Laser excitation of long-lived oscillatory states in a dusty-plasma trap

ISSN 1063�7761, Journal of Experimental and Theoretical Physics, 2012, Vol. 114, No. 6, pp. 1072–1077. © Pleiades Publishing, Inc., 2012.Original Russian Text © N.N. Antonov, A.V. Gavrikov, A.S. Ivanov, O.F. Petrov, R.A. Timirkhanov, V.E. Fortov, 2012, published in Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki,2012, Vol. 141, No. 6, pp. 1222–1227.

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1. INTRODUCTION

Using an artificial laboratory dusty plasma, it ispossible to create strongly nonideal model systems.The physical properties of these systems are of consid�erable interest in various fields of basic science andapplied research, including hydrodynamics, plasmaphysics, medicine, polymer science, etc. [1–14].Investigations of the dusty plasma properties are alsoimportant for various technologies, e.g., for the cre�ation of dispersed composite materials, separation ofsubmicron particles with respect to their dimensions,development of the new generation of engines forspace vehicles (colloid thruster), etc. [15, 16].

One of the key tasks in investigating a dusty plasmais to study the interactions of various types (includingelectric [17–19], gasdynamic [20, 21], and optical[22–24]) between particles and their beams [25, 26]and the resulting perturbations that arise in the dustyplasma. Earlier, we studied the effect of laser radiationon the character of volume flow of the dusty�plasmafluid and, in particular, on its viscoplastic properties[23, 24]. In the present work, we continue our investi�gation of the dynamics of particles in a dusty plasmaand consider the effect of laser radiation on the motionof a single particle in a dusty�plasma sheath, which ismanifested, in particular, by their self�excited verticaloscillations.

The oscillations of particles in the plane of a dusty�plasma sheath, which are related to stochastic heatingof a dusty plasma, have been studied earlier as reportedin [27–31], where the characteristic frequencies andamplitudes of these natural oscillations have been

determined in the absence of external driving action.In the present work, we experimentally study a laser�induced long�lived oscillatory state in which a particleoscillates at a large amplitude in the direction perpen�dicular to the plane of a dusty�plasma sheath. Theoscillation amplitude is retained for a long timedespite significant friction in the medium under con�sideration.

2. EXPERIMENTAL SETUP AND PROCEDURE

Experiments were performed using the setup sche�matically depicted in Fig. 1. An RF discharge at a fre�quency of 13.56 MHz and a power of 5 W was gener�ated in a vacuum chamber with a residual air pressureof 0.1 Torr between two flat disk electrodes with cen�tral holes. Macroscopic graphite particles with radiiwithin R = 28–30 μm were introduced from a specialcontainer through the hole in the upper electrode intothe region of discharge. Occurring in dischargeplasma, the dust particles acquired a negative chargeand levitated in the near�electrode layer. A potentialtrap was created in the horizontal layer at the lowerelectrode, where a monolayer dusty�plasma sheathwith a diameter of about 4 cm formed. The externalaction on a single dust particle was achieved using anapproximately 1�second�long pulse of the focusedradiation of an argon laser operating at a wavelength of514 nm at an output power of 1 W, the beam of whichwas directed perpendicular to the plane of the dusty�plasma sheath.

Laser Excitation of Long�Lived Oscillatory Statesin a Dusty�Plasma Trap

N. N. Antonova,b, A. V. Gavrikova,b,*, A. S. Ivanova, O. F. Petrova,b,R. A. Timirkhanova,b,**, and V. E. Fortova,b

aJoint Institute for High Temperatures, Russian Academy of Sciences, Moscow, 125412 RussiabMoscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow oblast, 141700 Russia

*e�mail: [email protected]**e�mail: [email protected]

Received June 20, 2011

Abstract—Vertical oscillations of a single particle in a monolayer dusty�plasma sheath have been excited bya laser radiation pulse. The particle oscillates in a self�sustained regime at a characteristic frequency of aboutν = 25 Hz. It is established that phenomena related to the particle recharge and the delay of its charge varia�tion relative to the equilibrium value cannot account for the self�sustained regime of oscillations. It is sug�gested that the experimentally observed vertical auto�oscillations of a single particle take place due to reso�nant pumping of the kinetic energy of the chaotic motion of particles in the dusty�plasma sheath to the energyof vertical oscillations.

DOI: 10.1134/S1063776112050081

STATISTICAL, NONLINEAR,AND SOFT MATTER PHYSICS

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 114 No. 6 2012

LASER EXCITATION OF LONG�LIVED OSCILLATORY STATES 1073

The motion of individual dust particles was moni�tored using a digital camera with the field of view inthe vertical plane (perpendicular to the plane of thedusty�plasma sheath). The particles were illuminatedby a solid�state laser sheet at a wavelength of 671 nm.The camera operated in the video imaging mode at aframe rate of 50 Hz.

The laser pulse action on a singe dust particle pro�duced its deviation from the equilibrium position andinduced undamped oscillations in the vertical direc�tion (perpendicular to the plane of the dusty�plasmasheath). It should be emphasized that a laser pulseonly produced the initial deviation, while the subse�quent motion of a macroscopic dust particle was freeof any action of the laser beam.

3. PROCESSING AND ANALYSISOF EXPERIMENTAL DATA

In the processing of video images (see Fig. 2a,which shows a typical frame) obtained in variousexperimental series, it was established that the ampli�tude L of the vertical oscillations of macroparticles inan unperturbed dusty�plasma sheath is lower than0.1 mm, whereas the amplitude L1 of laser�initiatedvertical oscillations of a single particle is about0.8 mm, so that L1 � L (see Fig. 2b). Video imageswere used to reconstruct the coordinates of a singlemacroparticle moving upon the action of the laserpulse. Figure 3 shows the typical time series of the par�ticle deviation from the equilibrium position, whichwere constructed based on these data. An analysis ofthese time series showed that the characteristic fre�quency of laser�initiated oscillations was ν ≈ 25 Hz.The amplitude of these vertical oscillations remainedunchanged for a rater long time (t > 20 s) and thenbegins to drop sharply with a characteristic decay timeof t1 ≈ 0.5 s (Fig. 3b). Thus, it was established that asingle dust particles can perform auto�oscillations ini�tiated by the laser pulse.

In the regime of self�sustained oscillations, theenergy dissipated in the course of particle motionshould be compensated by energy supply from anexternal source and the force field acting on the mac�roparticle turns out to be nonpotential [32]. The afore�mentioned investigations of self�excited oscillations ina low�temperature plasma indicated that this nonpo�tential character could be due to the delay of a realcharge on a macroparticle relative to its equilibriumcharge (i.e., the charge acquired by the particle uponleveling of the electron and ion fluxes to its surfaceduring infinitely slow motion in the plasma) [27]. Theequilibrium charge of a macroparticle is self�consis�tently related to the parameters of the surroundingplasma and, if the plasma is inhomogeneous, thischarge depends on the coordinates. In a horizontal flatdusty�plasma trap of an RF discharge, there is a verti�cal inhomogeneity with a profile that has been studiedby Tomme et al. [33]. The corresponding profiles of

Upperelectrode

RF generator

Digitalcamera

FilterMirror

Argonlaser pulse

Solid�statelasersheet

Lowerelectrode

Ringtrap

Vacuumchamber

Fig. 1. Schematic diagram of the experimental setup.

(b)

(а)

Fig. 2. Typical video images showing (a) unperturbeddusty�plasma sheath and (b) track of dust particle underaction of argon laser pulse.

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ANTONOV et al.

the floating potential UF of dust macroparticles werenumerically simulated by Ikkurthi et al. [34]. It wasestablished that macroparticles levitate in the region ofUF growth due to increasing electron energy on thebackground of a weak decrease in the electron density.The gradient of the particle charge that corresponds tothis variation of the floating potential is on the order of10–13 C/m. When a macroparticle moves in the inho�mogeneous plasma, the charge exhibits variations, butthe fluxes of charged particles to the particle surfaceare leveled with a certain time delay rather thaninstantaneously. As a result, the charge of the macro�particle will differ from the equilibrium value, this dif�ference increasing with the particle velocity and thedegree of plasma inhomogeneity. This mechanism canaccount for the fact that positive work of the electricfield on the moving macroparticle will exceed theabsolute value of negative work, thus creating condi�tions for a gain in the energy of the oscillatory motionof the particle.

In order to verify the validity of the hypothesis onthe possibility of maintaining the oscillatory motion ofa macroparticle via the aforementioned energy pump�ing channel, let us consider the following model. Theelectric field near the equilibrium position of levitatingmacroparticles can be expressed as follows:

(1)

where z is the coordinate of the macroparticle oscillat�ing along the vertical axis (directed downward) (z0 cor�responds to the equilibrium position), E0 is the electricfield strength at the equilibrium position, and (dE/dz)0

is the field gradient at the equilibrium position. Then,the force that acts on the macroparticle in the electricfield is, with allowance for the charging delay, as fol�lows:

(2)

where is the particle velocity and τ(z) is the chargingdelay time.

We express the particle charge as follows,

(3)

and since macroparticles levitate in the region ofmonotonic growth of UF over the entire trajectory ofoscillations, we describe Q(z) in the linear approxima�tion as

(4)

Here, Q0 is the equilibrium charge of the particle(in the equilibrium position), (dQ/dz)0 is the derivativeof equilibrium charge at z = 0, and T is the averagecharging delay time per oscillation period.

The equation of motion of the macroparticle in adusty plasma can be written as follows:

(5)

where m is the particle mass, g is the acceleration ofgravity, and β is the coefficient of particle fiction in thebuffer gas. Note that the proposed model ignores theforce of ion entrainment, since estimations of thisforce using the cold ion model [35] showed it was atleast two orders of magnitude lower than the force ofgravity.

Neglecting terms on the second order of smallnesswith respect to z, we can rewrite Eq. (5) as follows:

(6)

where

(7)

and (d(QE)/dz)0 is the derivative of the product equi�librium charge and electric field at z = 0. Using theobtained relations, it is possible to estimate the time τnecessary for operation of the proposed energy pump�

E z( ) E0dEdz��⎝ ⎠

⎛ ⎞0

z,+=

Fe z( ) Q z τ z( )z·–( )E z( ),=

z·

Q z( ) RUF=

Q z( ) Q0dQdz��⎝ ⎠

⎛ ⎞0

z dQdz��τz· .–+=

mz·· βz·– Q0dQdz��⎝ ⎠

⎛ ⎞0

z dQdz��⎝ ⎠

⎛ ⎞0

z·τ–+⎝ ⎠⎛ ⎞–=

× E0dEdz��⎝ ⎠

⎛ ⎞0

z+⎝ ⎠⎛ ⎞ mg,+

z·· 2δz· ω02z+ + 0,=

2δβ dQ

dz��⎝ ⎠

⎛ ⎞0

E0τ–

m��, ω0

2 d QE( )dz

��⎝ ⎠⎛ ⎞

0

1m��,= =

0

25.7t, s

0.8

25.9 26.125.5

−0.4

0.4

(b)

−0.8

1.2

0

21

0.8

22 2320

−0.4

0.4

(a)

−0.8

1.2z, mm

Fig. 3. Time series of (a) amplitude of particle oscillations(z = 0 corresponds to the unperturbed dusty�plasmasheath) with a lifetime of 23 s and (b) decay of oscillationswith a lifetime of 26.1 s.



ing mechanism and determine the frequency of auto�oscillations.

The condition of growth and maintenance of theoscillation amplitude (i.e., the condition of the exist�ence of the auto�oscillatory regime) is as follows:

(8)

In order to estimate the gradient of the macropar�ticle charge that is necessary for the maintenance ofauto�oscillations, let us rewrite this condition in thefollowing form:

(9)

which takes into account that E0Q0 = mg = 2.5 × 10–9 N.Taking into account the other experimental conditionsand using the Einstein formula [36], we estimate thefriction coefficient as

(10)

where η = 1.8 × 10–5 N s/m2 is the viscosity of air, andl = 5 × 10–4 m is the mean free path of molecules in thebuffer gas. The equilibrium charge Q0 ≈ 105e was esti�mated using the approximate theory of finite orbits[37] with the plasma electron and ion temperaturesassumed to be Te = 2 eV and Ti = 0.03 eV. Substitutingthe adopted values into formula (9) yields the followinglower limit of the particle recharge time: τ ≥ 2 × 10–3 s.However, under the real experimental conditions, thecharacteristic charging (and, hence, recharging) time ofa single dust particle amounts to approximately 10–5 s[38]. Therefore, the self�excitation of vertical oscilla�tions observed in experiment cannot be explained bythe proposed charging model.

Estimation of the oscillation frequency using rela�tions (7) yields ω ≈ 11 s–1, which is about one order ofmagnitude lower than the observed values. Thus, thecalculated values of parameters (τ, ω) are obviously atvariance with the measured data. These discrepanciesapparently evidence a significant disagreementbetween the characteristics of plasma inhomogeneityobtained in [33] and the real parameters of a dusty�plasma trap, or, they probably indicate that some othermechanism may account for the energy pumping to adust particle in the course of its vertical oscillations.It is interesting that, if the charge of a dust particle isformally set proportional to the electric field potential(Q = RU) at a given point, then the adopted approachgives values of the oscillation frequencies and charac�teristic charge delay times that agree quite well withthe experimental data.

δ 0, i.e., β dQdz��⎝ ⎠

⎛ ⎞0

E0τ– 0.≤ ≤

τβQ0

dQdz��⎝ ⎠

⎛ ⎞0

mg

��,≥

β 6πηD2

4 1.1l×�� 5.6 10 10–

N sm

��,×= =

One possible hypothesis that explains the appear�ance of self�sustained vertical oscillations is based onthe assumption about the interaction of an oscillatingmacroparticle with other particles of the given dusty�plasma sheath (Fig. 4). According to this, the otherparticles create an electric field that acts along the ver�tical axis z upon a macroparticle that is driven by alaser pulse out of the equilibrium position. During itsmotion along axis z, particle O interacts with particlesA and B which, in turn, interact with the rest of thesheath and acquire an additional energy dependent onthe kinetic energy (temperature) of the sheath. Thus, asituation is possible whereby particle O and particles Aand B move in phase downward and the former parti�cle gives a fraction of its momentum to particles A andB, and these are inelastically reflected from other par�ticles of the sheath so as to acquire an additionalmomentum dependent on the temperature of thesheath. When particle O moves in the reverse direc�tion, it receives increased momentum from particles Aand B. In other words, the resonance between particleO moving in the vertical direction and particles A andB moving in the lateral direction accounts for thepumping of kinetic energy from the chaotic motion ofparticles in the dusty�plasma sheath to the energy ofvertical oscillations of a single macroparticle. Theimportant role of the energy transfer between verticallyand horizontally moving particles was pointed out ininvestigations of the anomalous heating of dust parti�cles in plasma [39], where it was shown that, as theamplitude of horizontal oscillations approaches thatof vertical oscillations, a second parametric resonanceappears that “heats” the vertical oscillations due tohorizontal motions. Evidence in favor of the proposedhypothesis is again provided by the fact that the fre�quency of vertical oscillations observed in experimentsis about 25 Hz, which almost exactly coincides withthe oscillation frequency of dust particles in the sheath(νd ≈ 19 Hz) determined by the laser�excited reso�

FB

A B

FA

FA+B

Fig. 4. Particles of a monolayer dusty�plasma sheath createan electric field that acts in the vertical direction upon amacroparticle that moves through the sheath during laser�initiated oscillations.

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ANTONOV et al.

nance technique [40]. It should be noted that thebreakage of a self�oscillatory regime and the decay ofvertical oscillations can be explained by the proposedhypothesis via fluctuations that lead to violation of theresonance conditions.

4. CONCLUSIONS

We have established that vertical oscillations of asingle particle in a monolayer dusty�plasma sheath canbe excited by a laser radiation pulse. The particle oscil�lates in a self�sustained regime at a characteristic fre�quency of ν ≈ 25 Hz. It has been demonstrated thatphenomena related to the particle recharge and thedelay of its charge variation relative to the equilibriumvalue cannot account for the self�sustained regime ofoscillations using existing models with commonlyaccepted parameters. We have suggested that theexperimentally observed vertical auto�oscillations of asingle particle take place due to resonant pumping ofthe kinetic energy of the chaotic motion of particles inthe dusty�plasma sheath to the energy of vertical oscil�lations.

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Translated by P. Pozdeev

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Laser excitation of long-lived oscillatory states in a dusty-plasma trap