Shocks propagate in a 2D dusty plasma with less attenuation than that due to gas friction alone
aa r X i v : . [ phy s i c s . p l a s m - ph ] J u l Shocks propagate in a 2D dusty plasma with less attenuation than that dueto gas friction alone
Anton Kananovich and J. Goree Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa 52242 (Dated: 10 July 2020)
In a dusty plasma, an impulsively generated shock, i.e. , blast wave, was observed to decay less than wouldbe expected due to gas friction alone. In the experiment, a single layer of microparticles was levitated in aradio-frequency glow-discharge plasma. In this layer, the microparticles were self-organized as a 2D solid-likestrongly coupled plasma, which was perturbed by the piston-like mechanical movement of a wire. To excitea blast wave, the wire’s motion was abruptly stopped, so that the input of mechanical energy ceased at aknown time. It was seen that, as it propagated across the layer, the blast wave’s amplitude persisted withlittle decay. This result extends similar findings, in previous experiments with 3D microparticle clouds, to thecase of 2D clouds. In our cloud, out-of-plane displacements were observed, lending support to the possibilitythat an instability, driven by wakes in the ion flow, provides energy that sustains the blast wave’s amplitude,despite the presence of gas damping.
I. INTRODUCTION
A dusty plasma consists of electrons, ions, neutral gas,and microparticles.
Electrons and ions collect on thesurface of the microparticles, which accumulate a largecharge of thousands of elementary electron charges. Be-cause of that large charge, the microparticles can be-have as a strongly coupled plasma component, organizingin a crystalline microstructure.
Also because of thatlarge charge, large amplitudes are easily attained whencompressional waves propagate through a dusty plasma.Wave amplitudes are often great enough for the wave tobecome nonlinear, and in some cases they can have prop-erties of a shock.
Shocks can in general be classified as continuouslydriven ( i.e. , piston-generated) or impulsively driven ( i.e. ,blast waves). In the case of a continuously driven shockwave, there is a constant energy input into the system, sothat the amplitude of the propagating shock is sustainedand does not decay. In contrast, a blast wave results froman impulsive deposition of energy, which has a finite timeduration and occurs within a localized volume. Afterthis initial impulsive energy input, the blast wave propa-gates without any further input of energy from an exter-nal source. In this paper we report an experiment witha blast wave in dusty plasma; we previously reported adifferent experiment with a continuously driven shock. A shock’s energy generally experiences not only anexternal input, but also a dissipation. The dissipationmechanism that dominates laboratory dusty plasmas isusually friction exerted on the microparticles by neu-tral gas.
This gas-friction dissipation mechanism wasidentified experimentally as a cause of damping not onlyfor blast waves, but also other longitudinal and trans-verse waves and solitons as well. Because all labora-tory dusty plasmas include neutral gas, one might expecta blast wave to display an attenuation, with its amplitudedecaying as it propagates, after the initial impulsive en-ergy input.Unexpectedly, however, a lack of attenuation was ob- served, despite the dissipative effects of gas friction, ina previous dusty plasma experiment by Fortov et al. , with an impulsively generated large-amplitude pulse.Their cloud of microparticles filled a three-dimensional(3D) volume within a low-temperature DC-dischargeplasma. In that experiment, the large-amplitude com-pressional pulse wave was excited externally by a sud-den movement of gas, which was produced by a movingplunger. Fortov et al. reported that their high-amplitudepulse propagated for 500 ms without significant attenua-tion, even though a much stronger attenuation, requiringonly 15 ms, would be expected due to gas friction, giventhe high gas pressure used in that experiment.This unexpected lack of attenuation of their blast waverequires an explanation, as Fortov et al. noted. Be-cause their high gas density would be expected to causea “great damping,” they remarked that their observa-tion of an unattenuated waveform indicates that “thewave must have an energy source other than the ini-tial impulse.” In their conclusion, they suggested thattheir wave was driven by an instability, and they sug-gested the dust-acoustic instability could serve as themechanism for an energy source. (We note that thedust-acoustic instability’s energy source is provided bya flow of ions passing through the cloud of microparti-cles. ) This lack of attenuation, in the experiment ofFortov et al. , also captured the attention of other au-thors, Ivlev and Khrapak; they stated that the almost-undamped propagation suggests a mechanism of strongenergy influx, and hinted at an alternative mechanism,a modulational instability. Ivlev and Khrapak also re-viewed an earlier experiment by Samsonov et al. , whosimilarly used a momentary gas flow to disturb a 3D mi-croparticle cloud, but in an RF-discharge plasma. In thatexperiment, there might have also been an unexpectedlack of attenuation, according to Ivlev and Khrapak.In this paper we report another observation of little de-cay, for a blast wave, despite the dissipative effects of gasfriction. Our experiment used a different kind of dustyplasma, one with a single two-dimensional (2D) layer ofmicroparticles. As in the experiment of Fortov et al. , ourelectron-ion plasma filled a 3D volume, but unlike theirexperiment, we prepared our microparticle cloud to belimited to a 2D monolayer, not a 3D volume. Our ex-periment demonstrates that the previously reported re-sult for 3D microparticle clouds, that a shock’s decay ismuch less than would be expected if gas friction aloneacts on the microparticles, occurs also in 2D microparti-cle clouds.Experiments with two-dimensional clouds can be dif-ferent from those with 3D clouds, for several reasons. Im-portantly, the possible energy mechanisms for a 2D cloudcan be different from those in a 3D cloud. Although a 2Dcloud can sustain a compressional shock wave, just as a3D cloud can, the instabilities are not entirely the samein 2D and 3D clouds. In either case, instabilities can bedriven by ion flow, but the shock propagation cannot beparallel to the ion flow in a 2D cloud, which is unlikethe situation in a 3D cloud. In a 2D cloud, the ion flowmust be nearly perpendicular to the 2D layer, due to thedirection of the electric field that serves the dual purposeof levitating the microparticle cloud and driving the ionflow. For a 2D cloud, instabilities driven by ion flow havebeen well studied experimentally and theoretically, andthere is a theory for the Schweigert instability. Thisis different from the ion-acoustic instability in a 3D cloud,which involves ion flow that is aligned with the directionthat a wave propagates within a 3D volume.Another difference, for experiments with 2D clouds,is the greater ease of making a crystalline microstruc-ture than in 3D clouds. A stable crystalline microstruc-ture can in general be attained only by using enoughgas friction to suppress instabilities, and in a 2D cloudthis can be done at a much lower gas friction level thanis possible in a 3D cloud. Strong-coupling effects suchas a crystalline microstructure can be observed in a 2Dcloud even when the gas pressure is two orders ofmagnitude less than in a 3D cloud. Moreover, the mi-crostructure can be observed more easily in a 2D cloud,because it is possible to focus a camera so that it views allthe microparticles, which is not practical in a 3D cloudusing ordinary imaging.One feature of our blast-wave experiment is that ourmechanical energy input stops at a definitive time. Thisis important so that we can exclude the possibility that agreatly reduced attenuation is somehow due to a lingeringof this mechanical energy input, during the propagation.For example, in the case of previous impulsive experi-ments where the mechanical energy input came from agas puff, although it was known when the gas valve wasclosed, it was not necessarily known exactly when thegas has stopped circulating. To provide an impulsive me-chanical energy input that ceases at a definitive time, wewill use a moving wire as the exciter. The wire’s motionis precisely controlled by a stepper motor, and the timethat its motion stops is confirmed by our video imaging.
II. EXPERIMENTA. Dusty plasma
A single layer of microparticles was levitated above thelower electrode in our modified GEC chamber, Fig. 1.The chamber was configured as in Ref. , but with theaddition of a moving wire, for exciting the blast wave.Radio-frequency power, at 13.56 MHz, was capacitivelycoupled to the lower electrode, where the peak-to-peakand self-bias voltages were −
117 and 149 V, respec-tively. These voltage measurements remained steadywithin ± By introducing a limited quantity of thesemicroparticles, we were able to prepare a single-layertwo-dimensional (2D) cloud, located above the lowerelectrode. Our microparticles’ large diameter, 8.69 µ m,helped them to collect a large charge, which we measuredas Q = − . × e , by analyzing video recordings ofmicroparticle motion in a crystal, as in Ref. . Thisanalysis also yielded other parameters for the micropar-ticle cloud, including the longitudinal sound speed,which was determined to be 16 mm/s. This sound speedwill be used for calculating Mach numbers in our shockexperiments. B. Imaging
Our primary diagnostic tool was video microscopy.
Most of our data came from the top-view camera, whichwas a 12-bit high-speed Phantom Miro M120. This top-view camera was operated at 70 frames/s for our mea-surements of the crystal, and a faster rate of 800 frames/sfor our runs with blast waves. The spatial resolution was29.79 pixels/mm. The camera’s lens was covered by abandpass filter, blocking light at wavelengths differentfrom the 577-nm laser light, which was shaped into asheet to illuminate the microparticles. The sheet’s thick-ness was chosen to be 1.0 mm, which is wider than is typ-ically used in 2D dusty plasma experiments, so that mi-croparticles could not move out of the laser sheet, whichwas a difficulty in previous shock experiments. To allow the detection of out-of-plane displacements,we also used a side-view camera, unlike the 2D shockexperiment of Samsonov et al. Detecting out-of-planedisplacements is significant because they are essential forthe Schweigert instability. This instability arises fromion wakes downstream of a microparticle.
Thesewakes have the greatest effect when a microparticle islocated slightly above or below another microparticle, forthe case of a vertical ion flow.Our side-view camera was a Basler Pilot piA 1600 -35gm, the same as in the PK-4 flight instrument. FIG. 1. Experimental setup. (a) Schematic of the vacuumchamber, shown without flanges. The microparticles wereimaged simultaneously by a top-view camera with illumi-nation by a horizontal laser sheet, and a side-view camera(not shown) which used separate illumination optics. (b)Schematic of the microparticle cloud and the manipulationsetup. A horizontal exciter wire was propelled along the+ x direction. The wire and its support structure are shown toscale, while other features are not. The microparticle cloudwas levitated in a plasma sheath, which (as sketched by adashed line) had a curved edge, conforming to a shallow de-pression in the lower electrode. The argon plasma, whichfilled the chamber, was sustained by applying radio-frequencyvoltage between the lower electrode and the grounded cham-ber walls. This experimental setup was similar to the onein Ref. , except that the chamber, lower electrode, and thewire’s support structure were all different. To allow operating this camera at 100 frames/s, werecorded images not for the entire sensor, but only forthe 1600 ×
100 pixel portion of the sensor where the im-age of the microparticle layer was located. The imag-ing setup for this side-view camera used a vertical sheetof light, to illuminate the particles. This vertical sheetwas produced by a 632.8-nm HeNe laser, and a matchingbandpass filter was fitted to the camera. This verticalsheet, which illuminated a cross section of the horizontallayer of microparticles, had a thickness of about 1 mm,which was chosen to be larger than the interparticle spac-ing. Compared to our main camera, which viewed fromthe top, this side-view camera had a more limited spatial resolution of only 13.12 pixels/mm.The top-view and side-view cameras were synchronizedusing an external clock. Since the top-view camera wasoperated at a frame rate eight times greater than theside-view camera, the latter was triggered simultaneouslywith every ninth frame of the top-view camera. The twocameras were aligned and calibrated so that, in their im-ages, the x -coordinates correspond. This calibration wasdone by imaging a test object (see supplementary ma-terial ), which was placed in the field of view of bothcameras. C. Gas damping
The gas damping rate will be an important parameterlater, when we analyze our experimental results. Here wepresent a range of values for the theoretical gas dampingrate ν E , for our experimental conditions. The subscript E reflects the Epstein theory of gas damping, which weassume.We consider the theoretical case where gas friction isthe only force acting on a microparticle. The force actingon a particle is F g = ν E m p v p , (1)where m p and v p are the microparticle’s mass and speed,respectively. Calculating the ratio of the force and themicrosphere’s momentum m p v p , we obtain the dampingrate ν E = δ s m g πk B T g p g ρ p r p , (2)that would be expected if gas friction alone altered themicrosphere’s energy. Here, ρ p and r p are the mass den-sity and radius of the microparticle, while p g , m g and T g are respectively the pressure, atomic mass, and tem-perature of the gas. The leading coefficient in Eq. (2)must have a value in the range 1 ≤ δ ≤ . For our experimental conditions, the gas damping ratein Eq. (2) must be in the range2 . ≤ ν E ≤ .
19 s − . (3)The values that bracket the range in Eq. (3) correspondto the limiting values of 1 and 1.442 for the coefficient δ .As an experimental confirmation of the theoreticalrange of Eq. (3), we analyzed sloshing-mode data fromour experiment. Doing this, we found a gas damping rateof 2.9 s − , as explained in the supplementary material. D. Blast wave generation
In order to generate a blast wave, i.e. , an impulsivelydriven shock wave, we require an excitation mechanismthat has a finite time duration. For this purpose, in previ-ous dusty plasma experiments a momentary gas flow wasused in a 3D cloud, while an electrical pulse was appliedto a stationary wire to generate a blast wave in a 2D mi-croparticle cloud. Our approach with a 2D cloud alsoused a wire, but it was not stationary, but instead it wasmoved and then abruptly stopped. This approach hadthe advantage that the energy input stopped at a knowntime, because we detected the stopping of the wire’s mo-tion.An electrical repulsion, between the wire and the mi-croparticles, allowed us to manipulate the microparticles.The microparticles were floating electrically, so that theyhad a negative potential as compared to the surroundingplasma. Similarly, our wire was always allowed to float,as we did not apply an external voltage to it. As the wiremoved, it acted like a piston since nearby microparticleswere repelled from the wire. By abruptly stopping themotion of the wire, analogous to stopping the motionof a piston in a gas cylinder, we were able to excite ablast wave. This method of blast-wave generation differsfrom that of Ref. , where the wire was stationary, anda step-wise change in electrical potential was applied tothe wire to repel nearby particles impulsively. It is alsodifferent from the shock-wave excitation mechanism weused in our previous experiment, Ref. , where the wirewas moved steadily, analogous to a piston moving at asteady speed in a gas cylinder, so that a shock was drivencontinuously rather than as a blast wave.The mechanical structure of the exciter wire was sim-ilar to that in our previous experiment, Ref. . For thepresent experiment, the wire had the same diameter of0.41 mm but a greater length of 60 mm. The exciter wirewas propelled horizontally by the same motor drive as inRef. , although the motor drive was programmed differ-ently, so that the wire’s motion was stopped abruptly inthe present experiment. We also used a different plasmachamber. The chamber, and the wire’s structure, areshown in photographs in the supplementary material. The wire’s inward horizontal motion was abruptlystopped at a time that was well determined. Importantly,this time was known because the motor drive and cam-eras were synchronized, and the wire was visible in thecameras’ fields of view. As we mentioned earlier, sinceour wire’s motion was abruptly ceased, we would ex-pect the energy input from its mechanical motion to alsocease, so that the amplitude of the propagating compres-sional pulse would thereafter decay due to gas friction, ifother energy input mechanisms are absent.The sequence of experimental steps were as follows.Before the first experimental run, we allowed the mi-croparticle cloud to anneal into a crystalline structure,and during this unperturbed time we recorded its ran-dom thermal motion. Next, we performed a sequenceof experimental runs with manipulation. At the begin-ning of each run, the microparticle cloud had a crystallinestructure; the exciter wire was located outside the cloudand it was at rest. We then started the wire’s inward horizontal motion, at a steady speed having a desiredvalue. Video recording began, triggered by a photogatethat sensed the wire’s motion. During this manipula-tion, the wire moved toward the microparticle cloud, thecloud was disturbed, and a compressional pulse propa-gated away from the wire. This propagation started atthe cloud’s edge, and it continued in the same + x di-rection as the wire’s motion. As the moving wire ap-proached a specified position, approximately where thecloud’s outer edge was originally located, we brought thewire to a halt with a rapid acceleration of −
360 mm/s .After the wire’s motion was halted, the compressionalpulse continued propagating across the cloud, yieldingthe main data analyzed in this paper. The pulse propa-gated across the entire cloud, and then the video record-ing ended. To prepare for the next experimental run,the exciter wire was retracted to its original position,allowing the microparticle cloud to relax to its originallocation. We waited at least 15 min, allowing the cloudto anneal again into a crystalline microstructure, beforethe next run. After the sequence of experimental runswas completed, as a final step, the unperturbed randommotion of a crystalline structure was recorded again. III. RESULTS
Figure 2 shows data for the particle cloud, with imagesfrom the top-view and side-view cameras, and the den-sity profile. Images from the top-view camera, shown inthe upper panels, reveal the overall arrangement of mi-croparticles. These top-view images are cropped to showthe region of interest that we will analyze further. Imagesfrom the side-view camera are used to detect any out-of-plane displacements. The bottom panels of Fig. 2 areprofiles of the areal density n , obtained from top-viewimages by counting microparticles located within rect-angular bins. The bin width of 0.724 mm correspondsto three Wigner-Seitz radii. In each bin, the number ofcounts was weighted using a cloud-in-cell algorithm. A. Comparing unperturbed and perturbed conditions
In Figure 2, the panels on the left show the microparti-cle cloud at an early time, when it was still unperturbed,meaning that the exciter wire was still far from the cloud.The panel on the right shows the cloud at a later time (af-ter the wire’s motion had stopped) while a compressionalpulse was propagating in the + x direction.The unperturbed microparticle cloud had the mi-crostructure of a crystalline lattice, as can be seen inthe upper-left panel of Fig. 2. Microparticles were self-organized into a triangular lattice with six-fold symme-try, i.e. , it was hexagonal, with a lattice constant of about0.46 mm. In the central portion of the cloud ( x <
25 mmin Fig. 2), the number density was roughly uniform, witha variation of ±
10 %. Near the cloud’s edge ( x >
30 mm)
FIG. 2. Images of the microparticle cloud, and corresponding number-density profiles, in the region of interest. Data are shownin two columns; the unperturbed condition is at an early time, before the cloud was disturbed by the exciter wire, while theperturbed condition is at a time after the wire had approached the cloud and then stopped. In the perturbed condition, acompression in the cloud propagated to the right. In this pulse, the number density was compressed nearly two-fold, and itsprofile had a strong gradient on the peak’s leading edge. The side-view camera images confirm that the cloud was a singlelayer in the unperturbed condition. In the perturbed condition, although it remained as a single layer ahead of the peak, thecloud experienced out-of-plane displacements near the pulse’s peak and behind it. These data are for run 2, in which the wirewas stopped at t = 990 ms after moving at a speed of 50.8 mm/s. Data for all times in this run can be seen in a video in thesupplementary material. the number density diminished to zero, as is typical forexperiments with 2D microparticle clouds.The perturbed and unperturbed conditions differedin four significant ways. First, when it was perturbed,the cloud had a microstructure that was much less crys-talline, as seen in the upper-right panel of Fig. 2. Second,there was some out-of-plane displacement within the per-turbed cloud, as seen in the inset with 2 × magnification.Third, a significant compression for the perturbed condi-tion is seen in the top-view and side-view images, and inthe density profile as well. This significant compressionleads us to describe the pulse as having a high amplitude.Fourth, at x ≈
22 mm the profile has a sharp gradient,which is one of the characteristics that are required tocharacterize a compressional pulse as a shock, as we willdiscuss below.
B. Out-of-plane displacements
When the microparticle cloud was unperturbed, it hadonly a single layer. This single-layer structure is seen in the middle panel of Fig. 2. We never observed out-of-plane displacements in the unperturbed microparticlecloud.However, when the cloud had been perturbed, shortlyafter the wire’s movement ceased, we observed out-of-plane displacements in all six runs. These out-of-planedisplacements generally occurred either near the peakdensity, or behind the peak as in Fig. 2. We observedfewer out-of-plane displacements ahead of the peak.The width w of the region with detectable out-of-plane displacements was measured. In Table I we reportthis width, once for each run, when the density’s peak n peak was located near x peak = 20 mm ( i.e. x peak ≈
20 mm). These measurements were made manually byvisually inspecting the side-view camera images (such asthose in the middle panels of Figs. 2 and 3) and assessingwhether we could detect any out-of-plane displacement.The threshold of detection was essentially determined bythe resolution of the side-view camera.The magnitude of the out-of-plane displacement wasgenerally less than the horizontal distance between mi-croparticles. Nevertheless, despite their small size, these
TABLE I. Parameters for each run. After the exciter wire was stopped, measurements of the pulse were made. The valuesof n max correspond to the greatest density for an entire run ( i.e. , not for a specific frame). The width of the region whereout-of-plane displacements were observed is indicated as w , which was for the frame where the peak density was located at x = 20 mm, in runs 2-6. In run 1, out-of-plane displacements were sporadic, and did not appear in that particular frame.Exciter wire’s motion Measurements after exciter wire was stoppedRun Wirespeed(mm/s) Time wirestopped(ms) n max (mm − ) Compressionfactor n max /n Shockspeed(mm/s) Machnumber Width w (mm) Figure1 44.5 1140 10.8 2.0 47.8 3.0 -2 50.8 990 11.6 2.1 52.5 3.3 1.37 2,43 57.2 900 11.9 2.2 53.6 3.3 0.384 63.5 800 12.4 2.3 56.0 3.5 3.15 76.2 700 13.3 2.4 57.8 3.6 5.46 101.6 530 14.5 2.7 66.3 4.1 5.4 3FIG. 3. Images of the microparticle cloud, and correspondingnumber-density profiles, in the region of interest, for run 6.This was the run that used the fastest wire speed, so that thepulse had the highest amplitude of all our runs. The datashown here are from a sequence that is presented as a videoin the supplementary material. displacements may be physically significant for explain-ing an energy influx mechanism, as we will discuss later.To the best of our knowledge, ours is the first re-port of localized out-of-plane displacement, for a blastwave in a 2D dusty plasma. However, aside from blastwaves, there have been Mach-cone shock-wave experi-ments where there was a prominent report of out-of-planemotion that was not only detected but also measured. We should also mention that in blast waves that were re- ported for previous 2D experiments, even though the au-thors did not mention out-of-plane displacements, we be-lieve that we can identify some hints of this phenomenonin their reports. Samsonov et al. noted that in theirblast-wave experiment, within the horizontal sheet oflaser illumination, some particles disappeared. We be-lieve that this disappearance may have been due to out-of-plane displacements, which could have been confirmedif the experimenters had used a side-view camera likeours. C. Properties of the compressional pulse
Two quantities that we measure, using a pulse’s spatialprofile, are the peak’s amplitude n peak and its position x peak . These quantities are marked in the lower-rightpanel of Fig. 2, which was for run 2, which had a slowerinitial velocity for the wire. For comparison, in Fig. 3,a run with a higher initial velocity has a profile with agreater amplitude n peak .Measurements of the peak’s amplitude n peak as it de-veloped with time are presented in Fig. 4. The datapoints shown are experimental results, which we obtainedby selecting the greatest value of density in a given videoframe. Essentially, these data points describe the blastwave’s amplitude as the density measured at the wave’screst. While n peak is a peak density in a specific frame,we define n max as the greatest density registered in anentire run. The numerical values of n max are given inTable I. For example, for run 2, n max = 11 . − .Figure 4 is our chief result; it demonstrates a lack ofsignificant decay as the pulse propagates. During thetime interval shown, the exciter wire had already stoppedmoving, so that the input of mechanical energy had al-ready ceased. Despite the lack of this mechanical energyinput, we see in Fig. 4 that the pulse’s amplitude re-mained nearly constant. The amplitude diminished at aslow rate, ν pulse = 0 . ± .
02 s − , when fit to an ex-ponential ∝ exp( − ν pulse t ). We will discuss this result E = 2.21 s -1 for = 1 n p ea k ( mm - ) l og s ca l e time (ms) E = 3.19 s -1 for = 1.442 FIG. 4. Peak number density n peak in each video frame, ob-tained as in Fig. 2. The data shown were recorded in run 2, af-ter the exciter wire stopped at 990 ms. The axes are semilog-arithmic, so that an exponential decay would appear as aline, with a slope corresponding to a damping rate with unitsof s − . The experimentally observed decay, seen in the solidcurve fitting the data points, has a rate of ν pulse = 0 .
66 s − .For comparison, this value is much smaller than the theoret-ical rate in Eq. (3), which is bracketed by two values, shownhere as representative slopes in the lower lines; these theoret-ical values assume that neutral gas friction alone affected thekinetic energy of microparticles. shortly.Measurements of the pulse’s speed are presented in Ta-ble I. We obtained these values from measurements ofthe peak’s location x peak in each video frame, yielding atime series, x peak vs time. From this time series, we ob-tained the pulse’s propagation speed, using the methodof Ref. .Table I shows that the pulse propagated at a supersonicspeed in all six runs. This speed ranged from 48 mm/s to66 mm/s, which was much greater than the longitudinalsound speed c l = 16 mm/s. Also shown in Table I arevalues of the initial wire speed, the greatest measureddensity for every run n max , compression factor n max /n ,shock speed, shock Mach number, and the width w . IV. DISCUSSIONA. Attributes of a shock
We can classify our pulses as shock waves because theyhave three key attributes of shocks: supersonic speed,sharp gradient, and high amplitude. A supersonic speedof the pulse’s propagation was observed for the six ex-perimental runs, with Mach numbers ranging from 3.0 to4.1, as summarized in Table I. A sharp gradient of thepropagating pulse is apparent in the lower right panel ofFig. 2 and, most prominently, in Fig. 3. A high ampli- tude of a pulse is evident in comparing the lower panels ofFig. 2. The high amplitude is also observed in Fig. 3, andin the videos in the supplementary material. As a mea-sure of the amplitude, the compression factor n max /n ranged from 2.0 to 2.7 over our six runs. These valuesfor the compression factor all exceed what was previouslyreported in the literature for blast waves in 2D dustyplasmas. For example, n max /n was 1.2 in the experi-ment of Samsonov et al. In that paper, the pulses wereidentified as shocks.Our main result was a lack of significant decay ofthe shock’s amplitude in Fig. 4. After the exciter wirestopped moving, mechanical work by the wire on thecloud of microparticles ceased suddenly. Thereafter, dur-ing the time interval shown in Fig. 4, the shock’s ampli-tude remained almost constant, diminishing slowly withan experimentally measured rate ν pulse = 0 .
66 s − whenfit to an exponential.If gas damping were the only mechanism affecting theamplitude of the blast wave, after the wire stopped mov-ing, we would expect an exponential decay, with a pre-dictable time constant equal to 1 /ν E . The value of ν E ,for our experiment has limiting values of Eq. (3), whichare presented in Fig. 4 as representative straight lines toindicate slopes, since the axes are semilogarithmic.A great mismatch is seen in comparing the theoreticalslopes for gas-friction and the experimental dependence,in Fig. 4. The amplitude of the shock wave, as observedin the experiment, decays much slower than can be ac-counted for by gas drag only. B. Explaining how the shock can decay little
We suggest that the very low level of decay of our blastwave is explained by an energy influx from ion flow. Wefurther suggest that this energy influx could occur, forexample, through the Schweigert instability.
The Schweigert instability has been experimentallyconfirmed to convert energy in the ion flow into kineticenergy of the microparticles. As we mentioned earlier,the Schweigert instability is typically observed when anion flow passes perpendicular to a layer of microspheres,and it is most profound if the microspheres in fact do notrest strictly in a single 2D plane, but instead have notice-ably large out-of-plane displacements. As theions flow past a microparticle, they form a downstreamwake, where there is a spatially localized concentrationof positive space charge. That downstream wake can at-tract other microparticles, which are negatively charged.This attraction to the ion wake can lead to a particularlystrong energy transfer from the ion flow to the micropar-ticles, in what has been described as a nonre-ciprocal interaction.
Considerable kineticenergy can be added to microparticles, if they do not reston a single plane, but instead have out-of-plane displace-ments.For our experiment, it is significant that we detectedout-of-plane displacements. That detection was madeusing our side-view camera. Images, revealing thesedisplacements, can be seen in the right-middle panelof Fig. 2 and the middle panel of Fig. 3. They can also beseen in the videos in the supplementary material. Gen-erally, the out-of-plane displacements were observed in aregion beginning where the density gradient was greatest,near the shock’s density peak, and extending up to a fewmillimeters behind the peak. Since these displacementsare located so close the peak, it seems reasonable to sug-gest that the Schweigert instability could add energy thathelps sustain the pulse’s amplitude.
C. Excluding other mechanisms
The design of our experiment largely eliminates otherenergy input mechanisms, besides those arising from ionflow. We can identify two such mechanisms:The first mechanism that we avoid is gas flow,which was present in the previous 3D experiments ofSamsonov et al. and Fortov et al. In those experi-ments, a gas puff compressed the microparticle cloud,creating what appears to be a blast wave that did notdiminish much as it propagated. For those two experi-ments, the gas flow’s pattern and time scale for its dis-sipation were not measured, so that lingering effects ofthe gas puff cannot be excluded in explaining the lack ofdecay of the wave in their experiment. For our experi-ment, however, the gas flow was not a factor because itwas very small and never varied in time.The second mechanism that we avoid is whatSamsonov et al. termed a “tsunami effect.” This isan artifact of a density gradient in the undisturbedmicroparticle cloud. In an experiment with solitions,Samsonov et al. found that this effect was responsible forthe absence of amplitude decay, as the solitions propa-gated.
The unperturbed medium had a density pro-file that diminished, in the direction of pulse propagation,by 2.5-fold across the microparticle cloud, in that exper-iment. In our experiment, however, such a gradient wasnot a factor. A strong gradient appeared in our den-sity profile only at the extreme right edge of the particlecloud, at x >
30 mm. For the data in Fig. 4, our pulsewas located always at x <
25 mm, avoiding this so-calledtsunami effect.
V. CONCLUSIONS
In earlier experiments it was found that a shock wave’samplitude did not decay as much as would be expected,due to gas friction. Those experiments were performedwith a dust cloud that filled a 3D volume.
We havenow extended this finding to a dusty plasma that has a2D monolayer of microparticles. The amplitude of ourshock was observed to diminish very little, over an ex-tended time. If the only energy mechanism acting on the microparticle cloud were gas friction, there would be alarge and predictable exponential decay, which was notobserved. This experimental finding suggests a source ofenergy influx to the microparticle cloud, which largelycancelled the dissipative effects of gas friction.To help determine the source of the energy influx, wetake advantage of the design of the experiment, whichhad two advantages. Firstly, we prepared a micropar-ticle cloud that had no strong density gradients, in theregion of interest, before we manipulated the cloud withour moving wire. Secondly, and more importantly, weused a new method of blast-wave excitation, involvingthe movement of a thin wire, which allowed us to sud-denly stop the mechanical energy input by halting thewire’s motion. In this way, we avoided lingering sourcesof mechanical energy, as might occur in other experimen-tal schemes relying on excitation by a gas puff.
Thegas in our experiment was maintained at a steady condi-tion while the exciter wire was propelled at a supersonicspeed and then suddenly stopped.Ion flow is a source of energy in most laboratory exper-iments with dusty plasmas. Ion flow is capable of drivingseveral kinds of instabilities, depending on the conditionsin the dusty plasma. Because we prepared our dustyplasma as a 2D monolayer, which was perpendicular tothe ion flow, it is of particular interest to consider theSchweigert instability as a mechanism to couple the en-ergy of the ion flow to the motion of the microparticles.This instability can be great if there is some out-of-planedisplacements of the microparticles.To detect such displacements we used not only a top-view camera as in previous 2D shock experiments, but also a side-view camera. Although that side-viewcamera did not have a high spatial resolution, it was ca-pable of registering out-of-plane displacements. We de-tected such out-of-plane-displacements in all the runs re-ported here. The presence of these displacements lendssupport to the possibility that the Schweigert instabilityprovides an energy influx, in our experiment.
VI. ACKNOWLEDGMENTS
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