Suppression of a spontaneous dust density wave by modulation of ion streaming

Tonuj DEKA,Bidyut CHUTIA,Y BAILUNG,S K SHARMA and H BAILUNG

Physical Sciences Division,Institute of Advanced Study in Science and Technology (IASST),Paschim Boragaon,Garchuk,Guwahati,Assam 781035,India

Abstract

Keywords: dust density wave,suppression phenomena,strongly coupled dusty plasma,forced van der Pol equation

1.Introduction

Dusty plasma is a medium containing micrometer- or nanometer-size dust particles in a plasma background of electrons,ions,and neutrals.Interest in the field of dusty plasma has grown immensely in the last few decades,due to its natural abundance in space environments like interstellar clouds,comet tails,Saturn’s rings,etc.Dust formations occur in many laboratory devices such as in material processing industries and thermonuclear fusion devices [1–7].Dusty plasma is produced in the laboratory either by introducing the dust particles externally or by growing particles in the plasma using a mixture of noble gases with reactive gases such as SiH4,C2H2,CH4,etc [8–10].In the laboratory discharge plasma,dust grain floats with negative potential to maintain equilibrium between ion and electron fluxes onto the dust so that the surface current is zero.When the dust particle density in plasma is sufficiently high,Coulomb interaction between the dust particles dominates and the dusty plasma can support a variety of linear and nonlinear phenomena such as strong coupling effects,dust crystallization,low-frequency wave modes,instabilities,voids,vortex,solitary structures,etc [6,11–17].

Dust particles,being larger in size and heavier in mass,have a relatively low charge-to-mass ratio compared to ions and electrons.The spatial and time scales of their motion are suitable for direct observation of various kinds of waves and instabilities like dust acoustic waves (DAWs),dust acoustic(DA) shocks and solitons,and other nonlinear structures[18,19].In particular,the DAW is a longitudinal wave analogous to an ion-acoustic wave in which the heavy dust particles provide the inertia and the electrons and ions provide the restoring force to drive the wave.Rao et al[20]predicted the existence of this low-frequency wave mode in dusty plasma for the first time in 1990,which was experimentally verified by Barkan et al[21]in 1995.Since then,DAWs have been observed in many laboratory experiments [22–26] as well as in microgravity-based experiments [27,28].Most of the experiments are,however,carried out with micrometersized dust grains levitated in the plasma against the effect of gravity by the sheath electric field of the lower electrode.In experiments,dust acoustic perturbations are excited externally by using a voltage pulse(to the exciter or anode).However,in some situations,waves occur spontaneously in the presence of ion streaming (towards the sheath or from a high-electricfield region),and they are identified as dust density waves(DDWs).Self-excited DDWs are more frequently observed in dust clouds with nanometer-sized dust particles where the gravity effect is negligible [29–31].Self-excited dust density waves in nanodusty plasma were observed not only in radiofrequency (rf) discharges,but also in dc discharges [32].Hayashi observed a dust density wave in a dust cloud with 50 nm diameter Cu particles and measured the dispersion relation to investigate the spatial distribution and temporal evolution of dust charge[29].Tadsen et al[30]also observed a self-excited dust density wave in dusty plasma (with～196 nm radius carbon particles) and measured the dispersion relation of the wave.Recently Deka et al[31]observed a dust density wave in a dusty plasma with carbon particles(50 nm radius).The dust density wave originates from a void boundary above the live electrode,and the wave parameters are used to estimate the dusty plasma parameters.In these experiments,dust density waves are spontaneously generated due to ion streaming from the void.The criterion for the excitation is that the ion-streaming velocity must be larger than the ion thermal speed and the phase velocity of the wave.These self-excited dust density waves show complex behavior because of their three-dimensional oscillatory structure and varying growth and damping rates.Such self-excited density wave patterns resemble the self-organized structures in many natural situations and arise due to available free energy in the system [28].Recently,Pilch et al [33] reported the synchronization of self-excited dust density waves in an anodic dusty plasma with 0.97μm melamine formaldehyde particles.In the experiment,the self-excited dust density wave is superimposed by an external modulating signal with frequency fm～ (1–140) Hz around the natural frequency f ～ 40 Hz.In a similar experiment,Ruhunusiri et al [34] observed four synchronized states of unstable dust density waves with frequencies that were 1,2,3,and 1/2 times the driving frequency.The van der Pol oscillator model has been used as a quantitative model and as a qualitative reference to explain the characteristic behavior of synchronization in plasma waves,including dust density waves[35–37].The forced van der Pol oscillator also shows interesting results in the vicinity of the region of harmonic synchronization.The Mathieu equation has also been derived from the fluid model of plasma to investigate the synchronization of dust density waves due to periodic modulation of dust charge,density,etc [38].The van der Pol equation has also been used to explain the synchronization mechanism between two coupled dc glow discharge plasma sources [39].The external modulation technique has also been used for suppression of parametric decay instability in some other experiments [40,41].

Figure 1.Schematic diagram of the experimental setup.MN:matching network,PE: powered electrode,GE: ground electrode,E: grid exciter,LP: Langmuir probe,Amp: signal amplifier,FG:function generator,LS: laser source with cylindrical lens.

In this work,we present an experimental investigation of a self-excited dust density wave modulated by an externally applied sinusoidal signal with frequency close to the spontaneous signal frequency in a strongly coupled dusty plasma with nanometer-size dust particles.The 50 nm diameter nanoparticles have smaller mass (by less than two orders) as well as carrying very small charge (～75e) compared to micrometer-size particles [33,34].However,as the ratioZd2/mdremains in the same order,therefore the spontaneous frequency in this experiment is shifted only 2–3 times more than in earlier experiments with micrometer-size particles.The plasma potential profile has been measured to confirm the presence of an electric field that maintains the ion streaming from the void region towards the dust cloud,which is responsible for the excitation of the dust density wave.The formation of a dust void in the dust cloud has been explained and a quantitative description of the force balance at the void boundary has been presented with the help of the electric field obtained from the measured potential profile.The suppression of this spontaneous wave and growth of the external signal have been investigated under different parameter ranges.The synchronization phenomenon occurring through mode suppression is observed above a threshold amplitude of the forcing signal.Dusty plasma parameters such as dust density and average dust charge are obtained from spontaneous dust density wave parameters and relevant theoretical consideration to confirm that the nanodusty plasma fluid is in a strongly coupled state.The experimental results are explained with the help of the mathematical model of a forced van der Pol oscillator.

2.Experimental setup

The experiment is performed in a cylindrical glass chamber of diameter 3 cm and length 15 cm mounted horizontally[31].A schematic diagram of the experimental setup is shown in figure 1.A vacuum of the order of 10-3mbar is achieved inside the experimental chamber by using a rotary pump.Argon gas is then introduced into the chamber at a flow rate of 0.5 sccm to maintain the chamber at a working pressure of 0.015 mbar.Plasma is produced by applying rf power 10–15 W (13.56 MHz) from an rf power generator via a matching network.The powered electrode consists of a thin aluminium strip placed on the bottom outer surface of the glass chamber.The typical plasma parameters measured using an rf compensated cylindrical Langmuir probe of length 7 mm and diameter 0.2 mm (ALP,Impedans) are density 108–109cm-3and electron temperature ～5–7 eV.

Carbon nanopowder (average particle radius 50 nm) is sprayed at the bottom of the glass chamber.A stainless steel circular mesh grid is placed at the pumping end of the glass chamber to avoid the loss of dust particles.The grid is electrically grounded.As soon as the plasma is turned on,the dust particles are lifted up into the plasma.The floating dust particles form a cloud,and a void is formed in the region just above the powered electrode.In this region,a high positive plasma potential exists (because of the high ionization rate above the rf electrode),which creates an electric field in the outward direction.The outward electric field applies two different types of force on the negatively charged dust particles: the inward electric field force and the outward ion drag force.In the central region above the live electrode,outward ion drag is more effective due to the high electric field,and a dust-free void is generated.At the void boundary,outward ion drag force is balanced by the inward electric field force on the dust particles.The Langmuir probe is used to measure the axial variation of the ion density and the plasma potential from the void region towards the dust cloud.The electric field in the dust void region is obtained from the gradient of the measured potential profile.

The dust cloud is illuminated by laser light scattering using a sheet of green laser light (532 nm,50 mW).The nanodust particles are not individually traceable due to their small size.However,their collective dynamics are recorded in a high-resolution (1 megapixel) and high-speed digital video camera at ～240–420 frames per second (fps).A self-excited dust density wave is observed propagating in the outward direction originating from the void boundary.Image frames are extracted from the recorded videos and then analyzed using image-analyzing software.In order to examine the growth and interaction of the self-excited dust density wave,a finite-amplitude continuous sinusoidal signal is applied to an exciter grid.An exciter mesh grid(50 lpi)of diameter 10 mm is inserted into the experimental chamber.The exciter is placed in the dust void ～2 cm away from the void boundary and placed with its circular surface perpendicular to the cylindrical axis.

Sinusoidal signals of varying amplitude 1–15 V and frequencies ranging from 50–100 Hz are applied to the exciter through a signal amplifier (Krohnhite,USA) from a function generator.The external signal to the grid modulates the outward ion streaming,which excites another small-amplitude dust density wave at the frequency of the external signal.Both the external and spontaneous signals are first identified from fast Fourier transform (FFT) analysis of time-series data extracted from the recorded videos.The growth and coupling of the external signal and the suppression of the spontaneous signal are then analyzed for different parameter regimes of the external signal.

Figure 2.Snapshot of the observed self-excited DDW.The void boundary is shown by the arrows.The bright straight lines across the photograph are the laser reflection from the glass cylinder.The region of interest for calculation of wavelength and phase velocity is shown by the rectangle,and the vertical line shows the position at which time-series data is obtained.

3.Results and discussion

Figure 2 shows a typical snapshot (from a high-speed video@240 fps) of the observed spontaneous dust density wave in the dusty plasma cloud.The self-excited dust density wave originates from the boundary of the dust void and propagates axially outward along the dust cloud.The perturbation is seen throughout the dust cloud length ～5 cm.The wave frequency is measured by following a density compression(bright front)in successive frames in the video within the region of interest,and is found to be ～78 Hz.The average wavelength is measured to be ～0.27 cm and the phase speed of the perturbation is ～21 cm s-1.

The wavelength of the perturbation is,however,not constant and longer at the furthest end of the dust cloud.The dust void appears in the region just above the live rf electrode due to the presence of an electric field.In the region above the live rf electrode,a comparatively higher ionization rate leads to a higher plasma density.The presence of a density gradient in the plasma causes the anomalous ambipolar diffusion of plasma,which results in a higher plasma potential in order to maintain quasineutrality.The plasma potential,however,falls away from the region above the live electrode.The gradient of plasma potential results in an electric field in an outward direction.Under the influence of this electric field,ion streaming takes place.This ion streaming imparts a strong outward ion drag force on the dust particles,and therefore,a dust void is formed in the region.In an ideal case,a stable void boundary is maintained by a delicate force balance between the inward electric field force and outward ion drag force.However,the equilibrium can be destabilized,and a spontaneous dust density wave originates from the void boundary when the outward ion-streaming velocity is larger than the ion thermal speed.Other dusty plasma parameters such as dust density(～1.8 × 1010m-3),interparticle distance b (～2.3 × 10-4m) and Coulomb coupling parameter(～1.15) are estimated from the measured wave parameters [31].

Figure 3.Plasma potential Vp and ion density ni measured along the axis of the experimental chamber from the location of the rf live electrode.The rf live electrode extends from 0 cm to 4.5 cm.The positions of the void center and the void boundary are at 1 cm and 3 cm respectively.

The measured ion density and plasma potential at different positions along the central axis of the chamber using the Langmuir probe are shown in figure 3.It is seen that due to the higher ionization rate above the live rf electrode,the ion density is high here.The plasma potential is also high in this region due to the ambipolar diffusion.Both the ion density and plasma potential decrease along the dust cloud away from the region above the electrode.The plasma potential gradient creates an electric field that is directed away from the electrode position.This electric field results in the formation of the dust void in the region just above the live rf electrode.The positions of the void boundary and the void center are indicated by arrows in figure 3.The electric field at the position of the void boundary (～2 cm from the center of the void) is measured to be ～4.4 V cm-1.Beyond this point,the plasma potential decreases slowly and becomes constant.

The ion-streaming velocity( ui)is then calculated using the relation ui=μE whereμ is the ion mobility [42,43].Ion mobility is inversely proportional to neutral pressure,so uidepends on the ratio of the electric feild to the neutral pressure.The neutral pressure in the present experiment is 0.015 mbar(1.5 Pa),which is relatively low.The ion neutral collision frequency( νin)is calculated from the relationνin= Vth,innσin,whereVth,iis the thermal velocity of the ions,nnis the neutral number density andσinis the momentum transfer cross-section for ion–neutral collisions,and the value is found to be ～105s-1.The calculated value of ion mobility ( μ )is ～25 m2V-1s-1.The calculated value of the ion-streaming velocity( ui) ～ 1.1 × 106cm s-1is much larger than the phase velocity of the dust acoustic wave(21 cm s-1).It is also noted that the ionstreaming velocity( )uiobtained from the measured electric field is larger than the ion thermal speed (～2.7 × 104cm s-1).Thus the measured parameters indicate that the ion streaming provides sufficient free energy for the excitation of a spontaneous dust density wave.On the other hand,the electric field force(Fe= qE) on the dust particle is estimated considering the average charge on a dust particle q ～ 75e [31].The inward electric field force on a dust particle at the location of the void boundary is ～5.3 × 10-15N.

The ion drag force on a dust particle arises due to momentum transfer between the streaming ions and the dust particle [44,45].The ion drag force usually consists of two components: the collection force(Fcoll)and the orbital force( Forb).

The collection force is due to momentum transfer from all the ions collected by the dust particle,and is given by

where miis the ion mass.Here,is the mean speed of the ions withVth,ibeing their thermal velocity.The collection impact parameter at which the ions hit the dust particle is given byHere,a is the radius of the dust particle andφflis the floating potential of the dust particle.The floating potential for a 50 nm radius particle is estimated to be ～-2.15 V [31].

The orbital force represents the momentum transfer due to Coulomb scattering of the ions in the electric field of the dust particle,and is given by

Using equations (1)–(3),the ion drag force at the void boundary is calculated and is found to be Fi= 4.53 × 10-15N.It is therefore seen that the inward electric field force and the outward ion drag force on the dust particle nearly balance at the void boundary.

In order to obtain the time-series data (for spectral analysis) we extract the pixel intensity data from a longduration video (～5 s) at a fixed location (～1.5 cm from the void boundary) in a small spatial window (＜wavelength of the DDW).At least 1200 data points are generated from a typical video (@ 240 fps).The frequency spectrum is then obtained using FFT.A typical FFT signal of the spontaneous DDW is shown in figure 4.The spontaneous signal has a broad spectrum of frequencies with a dominant peak at ～78 Hz.

Figure 4.Typical FFT signal of the observed self-excited DDW showcasing the broad spectrum of frequencies with a prominent peak at 78 Hz.

The effect of external modulation is observed by applying a continuous sinusoidal signal of frequencies (fm～50–100 Hz) close to the frequency (f0) of the spontaneous signal.The external signal is applied to the circular grid exciter to modulate the streaming ions.Videos are then captured and time-series data are obtained to get the FFT spectrum.A typical set of FFT spectra is shown in figure 5 for varying frequency of the external signal with constant amplitude (Vm= 7.0 V).For fm～ 62–73 Hz (＜f0),both the spontaneous signal(～78 Hz)and the external signal appear in the FFT spectrum separately.Interaction of the external signal is seen through suppression of noise and slight enhancement of the amplitude of the spontaneous normal-mode frequency signal.When fmis very close to f0,the spontaneous signal is suppressed and fmgrows in amplitude.A typical example is shown for fm～ 81 Hz.When fm(～93 Hz) is higher than f0,coupling between the signals ceases and the spontaneous signal regains its amplitude while amplitude of the external signal diminishes.The suppression process is then examined for different amplitudes of the external signal for a wider range of fm.The amplitude ratio of the external signal (Im) to that of the spontaneous signal(I0)obtained from the FFT spectra is plotted in fgiure 6.It is seen that the interaction is weaker when the amplitude of the external signal is smaller (Vm= 5 V) and Δf=f0-fmis larger.The threshold amplitude for interaction is ～7 V and suppression of f0is more prominent with higher Vm(=10 V),as well as whenΔf=f0-fmis smaller.It is also noted that the bandwidth of interaction increases with higher amplitude of the external signal.

In figure 7,the dependence of Vmon the suppression process is presented.An example of a typical external signal with constant fm(～62 Hz) is shown here.For smaller amplitude(Vm= 4 V),fmand f0appear independently and the amplitude ratio Im/I0is very small (≪1).For Vm= 8 V,fmgrows with much higher amplitude compared to f0(Im/I0＞ 2).

On the other hand,a new peak at fm/2 appears.With further increase of Vm(= 12–15 V),complete suppression of the spontaneous signal occurs and period-doubling bifurcation becomes prominent.This dependence of Vmon the suppression process is further represented in figure 8.With the increase in the modulation amplitude,the amplitude ratio of the external to that of the spontaneous signal(Im/I0)increases and at Vm=14 V,the bifurcation is observed.

In order to explain the suppression phenomena,we consider the van der Pol equation,which describes a self-sustaining oscillation due to an external forcing term as given by [46]:

Figure 5.Typical FFT spectra of the modulated DDW with spontaneous frequency f0 ～ 78 Hz and the applied modulation frequencies fm ～ 62 Hz,73 Hz,81 Hz and 93 Hz at a fixed modulation voltage Vm = 7 V.

Figure 6.Variation of the amplitude ratio of the modulating signal to the spontaneous signal with the modulation frequency of the applied signal at three different modulation voltages of Vm = 5 V,7 V and 10 V.The frequency of the spontaneous DDW is f0 ～ 78 Hz.

wherenis the perturbed dust density,nmis the density modulation due to the external signal (proportional to Vm),αis the growth rate parameter,βandγare nonlinear saturation coefficients,ω0is the frequency of the normal dust mode andωmis the frequency of the applied external modulation.

Considering a solution of equation (4) as

and substituting it into relation equation (4),the following relations are obtained:

The external modulation frequencyωmis nearly equal to the spontaneous signal frequencySo,we consider the approximationZ= 2 (ω0-ωm) .Substituting a2from equation (6) into (7) leads to

Equation (9) indicates that with an increase in modulation amplitude nm,the amplitude b of the modulating signal(fm)also increases.This is experimentally observed as shown in figure 7.In equation (6),the spontaneous signal vanishes(a2= 0,i.e.the natural frequency is suppressed) when b2reachesa02/2.This consequence agrees well with the experimental results shown in figure 7.However,in our experiment,when a = 0,the amplitude of the modulating signal fmis larger thana0/√2,as the spontaneous wave is not monochromatic and has spectral width as seen in figure 4.

Figure 7.Typical FFT signal of the modulated DDW at modulating frequency ～62 Hz and varying modulation amplitude 4 V,8 V,12 V and 15 V with spontaneous frequency f0 ～ 78 Hz.The dependence of the suppression of the self-excited DDW on the amplitude of the modulation voltage is shown here.As amplitude of the modulation signal increases from Vm = (4–8) V,the amplitude of the self-excited DDW gradually decreases and the amplitude of the applied signal gradually increases.At Vm = (12–15) V,the spontaneous DDW is suppressed completely and a subharmonic signal appears at half the spontaneous frequency.

Figure 8.Variation of the relative amplitude of the modulated DDW with the modulation voltage at applied frequency fm ～ 62 Hz.The frequency of the spontaneous DDW is f0 ～ 78 Hz.Above the threshold value of the modulation voltage there is a sudden increase in the relative amplitude of the DDW.

4.Conclusion

The synchronization of a self-excited dust density wave in a dust cloud with 50 nm radius carbon particles is observed in a laboratory plasma.The ion density profile and the plasma potential profile along the axis of the chamber have been measured and indicate the existence of an electric field in the void region.The dust void has been observed at the central region of the plasma,just above the live electrode due to expulsion of dust particles by an outward ion drag force larger than the inward electrostatic force on the dust particles.The electric field force and the ion drag force on the dust particles at the void boundary are also estimated to correlate the force balance at the void boundary,which was earlier predicted in theoretical studies.The measured electric field at the void boundary (4.4 V cm-1) confirms that the ion-streaming velocity is much larger than the ion thermal velocity.Under such conditions the force balance at the void boundary becomes unstable,leading to generation of a dust density wave from the void boundary [22].The frequency of the spontaneous signal in the present experiment is ～78 Hz,which is 2–3 times larger than that of earlier experiments[33,34]because of the lower mass as well as smaller average dust charge of the nanometer-size dust particles.The measured dusty plasma parameters indicate that the Coulomb coupling parameter is greater than 1 and the nanodusty plasma is in a strongly coupled state.

Synchronization of the spontaneous signal is clearly observed when an external signal of frequency fmcloser to the frequency f0of the spontaneous DDW is applied to modulate the ion streaming.The spontaneous frequency f0of the wave is suppressed,and the external signal fmgrows in amplitude.The frequency bandwidth of the interaction is also estimated and is found to increase with higher amplitude of the external signal.It is observed that the amplitude of the spontaneous wave gradually decreases while that of the modulating signal gradually increases with the increase of the modulating voltage at a fixed external frequency.Beyond a threshold voltage,complete suppression of the spontaneous DDW and generation of subharmonics are observed.The phenomenon of synchronization,as observed in our experiment,is explained with the help of the forced van der Pol equation.The synchronization phenomenon has the signature of a suppression mechanism but not of a phase-locking mechanism.Additionally,the generation of chaotic states is not observed in our experiment.

Acknowledgments

One of the authors,Tonuj Deka,acknowledges the Council of Scientific and Industrial Research (CSIR),Govt.of India for CSIR-SRF fellowship (No: 09/835(0026)/2019-EMR-I).

ORCID iDs