work. Four pathways were identified based on the fission duration and the magnitude and dynamics of charge loss. The simplest pathway is “prompt” fission, where the fission event is characterized by a rapid drop in oscillation frequency. An example of this behavior is shown in Figure b for a 49 nm nanodrop that has an initial mass of 36.94 ± 0.25 MDa and a charge of 660.2e ± 1.5e. The relatively slow, constant frequency increase indicates that this nanodrop continuously lost water throughout the trapping period. However, a large, rapid drop in frequency occurs at 595 ms, indicating a prompt fission event. The decreases in mass and charge due to this fission event were 0.23 ± 0.71 MDa and 59.6e ± 2.5e, respectively.

Another fission pathway is “prefission,” where emission of a small but variable number of charges occurs one or more times prior to a much larger fission event where a significantly greater number of charges are lost. This pathway is illustrated in Figure c for a 38 nm nanodrop (16.88 ± 0.21 MDa; initial charge of 432.2e ± 4.1e). Three prefission events occur between 120 and 300 ms (Figure c, circled data). These are followed by a prompt fission event at ∼387 ms, in which the nanodrop loses 13.0% of its charge and 5.3% of its mass. In these prefission events, only a small number of charges are lost. Based on the magnitude of the frequency jump, these prefission events correspond to the loss of up to only a few charges in contrast to larger losses of ∼56 charges that occurred subsequently or the 60 charges lost in the previous case (Figure b). Similar low charge loss events occasionally occurred after a large fission event. However, there is insufficient data to differentiate these events from “prefission” events for a subsequent fission event for the same nanodrop.

A third common fission pathway is “sequential prompt” fission, where several prompt fissions occur, but less charge is lost in each fission event (average of 3.6% charge loss per event) than is typical for a single prompt fission process where an average of 9.6% of the charge is lost. An example of a sequential fission pathway is illustrated in Figure d for a 62 nm diameter nanodrop (initial mass of 73.66 ± 0.45 MDa; initial charge of 689.1e ± 1.6e). This nanodrop undergoes three consecutive prompt fission events at 190, 305, and 755 ms. These fission events correspond to the loss of 19.5e ± 2.4e, 14.5e ± 2.2e, and 34.1e ± 2.9e, respectively, and an overall mass loss of 3.91 ± 0.67 MDa. For the 37 nanodrops that exhibited this sequential prompt fission pathway, the time between consecutive fission events varied from 0.03 to 1.605 s. The time between sequential fission events does not appear to depend on the size or charge of the nanodrop (Figure S1).

A fourth distinct pathway is the “continuous” fission pathway, where sequential charge loss occurs in many small steps over a longer period of time, ranging from 0.33 to 0.56 s. This pathway is illustrated in Figure e for a 59 nm nanodrop (58.71 ± 0.16 MDa; initial charge of 806.3e ± 2.6e) that was trapped for 1 s. For this ion, fission starts at 255 ms and lasts until 535 ms where the nanodrop stops losing charge but continues to lose mass by neutral water loss. This fission process reduced the charge and the mass of the original nanodrop by 229.3e ± 4.3e (28.5% of the original charge) and 1.56 ± 0.47 MDa, respectively. This continuous fission pathway characteristically shows a more rapid initial loss of charge that decreases as the fission event progresses (Figures e and c).

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Evolution of the second harmonic frequency of motion of charged aqueous nanodrops with time inside an electrostatic ion trap for 500 ms trapping duration (top) and amplitude vs frequency plots (bottom) for (a) a 59 nm diameter nanodrop undergoing prompt fission and (b) the corresponding frequency-amplitude evolution over the time period indicated by the blue dashed lines in panel (a) analyzed using 2 ms STFT segment lengths. Red-yellow data indicates the frequency prior to fission, and blue-purple data indicates frequency after fission. (c) Second harmonic frequency of motion for a 49 nm diameter nanodrop undergoing a continuous fission event, and (d) the corresponding frequency-amplitude evolution over the time period indicated by the blue dashed lines in panel (c) analyzed using 4 ms STFT segment lengths. Red and blue data indicate the frequency immediately prior to and after fission, respectively. The number of peaks in between the initial and final frequencies indicates the minimum number of progeny droplets formed in these fission events.

The fission pathways that are described above and illustrated in Figure b–e have distinct defining characteristics that account for almost all fission events observed. However, there were 6 nanodrops that underwent fission events that were significantly more stochastic. These fission events were categorized as “complex.” One example of an ion in this category is shown in Figure f. This 51 nm nanodrop, with an initial mass of 42.5 ± 0.38 MDa and initial charge of 700.4e ± 5.8e, exhibited neutral water loss and small charge losses similar to prefission events intermixed with larger fission events over varying time scales for essentially the entire 1 s trapping period. In total, this ion lost 12% of its charge and 6.4% of its mass over the course of 11 clearly distinguishable fission events.

Another feature in the signal of some nanodrops is the slight curvature in the frequency at early times (e.g., such as those in Figure a,b), indicating that these nanodrops initially lose water at a slightly faster rate before reaching a steady-state loss of water. The signals for other nanodrops do not have this curvature (e.g., Figure d–f). Nanodrops are continuously introduced into and trapped in the quadrupole region during the preceding ions’ measurement period (1–4 s) in the electrostatic ion trap. Thus, nanodrops introduced to the quadrupole region at different times spend different lengths of time in vacuum prior to mass measurements. A small fraction of nanodrops can even pass right through the quadrupole region without being trapped if their entrance into the mass spectrometer is well-timed with the pulsed introduction of ions into the electrostatic ion trap. Evaporative cooling lowers the temperature of the nanodrops when they are initially introduced into vacuum, but they rapidly reach a steady-state internal energy owing to the absorption of blackbody radiation , as well as collisions in the higher-pressure quadrupole regions. For ions that pass through or are stored for only a short time in the quadrupole prior to injection into the electrostatic ion trap, the cooling process to a steady state of internal energy is incomplete. In the electrostatic ion trap, this incomplete cooling process is manifested by a faster evaporation rate during the beginning of the measurement period, which rapidly becomes more constant as the ions reach a steady-state internal energy distribution. Other nanodrops that are trapped for a longer time reach a steady-state internal energy distribution prior to the mass measurements, and these nanodrops do not show this characteristic initial curvature in their frequencies.

Fission Dynamics

The classification of fission pathways is largely based on the time scale of the fission events and the number of charges that are lost in such events. However, there is considerable heterogeneity within each classification. A more detailed examination of the dynamics of fission events can be performed by reducing the segment length of the short-time Fourier transform that is used to analyze these data. Shorter segment lengths improve the time resolution at the expense of the frequency resolution. Ultimately, this trade-off limits practical time resolution to ∼1 ms. A more detailed look into the fission dynamics of a nanodrop undergoing prompt fission is shown in Figure a,b. Figure a shows the frequency evolution (second harmonic shown) of a 59 nm nanodrop (∼63.87 MDa; ∼940e) that undergoes some prefission between 75 and 95 ms before undergoing prompt fission at 110 ms where ∼80 charges are lost. The evolution of the second harmonic frequency determined from the STFT data using a 2 ms segment length 5 ms prior to, during, and 5 ms after the large fission event is shown in Figure b. The second harmonic frequency is used to improve frequency resolution with these short STFT segment lengths. There is little change to the initial and final frequencies immediately prior to (red-yellow data) and after (blue-purple data) fission (Figure b), and what change does occur is due to water loss during these ∼4 ms. The amplitude of the nanodrop signal after fission is lower because it is proportional to the charge of the ion. There is a single peak at intermediate frequency (green data in Figure b) that occurs at a time that is between that of the initial and final frequencies. Both the width and intensity of the peak indicate that it is an intermediate species that is stable for ∼2 ms. This indicates that at least two progeny droplets were formed by this prompt fission process and that the overall fission event occurred in less than 4 ms.

An example of the dynamics for a nanodrop undergoing a continuous fission event is illustrated with a 49 nm aqueous nanodrop (∼36.81 MDa; ∼703e) that lost 27.6% of the initial charge over the course of ∼360 ms (Figure c). The frequency evolution of the progeny droplet immediately prior to, during, and immediately after fission (30–400 ms) is shown in Figure d (4 ms STFT segment lengths, second harmonic frequency). The initial and final frequencies (Figure d; peaks in red and blue, respectively) are essentially constant on this short time scale. However, a series of 40 distinct peaks occurs between these frequencies during the 360 ms fission event. This indicates that a minimum of 41 progeny droplets are formed in this process. The gradual decrease in peak spacing and decrease in peak amplitude indicate that the extent of charge lost with each sequential progeny droplet that is produced decreases as fission progresses.

A histogram of the duration of both the prompt and continuous fission processes illustrates the difference in how we have defined prompt versus continuous fission (Figure ). Prompt fission occurred over just a few ms, whereas continuous fission occurred for a variable duration ranging from 150 ms to 1550 ms. The most common duration for continuous fission was between 40 and 70 ms. The propensity of nanodrops to undergo these two different fission pathways is similar.

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Distribution of the duration of continuous (blue) and prompt (green) fission pathways.

Fission and Rayleigh Limit Charge

A measure of nanodrop charge density is the ratio of the nanodrop charge, z, to the Rayleigh limit charge, z R, (z/z R). A histogram of z/z R values for 122 representative nanodrops that underwent only evaporative water loss with no fission is shown in Figure a. These nonfissioning droplets have z/z R values that span a range from 0.57 to 1.41 with a median value of 1.15. For the 122 nanodrops that underwent fission, their z/z R values span a range from 0.40 to 1.58 with a median value of 1.20. Thus, the majority of droplets are charged above the Rayleigh limit, even those that do not undergo fission. However, the ones that do undergo fission tend to be more highly charged than those that do not; 76.2% of the droplets that fission had z/z R values above 1.00, i.e., nanodrops charged above the Rayleigh limit.

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Histograms of the number of droplets charged relative to the Rayleigh limit for a spherical droplet (z/z R) for (a) 122 aqueous nanodrops that undergo only evaporation of water and no fission and (b) 122 nanodrops that evaporate water molecules and undergo fission during the 1 s duration that these ions were trapped. These values were determined from the initial masses and charges on the nanodrops.

Charging of aqueous nanodrops above the Rayleigh limit has been reported for pure water and salt-containing nanodrops with masses above ∼10 MDa. For larger micron-sized microdroplets, fission has been reported to occur at, or more typically, below the Rayleigh limit for a wide range of solvents, although some microdroplet fission occurring above the Rayleigh limit was reported for methanol. Charging above the Rayleigh limit for the nanodrops may be due to a number of different factors. A value of the bulk surface tension of water was used in these calculations, but the surface tension in these relatively small nanodrops likely differs. Droplet elongation into a prolate or oblate shape reduces the Coulomb energy compared to that of a sphere and increases the number of charges necessary for fission to occur. , Kinetic effects may also delay droplet fission at low temperatures. Our results show that fission occurs over a much wider range of z/z R than has been previously reported for any solvent.

Interestingly, there is a broad range of z/z R values, even for the same nanodrop that undergoes multiple fission events. This is illustrated for a 59 nm nanodrop (65.7 ± 0.25 MDa; initial charge of 884.4e ± 2.1e) that was trapped for 3.8 s. This ion undergoes sequential prompt fission with four events, where a significant number of charges are lost (Figure a). These fissions occur at 0.565, 1.41, 2.42, and 3.11 s. There are also a number of “prefission” charge loss events where just a few charges are lost. The measured z/z R values for the same nanodrop (Figure a) are shown in Figure b. The nanodrop has an initial z/z R value of ∼1.34. This value increases to ∼1.36 after 565 ms due to water loss, reducing the size and mass of the nanodrop with no loss of charge (Figure a). The first sequential fission process reduces the z/z R to ∼1.26. This value again increases due to solvent loss until the second fission process occurs, which reduces z/z R from ∼1.29 to ∼1.26. The third fission occurs at a z/z R value of ∼1.30 and reduces this value to ∼1.24. The final fission event occurs at ∼1.25 and reduces this value to ∼1.22. Thus, the z/z R value where fission is observed can differ, even for the same nanodrop that undergoes multiple fission events.

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(a) Evolution of the frequency of ion motion for a 59 nm diameter aqueous nanodrop that underwent sequential fission over a time period of 3.8 s. (b) Corresponding charge of the same nanodrop shown in panel (a) relative to the Rayleigh limit charge as a function of time.

A key question remains. Does the charge density influence the fission pathway and dynamics? The frequency of the different fission pathways as a function of z/z R is shown in Figure . The majority of fission events occur above the Rayleigh limit, but all pathways except prefission are observed for nanodrops charged below the Rayleigh limit as well. The higher z/z R threshold for prefission indicates that charge–charge repulsion enhances charge losses by this pathway. Of the 180 fission events that occurred for the 154 nanodrops, 10 had prefission events. For the other pathways, there does not appear to be a relationship between the pathway and z/z R value (χ-square test results in p = 0.094; detailed discussion in Supporting Information (SI)), indicating that the charge density on the droplet is not a major factor in determining which fission pathway occurs. A similar correlation plot of pathways as a function of nanodrop diameter suggests that the size of an aqueous nanodrop is also not a direct indicator of a preferred fission pathway (Figure S3).

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Histograms of the number of aqueous nanodrops as a function of their charge relative to the Rayleigh limit (z/z R) for the different fission processes: sequential (purple), complex (light green), continuous (yellow), prompt (gray), and prefission (blue). The z/z R of a droplet was determined just before a fission event occurred in order to correlate the extent of charging with the type of fission process that each nanodrop undergoes.

Spontaneous Fission is Stochastic

The fraction of the nanodrop charge that is lost in a fission event can differ significantly, even for nanodrops that follow the same general fission pathway. To illustrate the extent of variability in nanodrop fission, the percentage of the original nanodrop charge that is lost as a result of fission by sequential, prompt, and continuous fission pathways is shown in Figure a. Sequential fission had the lowest variability (1–15%) and resulted in the lowest percentage of charge loss. Both prompt and continuous events spanned a range from 1% up to 35% of the nanodrop charge lost, with the average of both distributions at 9.3 and 13.4%, respectively. The percentage of charge lost does not depend on z/z R. (Figure b; a Pearson correlation coefficient of 0.15 indicates that there is not a significant correlation between these two variables; description in the SI). This indicates that the magnitude of the Coulombic repulsion between the precursor and progeny droplets is not a major factor in the number of charges that are lost.

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(a) Distribution of nanodrops as a function of the fraction of charge lost (%z) for sequential (blue), prompt (green), and continuous (yellow) fission processes. (b) Fraction of charge lost as a function of the Rayleigh limit (z/z R) showing no discernible correlation between these two parameters.

A single nanodrop can undergo fission by multiple pathways during the time that it is trapped, and the majority of nanodrops trapped for 4 s exhibited such behavior. For example, a 62 nm nanodrop (∼74 MDa; 717e) underwent two prompt fission events, one at 505 ms and the other at 3295 ms, leading to the loss of 23 and 52 charges, respectively (Figure a,b). The first prompt fission was followed by small prefission-like charge losses at 0.75 s, where only ∼2 charges were lost. This was followed by a period of neutral water loss between 0.75 and 2.16 s with no charge loss before the nanodrop underwent a continuous fission event at 2.16 s that lasted 0.675 s, and 77 charges were lost. A few small prefission events occur prior to the second prompt fission at 3.295 s. The evolution of the measured charge of this nanodrop is shown in Figure b. This observation of very different fission pathways for the exact same droplet indicates that fission events and their different pathways are uncorrelated. This also indicates that droplet deformations that lead to fission by one pathway do not necessarily persist after a fission event occurs. It is more likely that a new deformation process must initiate subsequent emission pathways.

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(a) Frequency and (b) charge evolution for a 62 nm aqueous nanodrop showing both prompt and continuous fission processes occur during the 4.0 s trap period. (c, d) Frequency evolution for two aqueous nanodrops that are the same size (59 nm diameter), charge (∼813 and 803e for (c, d), respectively), and z/z R (1.23) showing that the initial size and charge of an aqueous nanodrop is not a useful indicator of how or when it will undergo fission.

The stochastic nature of spontaneous fission events is further illustrated for two nanodrops that are the same size (∼59 nm), charge (∼813 and 803e), and z/z R (1.23). These ions lost nearly the same proportion of their initial charge (32% in Figure c and 29% in Figure d) over 4 s, but their charge losses occurred by a completely different process. The nanodrop in Figure c underwent a combination of continuous and prompt fissions over 4 s, with the longest period of neutral water loss lasting just 0.8 s. In striking contrast, the nanodrop shown in Figure d exhibited a few prefission charge losses, but the majority of the charge loss occurred during a continuous fission at 1.74 s that was preceded and followed by long periods of only neutral water loss. This example illustrates that the initial charge state and size of the nanodrop are not useful predictors of when fission will occur, the pathway by which fission occurs, or the overall dynamics of the fission process. A multinomial logistic regression was used to determine the probability that a nanodrop fissions by a given pathway at different z/z R values, and these data are provided in Figure S4.

Droplet Deformation Drives Spontaneous Fission of Nanodrops

In field-induced fission imaging experiments of large droplets by Leisner and co-workers, fission was nearly identical for each droplet. Fission of the ethylene glycol droplets occurred by the continuous ejection of ∼100 small, highly charged progeny. In contrast, the spontaneous fission of aqueous nanodrops is highly stochastic. Nanodrops that have the same charge and mass can undergo radically different fission processes over the 4 s that they are trapped. The percentage of charge lost over 4 s does not correlate with z/z R. Consecutive fission events for a single nanodrop can occur over a range of z/z R values and by different fission pathways that also occur over a wide range of z/z R values. The seemingly random nature of these fission events provides compelling evidence that these events are not initiated by the redistribution of charges on the droplet surface due to the relatively weak electric fields in our electrostatic ion trap but must be driven by statistical fluctuations. The strongest electric field that ions experience within the trap is 247 V/cm or 2.4 mV over the length of a 100 nm diameter droplet (SI). By comparison, the electric field at the surface of aqueous nanodrops is far greater (calculated to be ∼9 MV/cm) as is the field required to induce fission for much larger droplets that have much lower surface tension (over 20,000 V/cm for 225 μm diameter methanol droplets). With the exception of prefission events, the extent of Coulombic repulsion does not appear to be a significant factor in the timing or dynamics of the fission process.

Prefission events occurred over the narrowest range of average charge density and at the highest average value compared to that of the other pathways, although this difference was not statistically significant (Figure S5). Molecular dynamics simulations on nanodrops consisting of a few thousand water molecules indicate that the deformation process leading to the emission of a singly charged ion is limited to a relatively small area of an otherwise spherical nanodrop. The same is likely true for the much larger nanodrops investigated here. In contrast, more significant distortions are likely necessary for the emission of much larger progeny droplets that carry away a much greater charge. Even the continuous emission pathway in which a large number of charges are ultimately carried away by many progeny nanodrops likely involves a large distortion. We envision this distortion to be similar to that reported for field-induced fission where significant elongation of the droplet occurs, and a jet with a shape similar to that of a Taylor cone on an electrospray emitter is produced. ,, The prefission pathway may be initiated by smaller, much more frequent localized distortions on the nanodrop. The events are analogous to a relief valve-like mechanism where Coulombic repulsion is reduced slightly prior to a more significant droplet distortion and much greater loss of charge and mass. Our results indicate the initial deformation process is almost certainly stochastic and heterogeneous and that the droplet distortion, whether minor local distortions or more significant “global” distortions, leads to the wide range of fission processes and dynamics that are observed.

Rayleigh Fission and Earthquakes

There are some intriguing parallels between spontaneous fission of charged aqueous droplets and earthquakes, despite the fact that these two processes occur on vastly different time scales. Both involve a driving force that increases with time and ultimately initiates an event in which the force is reduced. For charged droplets, this driving force is the Coulombic repulsion between charges that increases due to solvent loss with no loss of charge. For earthquakes, the force is generated by the tectonic plate movement that occurs over time. The resistive force is the surface tension for droplets and the stability of the soils with respect to shear forces for plate movement. As is the case for spontaneous fission, earthquakes appear to be largely stochastic in nature, making it challenging to predict their exact timing, magnitude, and duration. Ultimately, both droplet fission and earthquakes lead to a release of energy. A common factor is a pathway for both processes that releases small amounts of energy that either precede or succeed larger energy release events. The prefission events that occur for charged aqueous nanodrops are analogous to foreshocks, which are small earthquakes that precede larger earthquakes. Both processes preferentially occur when the destabilizing force is the greatest. Foreshocks are not a common feature to all earthquakes nor are pre-emission events a common feature to all droplet fission. Recent results using more sensitive detectors indicate that foreshocks may occur more frequently than previously thought. Similarly, improved analysis methods may lead to detecting the emission of just a few charges closer in time to when a larger fission process occurs in droplets. An analogous relationship between postfission events and aftershocks may also be drawn. While the connection between spontaneous charged droplet fission and earthquakes is speculative at this time, it is our hope that a greater understanding of factors that influence charged droplet fission pathways and dynamics may lead to an improved understanding of earthquake events.

Conclusions

Charged aqueous droplets naturally occur and their chemistry and physics play an important role in environmental and atmospheric chemistry. Unusual chemical reactions and reaction rate acceleration have been reported in charged droplets, − but little is known about the physical behavior of much smaller aqueous droplets that are in the low or submicrometer size range. A detailed analysis of the fission of 846 aqueous nanodrops with diameters between 30 and 105 nm and charged between 44 and 158% of the Rayleigh limit reveals that asymmetric fission in which progeny droplets carry away a significant fraction of the initial precursor droplet charge but little mass occurs as is the case for much larger droplets. However, unlike field-induced fission events of larger droplets, spontaneous fission is stochastic and appears likely to be related to the detailed nature of spontaneous surface deformations. Despite the unique fissioning behavior of each nanodrop, four distinct fission pathways and their corresponding time scales were identified. Emission of just a few charges or hundreds of charges can occur with time scales as short as a few milliseconds to well over 150 ms. These data show that droplet fission is significantly more complex than previously known.

The nanodrops in this study are cold due to evaporative cooling that is balanced by blackbody absorption , and collisional heating. The form of the droplets in this study is not unambiguously known. However, the nature of these fission events, especially the continuous fission pathway and fission occurring near the Rayleigh limit, indicates that the surface of these nanodrops is liquid-like. Much colder nanodrops can have crystalline ice-like cores, and the cores of the fissioning droplets may also be ice-like. Master equation modeling of the water evaporation process for these large nanodrops is ongoing, and this modeling will provide more detailed information about the effective temperature of these nanodrops and the nature of the droplet surface.

Supplementary Material

ja5c02845_si_001.pdf (1.5MB, pdf)
ja5c02845_si_002.xlsx (372KB, xlsx)

Acknowledgments

This work was supported by the National Science Foundation Division of Chemistry (grant number CHE-2203907), the National Institutes of Health (grant number 5R01GM139338) for the development and construction of the CDMS instrument and data analysis methods used in this work, the Hearts to Humanity Eternal Graduate Research Grant (V.S.A.), the American Chemical Society through an Analytical Chemistry Graduate Research Fellowship sponsored by Eli Lilly and Company (Z.M.M.), and the Arnold and Mabel Beckman Foundation Postdoctoral Fellowship in Chemical Instrumentation (C.C.H.). The authors thank Professors Yoni Toker and Lutz Schweikhard for helpful discussions.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/jacs.5c02845.

  • SIMION simulation description, statistical tests’ description and plots, additional examples of fissioning droplets, and additional plots showing measured properties of nanodrops as a function of time (PDF)

  • Processed data depicting the evolution of frequency, charge, and mass of charged aqueous nanodrops (XLSX)

The authors declare no competing financial interest.

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Associated Data

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Supplementary Materials

ja5c02845_si_001.pdf (1.5MB, pdf)
ja5c02845_si_002.xlsx (372KB, xlsx)

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