Interaction force in a vertical ust chain insie a glass box Jie Kong, Ke Qiao, Lorin S. Matthews an Truell W. Hye Center for Astrophysics, Space Physics, an Engineering Research (CASPER) Baylor University Waco, Texas 76798-731 Abstract Small number ust particle clusters can be use as probes for plasma iagnostics. The number of ust particles as well as cluster size an shape can be easily controlle employing a glass box place within a GEC rf reference chamber to provie confinement of the ust. The plasma parameters insie this box an within the larger plasma chamber have not yet been aequately efine. Ajusting the rf power alters the plasma conitions causing structural changes of the cluster. This effect can be use to probe the relationship between the rf power an other plasma parameters. This experiment employs the sloshing an breathing moes of small cluster oscillations to examine the relationship between system rf power an the particle charge an plasma screening length insie the glass box. The experimental results provie inicate that both the screening length an ust charge ecrease as rf power insie the box increases. The ecrease in ust charge as power increases may inicate that ion trapping plays a significant role in the sheath. 1. Introuction Complex plasmas, partially ionize gases containing small, usually micron-size, ust particles, are of great interest in part because they provie an ieal an versatile analogue to the stuy of finite charge-particle systems. The ust particles collect ions an electrons from the plasma generally obtaining a negative charge ue to the higher mobility of the electrons. In laboratory experiments on Earth, gravity causes the ust particles to settle within the plasma, where they are then levitate by the electrostatic fiel above the lower powere electroe which is also negatively charge within the plasma chamber. Utilizing a glass box place on the lower electroe to provie strong horizontal confinement allows clusters with small numbers of particles to be easily manipulate, an multiple an/or single vertical chains to be forme. Previous experiments have shown that the structural transition between multiple vertical chains an a single vertical chain in the glass box is achieve by varying the
system rf power. This transition is assume to be ue to the variation in the plasma ionization rate that occurs uring such a change in system rf power [1]. The goal of this investigation is to experimentally explore the relationship between system rf power an global usty plasma parameters such as the Debye length an ust charge. The paper is structure as follows: Section presents the theoretical backgroun for the experiment an analysis. Section 3 escribes the experimental setup an presents the results. In Section 4, a etaile analysis of the experimental ata is provie. Finally, a iscussion of these results is given in Section 5 while conclusions are given in Section 6.. Theoretical backgroun The total energy of an N-particle ust cluster within an anisotropic, horizontally irecte confinement is given by [ 5], 1 Q e U m x y 4 r N i i (1) i1 pairs ij where x i, y i are the horizontal an vertical coorinates respectively, r ij is the separation istance between the i th an th j particles, m an r ij Q are the ust particle s mass an charge respectively, y is the frequency of oscillation in the vertical irection, x y is the anisotropy parameter ( is the frequency of oscillation in the horizontal irection) an 1 D x is the shieling parameter. The Yukawa potential seen in the secon term on the right han sie of Eq 1 represents the interaction between the ust particles. The equilibrium separation istance between ust particles Req an the relationship between their breathing an sloshing frequencies may be erive from the conitions ( U an U respectively). In this experiment, a vertically aligne two-particle system was chosen to allow examination of the interparticle interaction. A two-particle system was chosen ue to the fact that this structure provies the simplest form of Yukawa interaction an is therefore easiest to analyze. Accoring to Eq 1, the total potential energy of a vertically aligne two-particle system U is U R 1 R R Q e m 4 R () where R is the separation istance between the two particles. The equilibrium separation istance Req is etermine by setting U,
1 Q 1 Req U mreq e R Req 4 Req R eq (3) 1 while the breathing frequency is foun by efining U m R br, eq R Q eq e br 3 Req Req R eq m 4 (4) yieling a relationship between br an sl, the sloshing frequency. As shown, both sl an br are functions of the screening parameter an the ust charge 3. Experiment an results The experiment escribe here was conucte within one of two moifie Gaseous Electronics Conference (GEC) rf reference cells locate in the CASPER lab at Baylor University [6]. An open-ene glass box having imensions of 1 mm 1.5 mm (height with) an a glass wall thickness of mm was place on the powere lower electroe to provie enhance horizontal confinement for the ust particles (Fig. 1). Q.
Fig 1. Experimental setup. The open-ene glass box shown here has imensions of 1 mm 1.5 mm (height with). Dust particle oscillation was generate through moulation of an external DC bias introuce on the lower electroe. Plasma was prouce in Argon gas having a neutral gas pressure of Pa by proviing 4 W of rf power. Melamine formalehye ust particles 8.89 ±.9 μm in iameter were introuce into the plasma via shakers mounte above the hollow upper electroe. Uner these conitions, ust particles confine insie the glass box forme a turbulent clou. Slowly lowering the rf power cause the ust clou to stretch in the vertical irection while simultaneously shrinking in the horizontal irection, reucing the total number of trappe particles through particle loss to the lower electroe. Lowering the rf power further then create a single vertical ust chain. Once stable, the length of this ust chain coul be shortene by slightly lowering the rf power, which cause the lowest particle in the chain to enter a region with an unstable force balance, removing it from the chain [6]. On the other han, by slowly increasing the rf power the overall structural configuration of the chain can
be change starting from a single vertical chain, through a two-imensional zig-zag structure, to multiple chains exhibiting three-fol symmetry or above. One of the more interesting structures exhibiting three-fol symmetry is the helical structure shown below. See Fig (a e). Fig. Sie view of helical structures forme from 1 particles at varying rf powers using the technique escribe in the text. In all cases, the backgroun pressure is hel at Pa. (a) 1D single chain, rf power 1.3 W. (b) D zigzag structure, rf power 1.7 W. (c) 3D three-chain helical structure, rf power 1.3 W. () 3D four-chain helical structure, rf power 1.39 W. (e) 3D five-chain helical structure, rf power 1.8 W. Due to the with of the laser sheet use for illumination an the specific portion of the chain image by the camera, not every particle in each structure is visible in these sie view images. As state above, a two-particle vertical chain was chosen to test the valiity of Eqs 3 an 4, ue to its stable structure uring force oscillation uner ifferent rf powers. Fig 3 shows the relative vertical positions an interparticle separations for such a two-particle chain uner various rf powers.
Fig 3. Vertical height an interparticle separation variations for a two-particle chain uner various rf powers. From (a) to (e) the corresponing rf powers are 1.3, 1.7, 1.3, 1.39 an 1.8 W, respectively. These power settings are ientical to those use for transitioning from a single-chain through a five-chain structure when ten particles are confine within the glass box (Fig ). Force oscillation of the two-particle chain was achieve by moulating the DC bias on the lower electroe employing an externally generate, low frequency sinusoial signal. The riving amplitue of this signal was limite to1 mv (peak to peak) measure before a 5 B attenuator in orer to ensure all particle oscillations remaine in the linear regime. A sie mounte camera collecte ata on vertical chain oscillations at 15 frames per secon (fps). Fig 4 shows the frequency response spectra for both the sloshing an breathing moes occurring at 1.3 W rf power from riven oscillations. At all power settings, the peak sloshing amplitue is greater than that observe for the breathing moe. The relative ifference between the two moes increases as rf power increases (Fig 5a). This is in goo agreement with moe analysis from ata prouce by the ust thermal motion which shows the same results (Fig 5b) [7 9].
Fig 4. (a) Frequency response ata for vertically riven oscillation ata for a two-particle chain at a rf power of 1.3 W. Symbols represent experimental ata while soli lines represent the theoretical fit as explaine in the text. (b) Moe analysis showing the same values for sloshing an breathing oscillation frequencies as erive from the riven
oscillation metho. The four moes shown are the 1) horizontal relative, ) vertical sloshing, 3) horizontal sloshing an 4) vertical breathing moes. The moulation of the riving frequency was varie from 1 Hz to 3 Hz in orer to obtain the frequency response spectra necessary for analysis. The theoretical fit to the experimental ata shown in Fig 4a is etermine employing the formula for the amplitue of a ampe, riven oscillator [1, 11]. A a (5) where a is a constant, is the neutral rag coefficient, is the resonance frequency (i.e., epening on the reference system can represent either the sloshing or breathing frequency), an is the riving frequency. a, an are all etermine by fit to the experimental ata. Experimental results for the breathing an sloshing frequencies, their ratio an the equilibrium separation as a function of the rf power can each be etermine experimentally an are shown in Fig 5.
Fig 5. (a) Breathing an sloshing frequencies, (b) the square of the ratio of the breathing to sloshing frequency as a function of the rf power, an (c) the equilibrium separation istance between the two particles as a function of the rf power. Soli lines are inclue to guie the eye.
As shown, both breathing an sloshing frequencies increase as the rf power increases. While the square of the ratio of the two frequencies square also increases with rf power, the range is boune by br sl.7. Both the frequency ratio an Req approach limiting values as the power is increase. 4. Analysis Eqs 3 an 4 are functions both of experimentally etermine values, R eq, unknown values, sl an br, an Q an. For (the Coulomb interaction limit), a relationship between the breathing an sloshing frequency can be erive, 3 (6) br sl setting a lower limit on the breathing frequency. Fig 5(b) inicates that all experimentally etermine values for the frequency ratio.7 inicating that Eqs 3 an 4 are incomplete. Assuming small values of across the rf power settings use, br sl R eq, this iscrepancy between experimental results an theoretical formulation may be correcte by aing a linear force term, KR eq, to Eq 3, where K has yet to be ientifie. Eqs 3 an 4 may now be re-written as, 1 Q 1 Req U m Req e KR R Req eq 4 Req R eq (7) R Q eq e br U m K 3 m 4 Req Req R eq m (8) Unfortunately, with this aition there are now three unknowns, K, Q an in Eqs 7 an 8. This issue can be ameliorate in part by examining the system in the horizontal plane. In the horizontal irection, ust particles arrive at equilibrium upon achieving a balance between the confinement prouce by the glass walls an the Yukawa interaction between neighboring particles. Therefore, assuming the glass walls exert a horizontal Yukawa confining force to the ust particles, at equilibrium this confining force will be balance by the horizontal component of the interaction force prouce by the ust particles themselves, F exp R exp k Q r j (9) k Rk 4 j rj
In Eq 9, the left-han-sie represents the horizontal force prouce by the glass walls, with R k efine as the istance between the th k wall an the representative ust particle, r. The right-han-sie represents the horizontal component of the interaction force between the ust particle an all neighboring particles. For clusters with small total numbers of particles (for example N 1 as shown in the following figure) at the same rf power F an Q can be assume constant, i.e., F an Q are functions of the rf power alone. Representative clusters an corresponing rf powers are shown in Fig 6. Fig 6. Small clusters with (a) six particles at 1.69 W, (b) nine particles at 1.47 W an (c) twelve particles at 1.36 W. In this case, particle positions are measure experimentally for clusters having six, nine an twelve particles an forme at 1.36 W, 1.47 W an 1.69 W rf power. Applying Eq 9 to these clusters at the same rf power, the screening parameters may be etermine as shown in Fig 7.
Fig 7. Experimental result of the screening parameter as a function of the rf power. Using a linear fit for the Debye length as a function of the rf power, the Debye length for the conitions shown in Fig a e are given in Table 1. Substituting these values into Eqs 7 an 8 allows the ust charge Q an linear force constant K to be erive. These are shown in Fig 8. Table 1 RF Power (W) 1.3 1.7 1.3 1.39 1.8 λ D (µm) 64 63 576 539 3
Fig 8. (a) Dust charge, an (b) ratio of the linear force constant K to of the rf power. Fit lines serve to guie the eye. 5. Discussion m as a function As can be seen in Fig 7, the Debye length, 1, ecreases as the rf power increases. D Given the Debye length is inversely relate to the plasma ensity, such a ecrease in Debye length an increase in electron an ion ensities can best be explaine by the increase in ionization occurring with increasing rf power. sl
Therefore, a ecrease in rf power will prouce an increase in Debye length, proviing less shieling to the ust particles within the box from the potential on the glass walls comprising the box. (This assumes a Yukawa type confinement force.) This is the primary reason for the structural transition from multi-chain to single chain, which is observe as the rf power ecreases as shown in Fig. Fig 8b shows that the linear force term constant K as efine for Eqs 7 an 8 remains relatively uniform at approximately % of the value of the sloshing frequency, m, over the rf power regime where multiple chains form. However, as the rf power ecreases below 1.4 W, the rf power regime where single chains form, K rapily increases to 4% of the sloshing frequency. As such, this force provies a significant contribution to Eqs 7 an 8. One possible reason for this result has to o with the potential on the surface of the glass walls. The force prouce by the glass walls establishes much of the horizontal confinement acting on the ust particles. However, previous experiments have shown that the glass box also provies a vertical force on the ust particles [11]. This vertical force from the glass box is the primary contributor to the linear term KR eq. A secon possible contributor is the ion focusing effect. The wake effect ue to streaming ions has been shown to create a charge ifference between two vertically aligne ust particles an to create a positive charge space region below the upper particle [1]. Thus the ion focusing effect introuces an aitional attractive force between the ust particles [13, 14]. Finally, it has been note that ion-neutral charge exchange collisions can lea to trappe ions aroun ust grains, which reuces the grain charge substantially in laboratory plasmas ue to the increase in ion-neutral collisions with increasing plasma ensity (15, 16, 17, 18). However, all previous stuies examine only the effect ue to trappe ions in an isotropic plasma, neglecting the effects of flowing ions or graients in the electric potential, both of which exist in the plasma sheath. The fact that the measure charge on a ust grain in this experiment ecreases with increasing plasma power inicates that such collisional effects may well play a significant role in grain charging, even in the sheath. 6. Conclusions The total potential energy of a vertically aligne, two-particle ust cluster system an its first an secon erivatives were employe to investigate the relationship between the system s sloshing an breathing frequencies. These were then employe to etermine the Debye length an ust charge as a function of the rf power. In orer to moel the relationship between the frequencies of the breathing an sloshing moes, an aitional linear force term (see Eqs 7 an 8), KR, was introuce to solve the sl
iscrepancy between the values preicte theoretically, (see Eqs 3 an 4) an the experimental results. The origin of this linear force term was examine an two possible mechanisms, the glass box potential an ion focusing effect, were propose. The experimental results provie show that the Debye length ecreases as the rf power increases, explaining the transition observe for ust particle bunles confine within a glass box from vertically aligne single chains to multiple chains as the rf power increases. The relationship between the ust charge an the system rf power was also examine an is shown in Fig 8a. These results inicate that ion-neutral collisions can lea to trappe ions aroun the ust grains an play a significant role in the grain charging process insie the glass box. References 1. Truell W. Hye, Jie Kong, an Lorin S. Matthews, Helical structures in vertically aligne ust particle chains in a complex plasma, Phys. Rev. E87, 5316, 13.. A. Melzer, Zigzag transition of finite ust clusters, Phys. Rev. E, 73, 5644, 6. 3. Lair Canioy, Jose-Pero Rinoy, Nelson Stuarty an Francois M Peetersz, The structure an spectrum of the anisotropically confine two-imensional Yukawa system, J. Phys. Conens. Matter, 1, 1167 11644, 1998. 4. T. E. Sherian, an K. D. Wells, Dimension phase transition in small Yukawa clusters, Phys. Rev. E, 81, 1644, 1. 5. T. E. Sherian, an Anrew L. Magyar, Power law behavior of the zigzag transition in Yukawa clusters, Phys. Plasmas, 17, 11373, 1. 6. Jie Kong, Truell W. Hye, Lorin Matthews, Ke Qiao, Zhuanhao Zhang, an Angela Douglass, One-imensional vertical ust strings in a glass box, Phys. Rev. E84, 16411, 11. 7. A. Melzer, Moe spectra of thermally excite two-imensional ust Coulomb clusters, Phys. Rev. E67, 16411, 13. 8. Ke Qiao, Jie Kong, Zhuanhao Zhang, Lorin S. Matthews, an Truell W. Hye, Moe Couplings an Conversions for Horizontal Dust Particle Pairs in Complex Plasmas, IEEE Trans. Plasma Sci., 41, 741 753, 13. 9. Ke Qiao, Jie Kong, Eric Van Oeveren, Lorin S. Matthews, an Truell W. Hye, Moe couplings an resonance instabilities in ust clusters, Phys. Rev. E88, 4313, 13. 1. N. J. Prior, L. W. Mitchell an A. A. Samarian, Determination of charge on vertically aligne particles in a complex plasma using laser excitations, J. Phys. D: Appl. Phys., 36, 149, 3. 11. Jie Kong, Ke Qiao, Jorge Carmona-Reyes, Angela Douglass, Zhuanhao Zhang, Lorin S. Matthews, Truell W. Hye, Vertical interaction between ust particles confine in a glass box in a complex plasma, IEEE Trans. Plasma Sci., 41, 794 798, 13.
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