Controlling the charge of dust particles in a plasma afterglow by timed switching of an electrode voltage

A method is demonstrated for controlling the charge of a dust particle in a plasma afterglow, allowing a wider range of outcomes than an earlier method. As in the earlier method, the dust particles are located near an electrode that has a DC voltage during the afterglow. Here, that DC voltage is switched to a positive value at a specified delay time, instead of maintaining a constant negative voltage as in the earlier method. Adjusting the timing of this switching allows one to control the residual charge gradually over a wide range that includes both negative and positive values of charge. For comparison, only positive residual charges were attained in the earlier method. We were able to adjust the residual charge from about −2000 e to +10 000 e, for our experimental parameters (8.35 µm particles, 8 mTorr argon pressure, and a DC voltage that was switched from −150 V to +125 V within the first two milliseconds of the afterglow). The plasma conditions near the dust particles changed from ion-rich to electron-rich, when the electrode was switched from cathodic to anodic. Making this change at a specified time, as the electrons and ions decay in the afterglow, provides this control capability. These results also give insight into the time development of a dust particle’s charge in the afterglow, on a sub-millisecond time scale.


Introduction
In a dusty plasma, small solid particles are immersed in a low-temperature plasma [1][2][3][4][5][6][7]. In a laboratory plasma, dust particles generally have a negative charge while the plasma * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. operated under steady conditions . That negative charge allows micron-size dust particles to be levitated just above a lower electrode, in the presence of a downward DC electric field. This levitation can persist as long as the plasma power remains on.
When the plasma power is turned off, electrons and ions do not immediately recombine, but instead gradually diminish in number, in what is called a temporal afterglow [34][35][36][37][38][39][40]. The ions and electrons are transported to the chamber walls, during the afterglow, and the timing of this transport has been a topic of renewed interest because of dust particle charging [8,[11][12][13]. A dust particle will retain a charge when the plasma power is turned off, but that charge may be quite different from what it was during steady operation of the plasma. We will show in this paper that the charge develops over the course of about one or two milliseconds in a typical laboratory plasma chamber. At later times, when electron and ion densities are too small for the dust charge to change meaningfully, the charge is said to be frozen. This frozen charge has a value called the dust residual charge, Q res , which remains indefinitely until the dust particle impacts on a surface.
Chaubey et al reported an earlier method of controlling the residual charge of dust particles, after the power is turned off for a capacitively coupled radio-frequency (RF) plasma [68]. It was found in [68] that the dust residual charge reversed during the afterglow, attaining a large positive value [68]. This reversal was attributed to the collection of ions, in the absence of electrons. This non-neutral plasma condition occurred near the lower electrode because it had a DC negative voltage. By varying that negative voltage, it was found that the positive residual charge was controllable. Having a method of control would be useful in the semiconductor industry, where particulate contaminants can have a serious impact on manufacturing yield [65].
The setup for this earlier method relied on a negative DC voltage that remained essentially constant throughout the afterglow [68]. Later, we improved that setup by adding a transistor switch to allow controlling the lower electrode's DC voltage. In particular, this improvement allows switching the electrode's voltage to a positive value, at a desired time in the afterglow [69]. In that experiment, we switched the electrode's voltage after a delay of 2 ms, when the dust particle's positive residual charge was already fully developed and frozen. We did that in order to apply an upward electric force to slow the fall of the dust, without affecting its charge.
In this paper, we report another method, using the same setup as in [69], but switching at earlier times in the afterglow than in [69]. This approach allows us to control the charge, by altering the electron and ion conditions at a time when a dust particle's charge is still developing. At first, the lower electrode has a negative DC voltage, so that electrons were pushed far away from the dust particles, so that only positive ions collect on the dust. Then, with a time resolution of about 30 µs, we switch the lower electrode's voltage to a positive value while there are still numerous ions and electrons in the chamber. The ions are then pushed far away from the electrode, and therefore far away from the dust particles, while electrons are brought near. As a result, dust particles collect electrons instead of ions, starting when the electrode is switched to a positive voltage.
A result of this approach is that we found that we can control the dust residual charge, so that instead of always being positive, it can be made negative. This negative residual charge can be attained by adjusting the timing of switching the lower electrode's DC voltage. By switching it at a sufficiently early time in the afterglow, we will find that the residual charge ends up with a negative value.
Another result is that our measurements allow us to infer how a dust particle's charge develops in time. We do this over the course of the first one or two milliseconds of a temporal afterglow plasma. The result of this inference is a time series for the charge. While that time series is largely qualitative, it is nevertheless a useful development because the charging process in afterglow plasmas is in general not yet well understood, as was pointed out recently by van Huijstee et al [65]. Advances in experiments, like the one reported here, are helpful in defining the conditions that determine the dust charge over time, during an afterglow.

Experiment
The experimental setup shown in figure 1 is similar to previous paper [69]. The procedure and parameters are also similar, except for the timing. Instead of switching the lower electrode voltage at a fixed delay time of 2 ms, as in [69], here we chose earlier delay times, which we varied. Some of the experimental details are presented below; other details can be found in [69].
A plasma was operated steadily until its power was turned off at time t = 0. The power was provided by a 13 MHz RF waveform applied between the lower electrode and the grounded chamber. As a result, the argon gas at 8 mTorr was partially ionized while the RF power was on. A 50 nF coupling capacitor was located between the matching network and the lower electrode, so that during plasma operation the lower electrode developed a DC self bias, which was −150 V. This negative DC voltage did not cease when the RF power plasma was turned off, but was sustained by the coupling capacitor. The plasma glow, which was due to the impact of energetic electrons when the RF power was on, became undetectable within a few microseconds, as the population of energetic electrons vanished. However, colder electrons and ions remained in this afterglow for a much longer time, as we will estimate in this paper.
As in the experiments of [68][69][70], we used 8.69 µm diameter melamine formaldehyde microspheres. During plasma operation, these particles were levitated in a single layer, about 14 mm above the lower electrode. A vertical sheet of laser light illuminated the microparticles, and they were imaged using a 12-bit camera operated at 1000 frames per second. The video recording began at the time that the RF power was turned off. The imaging resolution was 0.037 mm per pixel.
As a manipulation of the afterglow, the voltage on the lower electrode was controlled. At first, due to the coupling capacitor, the electrode was cathodic, maintaining nearly the same negative voltage of −150 V that it had while the RF power was on. Then, at a time t delay governed by a delay generator, a transistor switch was closed, causing the lower electrode's DC voltage to become anodic, rising to a potential V bias = +125 V.
The time resolution for our experiment was determined by a finite switching time, for the polarity of the lower electrode. This switching time can be seen in figure 2(b), which is the voltage waveform measured on the lower electrode. Although the transistor itself closed very rapidly, on a time scale of 40 ns, it took much longer for the electrode's voltage to change, due to the overall capacitance including that of the coupling capacitor. A DC power supply sourced a current that gradually changed the charge stored in the capacitance, until the Side-view sketch of the setup, which is the same as in [69]. During plasma operation, a layer of dust grains was levitated near the lower electrode, which had a radio-frequency voltage as well as a DC voltage. A side-view camera imaged a vertical cross-section of the volume (illuminated by a vertical sheet of 671 nm laser light, not shown here), to capture the motion of dust particles. During the afterglow, there was an upward electric field, which was due to our application of a positive DC bias V bias to the lower electrode. At a time controlled by the delay generator, the transistor switch in this circuit was engaged so that the DC power supply could then overwhelm the charge stored in the coupling capacitor C coupl and bring the lower electrode to a positive voltage. electrode reached its final bias of V bias , determined by the DC power supply. In figure 2(b) we see that the lower electrode polarity became positive after a time t switch = 30 µs, which includes a 7 µs lag time. Another measure of the finite time scale for the electrode voltage to change is the 40 µs rise-time, measured from the 10% level to the 90% level, as indicated in figure 2 Unlike the experiment reported in [69], where t delay was always 2 ms, here we adjusted this delay. Performing 49 experimental runs, each with a different delay, yielded data that we will interpret in section 5 to describe the time development of a dust particle's charge in an afterglow. Such a description could not be obtained from our previous experiment [69] because it had only a single delay time, which was after the dust charge was already frozen.

Obtaining the residual charge
As in the previous experiments [68][69][70], we obtained the dust residual charge Q res from measurements of the acceleration of falling dust particles, in the afterglow. The acceleration a was calculated as explained in [68], starting with the positions of the dust particles obtained from a sequence of side-view images. The acceleration was steady over the entire time that dust particles fell.
The formula for obtaining the residual charge from the acceleration, derived in [68], is At t = 0 the RF voltage was turned off. Next, during the interval 0 < t < t delay , the lower electrode remained cathodic, with a natural negative voltage due to capacitor Ccouple. At t delay , which was 1000 µs for the run shown here, the lower electrode voltage was changed to V bias = +125 V by closing the transistor switch. (b) The same time-series data, magnified to show the transition, beginning when the transistor switch was closed at t delay = 1000 µs. The electrode voltage began to change at t = 1007 µs, i.e. with 7 µs lag after t delay . Overall, the lower electrode's voltage changed polarity after a time interval t switch = 30 µs. Another measure of the finite time scale is the 40 µs risetime, measured from the 10% to 90% levels of the waveform.
In this expression, E falling is the electric field during the time interval when the particles are observed falling, not an earlier time (roughly the first millisecond) when electrons and ions are still present in sufficient quantity to change the particle's charge. Our convention here, for the sign of both a and E falling , is that they are defined to be positive in the downward direction, and g is defined as a positive scalar value. The signs of these quantities are of interest because it is important whether the residual charge Q res is positive or negative. As an example, if E falling is downward, then a positive charge Q res is indicated if a downward acceleration is observed with a magnitude a > g.

Experimental results
Our main result, figure 3, presents values of the final charge Q res , for various experimental runs. These include 49 runs with the lower electrode switched from cathodic to anodic, at a delay time t switch . For comparison, we also include a control run, without any switching, plotted as a triangular data point. During that control run, the electrode remained cathodic at all times in the afterglow, as in the earlier method of [68].

Control run with cathodic lower electrode during entire afterglow
The dust residual charge for the control run was found to be Q res = +10 840 e, where e is the elementary charge. This value has an uncertainty of ±2% due to particle-to-particle variation in the observed acceleration. The scatter in the data points in figure 3 is roughly 15 %, which may indicate some runto-run variations. For this run, the dust particles were in the vicinity of a cathodic electrode at all times in the afterglow. As explained in [68], under these conditions the value of this charge is determined by the energy of ions as they drift through the gas in the presence of a DC electric field. That electric field was sustained by the nearly steady voltage of −150 V on the lower electrode, while the rest of the chamber walls were grounded.

Runs with switching of the lower electrode voltage, after a time delay
Results for the 49 runs with manipulation are shown as circular data points in figure 3. All of these 49 runs had the same plasma conditions and the same levitation height for the dust particles. The only difference between runs was the switching time t switch , when the lower electrode voltage changed from cathodic to anodic.
In examining figure 3, one must keep in mind that this is not a time-series graph showing how a particular parameter varied during the afterglow. Instead, it is a presentation of different final outcomes, i.e. different residual charges, each for a different experimental run. The usefulness of this graph is that it shows how the residual charge depends on a control parameter, t switch . The physical reason that we expect t switch to matter is that a dust particle located near the lower electrode will be exposed mainly to ions for t < t switch , or mainly to electrons for t > t switch , when the lower electrode is cathodic or anodic, respectively. The earlier that we switch the electrode polarity, the less time the dust particle can collect positive ions before switching to collecting electrons.
Several distinct time intervals are noticeable in figure 3. For runs with t switch < 500 µs, the residual charge was negative, indicating that electrons were still present in significant numbers at the location of the dust grains at those early times. When the electrode was switched in that time interval, which  (1) with an input of the experimentally measured particle acceleration. The horizontal axis of this graph is labeled two ways: on the top is the delay time t delay , which is selected by the experimenter, while on the bottom is the switch time t switch which occurred later, when the measured electrode voltage became positive, i.e. anodic. For comparison, Qres for a control run, with the electrode voltage always cathodic, is shown as a triangle. The time intervals N, X, P1, and P2 are our phenomenological labels for distinctive portions of the graph, which are interpreted physically in section 5. is labeled N in figure 3, the residual charge 1 was typically about −2000 e. Conversely, for runs that were switched later, at t switch < 650 µs, the residual charge was positive, indicating that electrons were less numerous than ions so that the grains could charge positively; this time interval is labeled P in figure 3. There is also a short transition, for 500 < t < 650 µs, labeled X. We will further discuss the role of electrons and ions during these time intervals in section 5.

Discussion
In this section, we interpret figure 3, taking advantage of the sub-millisecond time resolution that it provides. This time resolution was made possible by our use of switching the electrode voltage, at a specified time in the afterglow, which we did not vary in our previous experiment [69].

Finding that the dust residual charge in an afterglow can be controlled
Among the observations that we can make, by examining figure 3, is that it is possible to change the final outcome of the dust charge by switching the electrode voltage at a particular time. The earlier that the switching is done, the less positive the residual charge of the dust particle. For the shortest delay times, it is even possible to make the dust residual charge negative, which was not an outcome that we previously observed [69].

Explaining the control of the dust residual charge
Explaining the physical processes behind this method of control requires an understanding of the electron and ion conditions near the lower electrode. We will describe these conditions for three situations: plasma on, afterglow with a cathodic lower electrode, and afterglow with an anodic lower electrode.

Conditions with the plasma on.
When the capacitively coupled RF power is applied, the electrons and ions behave very differently, in the vicinity of the lower electrode. The voltage waveform on the lower electrode is an oscillation superimposed on a large negative DC voltage. The latter voltage, commonly called a self bias, is possible because of the coupling capacitor C coupl . Electrons and ions respond differently, when the RF frequency is of order 13 MHz. The electrons move rapidly enough to respond to the radio frequency, so that during the most positive portion of the RF cycle, electrons penetrate all the way to the lower electrode. During that portion of the RF cycle, dust particles near the lower electrode are exposed to electrons and collect them. At other portions of the RF cycle, electrons are kept farther from the lower electrode. Ions on the other hand cannot respond at 13 MHz, and respond instead to just the time-averaged sheath above the lower electrode. Thus, the ions stream toward the lower electrode throughout the RF cycle. For the dust particles, the combination of collecting speedy electrons during a portion of the RF cycle, and collecting much slower ions during the entire RF cycle, works out to cause the dust particle to have a negative charge.

Conditions in the afterglow.
In the afterglow, the situation is less complicated than when the strong RF oscillation of the lower electrode's voltage was present. The lower electrode has a nearly steady voltage, which greatly simplifies the behavior of the electrons and ions in its vicinity. We will consider the cathodic and anodic cases separately.
Near a cathodic electrode in the afterglow, there will be a cathode sheath, as sketched in figure 4(a). Within that cathodic sheath, there will be mainly ions, and no electrons. Dust particles located near the lower electrode will thus be exposed mainly to ions. This ion exposure is the reason that the dust charge can reverse from the negative value it had when the plasma was on, and become positive in the afterglow, as explained in [68]. The value of that positive charge is determined by the energy of ions, as they stream in the presence of the DC electric field of the cathodic sheath. The streaming velocity of ions is mobility limited, because they undergo collisions with neutral atoms as they transit the sheath (supplementary material). (a) when the lower electrode has a DC cathodic voltage, dust particles are exposed mainly to ions streaming downward so that positive charging of the dust occurs, but in (b) when the lower electrode has a DC anodic voltage, dust particles are exposed mainly to electrons and negative charging occurs. In our experiment, a transition between these two conditions started at t switch , when the lower electrode voltage reversed from cathodic to anodic.
Near an anodic electrode in the afterglow, the electron and ion conditions will be entirely different. As sketched in figure 4(b), electrons will be present near the anodic electrode, but positive ions will be absent. This situation is reversed from that of a cathodic sheath. Dust particles near this anodic electrode will collect mainly electrons.

Temporal development of a dust particle's charge, in the afterglow
We now discuss how the residual charge can be controlled, using switching of the lower electrode's DC voltage, to be negative instead of positive. We observed a negative residual charge for the first 21 data points in figure 3, for an early switching of the lower electrode's voltage, t switch ≲ 505 µs. For a later switching time, i.e. beyond the first 21 data points, the residual charge was positive. In other words, by applying a positive voltage to the lower electrode at a sufficiently early time, the dust particle's residual charge can be controlled to be negative.
To grasp how this controlling of the charge occurs, it would be valuable to obtain a time-series graph of the dust particle's charge in the afterglow, Q(t). However, figure 3 does not show Q(t); instead, it shows final outcomes Q res for various values of the control parameter t switch . We actually have no direct measure of Q(t), but we can at least infer a description of Q(t). We make that inference by relying on a combination of three inputs: first figure 3, second the experimental result of Staps et al [40] for the survival of electrons, and third what is known in general about the finite charging time for a dust particle. We elaborate on these three inputs, next.
First figure 3, allow us to identify four distinctive time intervals. Within the first interval, labeled N, switching the electrode's voltage led to a negative residual charge, and not positive as in the control run. This negative value indicates one of two possibilities: either Q(t) never reversed in the afterglow to become positive, or it did reverse to become positive but later underwent a second reversal to become negative again.
We will be unable to distinguish these two possibilities, based on our data, but in general either could happen. The second time interval, labeled X, has a steep curve for Q res , with a rather sharp transition between a negative versus positive Q res . The third time interval, P1, has a positive value that changes only gradually as the switching time is further increased. Finally, for a switching time in the time interval P2, the positive value of Q res nearly reaches the asymptotic value that was indicated by the control run.
Second, regarding how long electrons survive in the chamber, we will rely on the experimental observations of Staps et al [40]. They detected electrons using microwave cavity resonance, in an afterglow. They operated an argon plasma with capacitively coupled RF power, and then suddenly switched the power off, as in our control run. In the afterglow, they found that electrons survived at a density exceeding their minimum detectable level, roughly 10 5 cm −3 , at a time as late as several milliseconds. That time, for electrons in their chamber to survive at a detectable level, was found to diminish with gas pressure. Our argon pressure of 8 mTorr was lower than in their experiment, leading us to extrapolate their data. As explained in (supplementary material), this extrapolation yielded 1.5 ms, as our estimate of the survival time for electrons to remain in our chamber, at the minimum detectable density of Staps et al [40]. We will use that value, 1.5 ms, as the second of our three inputs in our inference, to describe the general shape of Q(t), for various runs.
Third, we will use what is known about the charging time of a dust particle, and this requires some discussion and calculation. The time development of a particle's charge is, in general, modeled by analyzing the currents collected on the particle [71]. In a steady-state neutral plasma, the electron and ion currents will balance when the particle has charged to a value Q steady , which depends on the electron and ion parameters. This well-known steady state was described for example by Spitzer [72] and Whipple [73]. However, when the plasma conditions are not steady, the charge can change gradually, with an e-folding time called the charging time, t c . A common illustration of this temporal change in the charge is to assume that a dust particle somehow initially had a zero charge, and then at t ′ = 0 became suddenly exposed to a plasma with steady conditions; in this scenario the charge would evolve as The charging time t c that appears in equation (2) can be obtained by analyzing the currents [71]. The resulting value of t c is for a dust particle of radius r immersed in a non-drifting neutral plasma with electron temperature T e . The coefficient K τ has values that are tabulated for hydrogen and argon in [71]. We note in equation (3) that t c varies inversely with the electron and ion density n. That density dependence is important in the afterglow because there is a gradual diminishment of the density of electrons and ions as they escape the plasma volume, eventually reaching a density of zero.
When the plasma was on, the charging time in our experiment is estimated to be t c = 0.3 µs. To obtain this value, we used equation (3), with inputs of a density 1.8 × 10 9 cm −3 as reported in [68], along with typical values T e = 1 eV and T e /T i = 20 for an argon plasma. This result of t c = 0.3 µs is for the plasma when it was powered, but that value will actually not be applicable in the afterglow.
In the afterglow, the charge will change on a slower time scale than was possible when the plasma was powered. This charging time in the afterglow will gradually become slower and slower, as the densities of electrons and ions diminish. For the case of dust particles located within a DC cathodic sheath, speedy electrons will be absent, and the charging time will be solely due to the slower ions. For our experimental conditions, we estimated the charging time will be of order 45 µs in the very early afterglow, as explained in (supplementary material). Later in the afterglow, the charging time will become gradually longer than this estimate of 45 µs.
Having now introduced the three inputs to our inference, we present a sketch in figure 5 of the time series of Q(t), for various experimental runs at different switching times. This sketch is intended to be only qualitative; the only values that are known precisely are the switch time and the residual charge Q res at large t. We will next discuss each of the six curves of figure 5.
The curve in figure 5(a) is for the control run, which is essentially identical to the experiment in [68]. The curve starts at t = 0 with a large negative charge. (That initial value was not measured in the experiment, but we know that the initial charge was of order −10 4 based on a phonon-spectrum measurement in a previous experiment in the same chamber with the same particle diameter [27]). After the RF is turned off at t = 0, the lower electrode will sustain its negative voltage due to the coupling capacitor, so that there will be a cathodic DC sheath present above that electrode, as sketched in figure 4(a). For the electrons, this DC cathodic sheath is quite different from the conditions during plasma operation; electrons will no longer have any opportunity to move near the lower electrode, as they did before t = 0 when the RF waveform was superimposed on the negative DC voltage of that electrode. Thus, for the control run, at t > 0 we expect dust particles located just inside the sheath edge to no longer collect electrons, but instead to collect only ions for the most part. The dust particle's charge will respond to this sudden change in conditions at t = 0 with an exponential time dependence developing somewhat as in equation (2). The e-folding time constant for that response was estimated above to be about 45 µs. Ultimately, the dust particle will attain its final value, Q res , which was measured in the experiment, and presented in figure 3. Combining these concepts, we arrive at the curve sketched in figure 5(a). This curve is qualitative, except for the final value Q res which was measured. Moreover, for clarity we have not attempted to draw the curve to scale, because it would otherwise be difficult to see the rather quick charging time of about 45 µs, when the RF power is first turned off. The reversal of the charge, as marked in figure 5(a), is a particularly notable feature in the time series Q(t). The timing of this first reversal cannot be detected by our method, but we expect that it occurs within tens The final value for each charge of Q(t), at large t, is the residual charge Qres. The shape of each six curves is qualitative, reflecting our inference, based on the residual charge data in figure 3 along with what is known about charging times and the survival of electrons, as described in the text. Each curve represents a particular time interval for the switching time, as marked in figure 3. For the switching-time interval N, we present two scenarios, (b) and (c), that can lead to a negative residual charge. In (b) the reversal of the dust charge is suppressed, whereas in (c) there is a double reversal. In (d)-(f) the switching occurred later so that the remaining electron density was insufficient to cause a second reversal, leading to a positive residual charge as in the control run (a). of microseconds, based on the 45 µs estimate of the e-folding time.
The curve in figure 5(b) is for runs when the electrode voltage was switched at a particularly early time, so that the charge remained negative at all times. In other words, the charge reversal was suppressed. After the switching time, the curve Q(t) could have remained constant if electrons were absent, but it seems more likely that a particle would collect significant electrons after such an early switching time, based on the electron-survival time estimated above to be 1.5 ms, which is much longer than the ≊ 500 µs duration of interval N in figure 3.
The curve in figure 5(c) is a scenario that should be considered along with that in figure 5(b). Both are intended to describe the time interval N in figure 3. The difference is that Q(t) reversed twice in figure 5(c), versus no reversal at all in figure 5(b). These two scenarios both result in final charges Q res < 0, and as we mentioned earlier, we are unable to determine which of these scenarios occurred, for a given run, based on the data available to us.
The curves in figures 5(d) and (e) both end with a positive residual charge Q res , instead of one that is negative. In figure 5(d), the residual charge has kept a small positive value, corresponding to time interval X in figure 3, while figure 5(e) is for interval P1, where the residual charge became a larger positive value.
The curve in figure 5(f) is intended to describe the dust charge time series corresponding to the latest interval P2 of figure 3. The positive residual charge has almost reached its asymptotic value, which would be +10 800 e as measured in the control run.

Summary
A new means of controlling the residual charge of dust particles in a temporal afterglow was demonstrated in an experiment, with a DC voltage that was switched in polarity to a positive value, at a specified time in the afterglow. Differently from a previous experiment [69], the timing of this switching was varied, for different experimental runs. We found that adjusting the switch time allowed us to control the outcome of a dust particle's residual charge, as shown in figure 3. The residual charge could be made to be negative, instead of positive as in the case of a control run where the dust particle attained a positive residual charge. To attain a negative value, the experimenter needs to apply the positive voltage at a sufficiently early time, which was t switch ≲ 505 µs for our argon gas pressure and chamber dimensions.
As an additional result, we obtained a description of the time development Q(t) of a dust particle, during the entire duration of the afterglow. This description, sketched qualitatively in figure 5, was made possible by the time resolution of our results. That time resolution was tens of microseconds or better, for the switching of the electrode, which is much faster than the millisecond time scale for the movement of dust particles.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).