A dusty plasma, also known as a complex plasma, contains small solid particles as well as electrons, ions, and gas [1–4]. The solid particles usually collect more electrons than ions, so that they gain a negative electrical charge. When the particles are microns in size, the charge can be thousands of elementary charges [1–4]. Due to the large charge, the Coulomb interactions between two micron-size solid particles can be so strong that the interparticle potential energy exceeds the particle kinetic energy. In that situation, as a charged component of the plasma, the solid particles are said to be strongly coupled [1–12], even when the electrons and ions remain weakly coupled.

When they are strongly coupled, charged particles can arrange themselves in an organized structure. When their potential energy is sufficiently large, this structure can be crystalline, with particles arranged in rows with a nearly equal spacing. Such dust crystals are common in laboratory experiments with dusty plasmas, especially when the solid particles are all equally charged, or nearly so. To assure that charges are nearly equal, the particles that are used are often precision microspheres, made of silica or a polymer, melamine formaldehyde. Such microspheres are available commercially, with a size dispersion as small as a few percent, in terms of their diameter.

Experiments with crystalline structure in dusty plasmas began nearly 30 years ago [13–16], and since then there have been numerous publications, including [5, 9, 10, 17–40] to mention just a few. These experiments were generally done with the plasma in a powered state, with a high voltage applied to electrodes to sustain the ionization of the gas. Operating with steady power allows the experimenter to levitate the microspheres, using an upward electrical force due to a negative potential on a lower electrode. When the power is switched off, this levitation abruptly ceases, and the microspheres fall, so that they land on the lower electrode within milliseconds. Until recently, the situation when the power had been switched off was usually considered by experimenters to be just the end of an experiment, and not the experiment itself. Now, however, the response of microspheres to afterglow-plasma conditions, immediately after the plasma power is switched off, is gaining more attention [41–66].

In the first one or 2 ms of an afterglow plasma, the electrons and ions leave the plasma volume, and their charges are absorbed on the chamber walls [49, 59]. The electrons leave somewhat more rapidly, so that it is possible for the microsphere to collect an excess of ions. With this excess collection of ions, the microsphere’s charge can reverse polarity, becoming positive and remaining so during the rest of the afterglow [41, 57, 62–64, 67]. With this positive charge, the microspheres experience a vertical electric force, due to the potential on the lower electrode. If the lower electrode retains a negative potential at all times, the newly positive microspheres are pushed downward, so that they fall with a larger acceleration than due to gravity alone, as reported recently [41].

In this paper, we report an experiment to investigate what happens to the crystalline structure of a collection of microspheres, as they fall toward the lower electrode. Before performing this experiment, we expected that we would observe a strong horizontal repulsion among the microspheres, as they fell, so that the crystalline structure would be lost. In fact, however, we found that the crystalline structure was preserved as the cloud of microspheres fell, with no significant disorder developing until the microspheres landed on the lower electrode. After they landed, at a velocity of 712 mm/s, the microspheres bounced, and only then was the structure lost. These results for a falling dust crystal were made possible by imaging the microspheres not only from the side, as in the previous experiment [41], but also from the top so that the crystalline structure could be viewed before and after switching off the power that sustained the plasma.

An experiment was performed in the afterglow of a dusty plasma, using instrumentation based on that of Ref. [41]. During the steady application of radio-frequency power, a single layer of microspheres was levitated above a horizontal electrode. These microspheres self-organized into a dust crystal. Later, the radio-frequency power was abruptly switched off. Both before and after this event we imaged the microspheres using not only a side-view camera as in Ref. [41], but also a top-view camera to allow us to characterize the structure.

The chamber, sketched in Figure 1, was powered by a 13 MHz capacitively coupled radio-frequency waveform, with 336 V_pp amplitude. The horizontal lower electrode, which was 16.25 cm diameter, was powered, while the remainder of the chamber was grounded. A dc self-bias of −171 V developed naturally on the lower electrode, while the plasma was powered. Argon at a steady pressure of 8 mTorr was used, with minimal gas flow.

The microspheres used were melamine-formaldehyde (MF) polymer. The manufacturer’s [68] specifications were 8.69 μm diameter and 1.51 g/cm³ density. After turning on the plasma, we introduced about 2,500 microspheres by agitating a dispenser located above the plasma. They settled into a horizontal layer, 13.9 mm above the lower electrode. At that time, the microspheres were negative charge estimated as -14,000 e, as explained in Ref. [41] where the conditions were nearly identical to those of the present experiment. Due to this large negative charge, the microspheres were levitated upward by the negative dc bias on the lower electrode. Within seconds of being introduced into the plasma chamber, the cloud of microspheres self-organized for form a 2D crystalline structure, with a lattice constant [8, 69] of typically b = 0.58 mm.

To produce an afterglow plasma, a square-wave modulation was applied to the 13 MHz oscillator, causing it to suddenly switch off at time t = 0. Thereafter, the lower electrode no longer had an oscillating voltage that could accelerate electrons and thereby sustain the ionization, and the background plasma extinguished within 2 ms. Even after the background plasma of electrons and ions was gone, the lower electrode retained a large negative bias, due to the coupling capacitor C _coupl shown in Figure 1.

As explained in detail in Ref. [41], there is an evolution, with different time scales, for the abundance and energy of electrons and ions, after turning off the rf power. Within a first few microseconds, the energetic electrons are cooled by colliding with the neutral gas atoms. And after that, in next few hundreds of microseconds, the slow electrons are lost to the chamber walls. Thereafter, only ions remain in the plasma chamber that too lost to the chamber walls within a time of t < 2 ms.

The microspheres can develop a positive charge during the time interval when only ions are present (after the first few hundreds of microseconds and before t < 2 ms). Having this positive charge, the microspheres were accelerated downward more forcefully than by gravity alone, due to attraction toward the lower electrode, which still had a negative bias. The value of the positive charge was +12,500 e, as determined by measuring the downward acceleration, using the method developed in Ref. [41]. It took 40 ms for the microspheres to fall 13.9 mm and impact on the lower electrode.

Two cameras were used to image the cloud of microspheres. These were both 12-bit Phantom v5.2 cameras. A side-view camera was used as in Ref. [41], operated at 600 frames/sec, allowing us to image the cloud of microspheres as it fell. For this experiment, we also used a top-view camera, to image the 2D structure of the cloud. This camera was mounted on a vertical translation stage, so that we could adjust the height of its image plane.

Laser sheets were used to illuminate the cloud. A vertical sheet of 671 nm was used for the side-view camera, while the horizontal sheet was 532 nm. The horizontal sheet was for the top-view camera, and the optics that produced this sheet included a vertical translation stage, matching the one for the camera, so that the camera remained in focus, with the same magnification, as the horizontal sheet was moved. The lenses of both cameras were fitted with interference filters, so that only the desired wavelength of light was imaged. The imaging resolution was 0.037 mm and 0.034 mm per pixel, for the top and side-view cameras, respectively.

The horizontal laser sheet was at first positioned to illuminate a plane at 13.9 mm above the lower electrode, where the microspheres were levitated during steady plasma operation. After recording an image, we then lowered the laser sheet to 0.9 mm (which is the lowest that was practical for our setup) above the lower electrode, and moved the top-view camera downward by the same displacement. This re-positioning of the optics required about 1 min, and it was performed while the plasma continued to be steadily powered.

We then triggered the modulation, turning off the rf power and causing the plasma to extinguish. As the cloud of microspheres were falling, we captured an image when they were at a height of 0.9 mm.

The procedure described above allowed capturing, in a single experimental run, an image of the microspheres about a minute before they were dropped, and then again as they were falling. We verified that additional runs were consistent. For example, in some of these additional runs, we imaged the microspheres at other heights during the fall.

Beginning at t = 0, when the rf power was extinguished, the side-view camera recorded the motion of the microspheres as they fell to the lower electrode, and then bounced. This motion was characterized using the side-view camera that recorded a total of 250 ms at 600 frames/sec. A movie is provided in the Media, Supplementary Material.

We verified that additional runs were consistent. For example, in some of these additional runs, we imaged the microspheres at other heights during the fall.

During steady plasma operation, with the radio-frequency power on, the microspheres were arranged in a crystalline pattern, seen in Figure 4A. This bit-map image of a dust crystal was recorded by the top-view camera, when it was adjusted to image at a height of 13.9 mm.

Using this bit-map image as an input, we obtained the x-y coordinates of individual microspheres. This image analysis was done using ImageJ software [70], following the prescription of Ref. [71] for the moment method of measurement.

While the plasma was steadily powered, the structure in Figure 4A is mainly hexagonal, which is to say that there is a six-fold symmetry to the arrangement of neighboring microspheres. A few microspheres had five or seven nearest neighbors, as seen in the Voronoi diagram in Figure 4B. A Voronoi diagram is made by first identifying the line segments, called bonds, between nearest neighbors, so that they are arranged as triangles. A cell of the Voronoi diagram is then constructed as a polygon by bisecting the bonds [9, 10, 72]. We calculated Voronoi diagrams using IDL software, with an input of x-y coordinates of the microspheres. The defect fraction was 5.7%, meaning that 5.7% of the microspheres had either five or seven nearest neighbors. These defects, which are colored differently in the Voronoi diagram, were especially concentrated at the outer perimeter of the cloud, while the central region had a lesser abundance of defects.

Later, in the afterglow conditions at t > 0, we recorded another image of the microspheres as they fell. This image, recorded at a height of 0.9 mm, is shown in Figure 5A, which reveals that the microspheres still had the same hexagonal structure, even after the plasma had been switched off and the microspheres had almost completed their fall to the lower electrode.

Along with the bit-map image of the cloud of microspheres in Figure 5A, we also present the corresponding Voronoi diagram in Figure 5B. Once again, there are only a few defects in the crystalline structure of the cloud of microspheres. The defect fraction of 5.2% was almost unchanged as the microspheres fell. We can see that the crystalline structure was largely preserved, as dust crystal fell.

For a better comparison of the microspheres as viewed from the top, in Figures 4A, 5A, the two images are superimposed in Figure 6. From this figure, we can see how the microspheres were displaced horizontally very little, as they fell toward the lower electrode. This result, which is consistent with the observations from the side-view imaging, is due to the brevity of the time of fall, as we explained in Section 3.1.

A close examination of Figure 6 reveals that the diameter of the cloud expanded slightly during the fall. Aside from that expansion, other small differences in the two images are detectable mainly in regions where defects are numerous. Defects are generally known to cause an increase in horizontal motion, even during steady operation of the plasma [10, 13, 16, 37, 73, 74], which is a phenomenon described as dynamical heterogeneity [75, 76].

As an additional indicator of structure, we present a pair correlation function, g(r), in Figure 7. This function, also known as the radial distribution function, represents the probability of finding a particle at a given distance from another [10, 20, 34, 73, 77, 78]. For a perfect 2D crystal at a zero temperature, g(r) would be a series of delta functions as a function of distance. These peaks will broaden and diminish in height as the temperature increases, and as translational order diminishes [37, 73, 78].

To obtain Figure 7, we calculated g(r) using an input of measured coordinates of microspheres. Those coordinates were obtained by analyzing two images, Figure 4A with the plasma power on, and Figure 5A as the microspheres were falling after the plasma was switched off. A standard method was used, counting microspheres within annual rings of radius r, which were centered on the location of a microsphere, and dividing by the area of the ring [37].

The pair correlation function in Figure 7 confirms that the structure was largely unchanged, as the cloud of microspheres fell. If the structure had instead become more disorganized, the peaks in g(r) would have broadened. Instead, the peaks actually became higher and more distinctive, indicating that the translational order of the microspheres did not decrease, but actually increased, as they fell.

Crystalline arrangements of microspheres are commonly observed in dusty plasmas [5, 9, 10, 13–40]. These dust crystals have, until now, usually been observed with the electrical power switched on, to sustain the background plasma of electrons and ions. That background plasma serves the purposes of sustaining a charge on each microsphere, and providing a sheath electric field that levitates and confines the microspheres. When the plasma is switched off, the levitation ends abruptly, and the cloud of microspheres falls.

The experiment reported here answers two questions. First, is the crystalline structure preserved as the microspheres fall? Second, when microspheres land on the lower electrode, do they bounce, or do they stick to the solid surface?

By imaging a 2D cloud of microspheres from both the top and side, we found that the cloud of microspheres sustained its crystalline microscopic structure as it fell, when the radio-frequency power that sustained the plasma was switched off. The transverse displacement of microspheres during the fall was minimal, in this particular experiment, with a downward acceleration of 17.8 m/s². After impact on the lower electrode, we found that the microspheres bounced from its surface, and scattered laterally so that the crystalline arrangement ceased after impact.