Channeling of particles and associated anomalous transport in a 2D complex plasma crystal
Cheng-Ran Du, Vladimir Nosenko, Sergey Zhdanov, Hubertus M. Thomas, Gregor E. Morfill
CChanneling of particles and associated anomalous transport in a 2D complex plasmacrystal
Cheng-Ran Du, ∗ Vladimir Nosenko, Sergey Zhdanov, Hubertus M. Thomas, and Gregor E. Morfill
Max-Planck-Institut f¨ur extraterrestrische Physik, D-85741 Garching, Germany (Dated: September 15, 2018)Implications of recently discovered effect of channeling of upstream extra particles for transportphenomena in a two-dimensional plasma crystal are discussed. Upstream particles levitated abovethe lattice layer and tended to move between the rows of lattice particles. An example of heattransport is considered, where upstream particles act as moving heat sources, which may leadto anomalous heat transport. The average channeling length observed was 15 −
20 interparticledistances. New features of the channeling process are also reported.
PACS numbers: 52.27.Lw, 52.27.Gr, 82.70.Dd
Complex, or dusty plasmas exist in different environ-ments and take various forms [1, 2]. One interesting va-riety is a two-dimensional (2D) plasma crystal [3–18].It is a single-layer suspension of micron-size solid par-ticles in a weakly ionized gas. Due to the high thermalspeed of plasma electrons, microparticles become nega-tively charged and interact with each other via a screenedCoulomb (Yukawa) pair potential. Under certain condi-tions, these charged particles can self-organize in a tri-angular lattice with hexagonal symmetry, forming a 2Dplasma crystal. Individual particle motion can be easilyrecorded in real time using video microscopy. This makes2D complex plasmas excellent model systems where dy-namics can be studied at the level of individual particleswhich can be regarded as proxy “atoms.”Various aspects of 2D complex plasmas that were re-cently studied in experiments include phase transitions[3–5], dynamics of defects [6, 7], microstructure [8], waves[9, 10], mode coupling instability [11], and transport phe-nomena: momentum [12–14] and heat [15, 16] transport,self-diffusion [17], superdiffusion [18]. Systems other thancomplex plasmas were also reported to allow superdiffu-sion: supercooled liquids [19], solid surfaces [20], granularmedia [21].It was recently discovered that single-layer plasmacrystals sometimes include extra particles levitatingslightly above the main layer [22]. Such particles werecalled upstream particles since they are upstream of theflow of ions in the plasma sheath, with respect to thelattice particles. These extra particles move about andcreate disturbances in the lattice layer by both Coulombrepulsion and the effective attraction due to the ion wakeeffect. If a particle moves faster than the sound speedof the plasma crystal, it creates a Mach cone behind it.Compared to other disturbance sources such as a mov-ing extra particle beneath the lattice layer [23] or a laserbeam [24], the disturbance is rather weak and thus hardto notice. In certain conditions, the upstream particlestend to travel between the rows of lattice particles, anal- ∗ Electronic address: [email protected] ogous to channeling of energetic particles in regular crys-tals [25]. θ xy FIG. 1: (Color) Channeling of an upstream extra particle in a2D plasma crystal (a movie is also available [26]). The figureis an overlay of a sequence of thresholded images recorded bya top-view video camera. The upstream particle trajectoryis the curve across the figure starting at the red cross mark.The inset is a zoom-in of the area where the upstream particleencounters a chain of dislocations. Black dots represent thelattice particles in a snapshot taken 1 . . In this paper, we study channeling of upstream extraparticles in a 2D plasma crystal in more detail and showits implications for experiments on transport phenomena.The experiments were performed in a modifiedGaseous Electronics Conference (GEC) rf reference cell[8]. Argon plasma was sustained using a capacitivelycoupled rf discharge at 13 .
56 MHz. The input power wasset at 20 W. We used monodisperse polystyrene (PS) a r X i v : . [ phy s i c s . p l a s m - ph ] N ov particles to create a 2D plasma crystal suspended abovethe bottom rf electrode. The particles have a diameter of11 . ± . µ m and mass density of 1 .
05 g/cm . The gaspressure was maintained at 0 .
65 Pa, corresponding to thegas friction rate γ (cid:39) .
91 s − [27]. The lattice layer wasilluminated by a horizontal laser sheet shining through aside window of the chamber. A high-resolution videocamera (Photron FASTCAM 1024 PCI) was mountedabove the chamber, capturing a top view with a size of42 . × . . The recording rate was set at 250 framesper second.We prepared 2D plasma crystals using a standard tech-nique [7]. After the particles were injected into plasma,they formed a single-layer suspension; in addition, someheavier particles always levitated beneath the main layer[23, 28]. These were mainly agglomerations of two [29]or more particles [30]. We removed these particles fromthe suspension by dropping them on the lower rf elec-trode while reducing the discharge power. However,even after all heavier particles were removed, we some-times observed moving disturbances in the lattice. Thesewere caused by the upstream extra particles that levi-tated above the main layer as was recently discovered inRef. [22]. In the present paper, we study in more detailthe dynamics of upstream extra particles and their ef-fect on the transport phenomena in the main layer. Tomaximize the rate of occurrence of upstream extra par-ticles, we used polystyrene (PS) particles in this study.The reason why plasma crystals composed of PS parti-cles contained upstream extra particles more often thanthose composed of melamine formaldehyde (MF) parti-cles is not clear.An upstream extra particle usually moved about ina horizontal plane approximately 0 .
25∆ higher than themain layer, where ∆ is the in-plane interparticle distance.Since a moving particle must overcome the ambient gasfriction, there must be a driving force acting on it; the ori-gin of this force is not clear. Possible candidates are thewake-field interaction [28] and photophoretic force fromthe illuminating laser [31]. From a different perspective,upstream particles can be regarded as self-propelled (ac-tive) particles [32]. They may be smaller particles fromwithin the natural size distribution or particles with somedefects on their surface. The rather small vertical sepa-ration between the main layer and the upstream particlesallowed us to illuminate and observe them simultaneouslyby properly adjusting the height of the illuminating lasersheet.A particularly long trajectory of an upstream particleis shown in Fig. 1. By tracking its motion and analyz-ing its interaction with the lattice particles, we observedseveral new features of the channeling process. First, thechanneling path is not necessarily straight. As the up-stream particle follows the channel formed by the latticeparticles, it is constantly redirected and the resulting tra-jectory may be curved as in Fig. 1. Here, the total angleof deflection is about 50 ◦ , see Fig. 2(a). The rows of lat-tice particles are curved due to the presence of defects in (c)0 5 10 15 20 25 30 35S (mm)0.60.70.8 ∆ ( mm ) (a)0.00.51.01.52.0 θ (r ad ) (b)0.60.70.80.91.0 < | ψ | > −0.1 0.0 0.1d (mm) −400−2000200400 a ( mm / s ) global bendingdirection of motion ∆ t = − ∆ t = 0 ∆ t = 12 ms ∆ t = 40 ms ∆ t = 80 ms FIG. 2: (Color) Analysis of the channeling event shown inFig. 1. The S -axis represents a natural coordinate systemalong the upstream particle trajectory, where S = 0 is markedby a red cross in Fig. 1. Panel (a) shows the instantaneousdirection of motion θ (black) and the global bending angle(red) of the upstream particle. The inset shows the particletransverse acceleration a as a function of its deviation d fromthe central line of the channel. Color-coding from blue toorange represents timing during 1 .
85 s. Panel (b) presentsthe local bond-orientational order parameter (cid:104)| ψ |(cid:105) (see text)of the lattice at different delay times ∆ t . Panel (c) shows thelocal interparticle separation. The gray stripes correspond tothe chain of dislocations marked in Fig. 1. the lattice, e.g. a chain of dislocations marked by a greystripe in Fig. 1 [38]. This situation is analogous to thechanneling effect in regular crystals. It is well known thatbent crystals are used to redirect energetic particle beams[25]. Despite the totally different energy, space, and timescales involved, our experiment presents the first directobservation of the dynamics of such process.Second, we found that the upstream particle scatteringangle is determined by two competing factors: numberdensity and regularity of the lattice. As an upstreamparticle travels in the channel, it bounces on the chan-nel walls and its trajectory has a zig-zag pattern. Thescattering angle is related to the lattice particle number (a)0 1 2v/c s c oun t (b)13 19 25 31 37
12 mm, see Fig. 3(c). Very long trajecto- ries like that in Fig. 1 were rare. Note, however, that (cid:104) L ch (cid:105) (cid:29) ∆ (the latter was 0 . − .
75 mm). This maypresent a problem for studying transport phenomena inthis system. For example, if one intends to study diffu-sion, the existence of such long particle trajectories ap-parently contradicts the basic assumption that diffusionresults from the random walk of particles.The diffusion coefficient D is usually calculated bymeasuring the mean squared displacement (MSD) of par-ticles in an ensemble [34]: D = lim t →∞ (2 nt ) − (cid:104)| r i ( t ) − r i (0) | (cid:105) , (1)where r i ( t ) is the position of the i th particle and n is thespace dimensionality ( n = 2 in our case). If upstreamparticles are accidentally recorded in an experiment andenter the calculation of MSD in Eq. (1) along with theregular particles, their high velocity (on the order of C s )will lead to a spurious rise in the diffusion coefficient. Infact, in this case MSD will not even have a ∝ t asymptoteat large times, since the displacement of an upstreamparticle is (roughly) proportional to time. Therefore, anaccurate measurement of diffusion coefficient under thesecircumstances is problematic.Furthermore, a moving upstream particle interactswith the lattice particles, particularly those in the chan-nel walls, and transfers some of its momentum and energyto the surrounding particles. The momentum and energywill then spread further in the particle suspension. Thismay affect any measurements of the transport properties.To quantify the disturbance created by an up-stream particle in the lattice, we use the bond-orientational order parameter defined at each site as ψ = n − Σ nj =1 exp(6 i Θ j ), where Θ j are bond orientationangles for n nearest neighbors [35]. In a perfect hexago-nal structure, | ψ | = 1. The smaller | ψ | <
1, the moredisordered the structure is, e.g. | ψ | ≈ .
35 in a disloca-tion core [6]. Here, we average | ψ | within a small circulararea with a radius of 1 . t is the delay time of the mea-surement, where negative and positive values correspondrespectively to the time before and after the upstreamparticle reaches the corresponding site along the S -axis.As can be seen in Fig. 2(b), a dip in (cid:104)| ψ |(cid:105) at S ≈
15 mm corresponds to a chain of dislocations that waspresent in the crystal even before the arrival of the up-stream particle. As the upstream particle travels in thechannel between S ≈ ≈
13 mm, the locallattice structure is slightly disturbed with a decreaseof (cid:104)| ψ |(cid:105) from 0 .
99 to 0 .
93. However, when the up-stream particle encounters the chain of defects, it startsto move more irregularly and the scattering angle in-creases. Though it is still confined in the channel, thedisturbance in the lattice becomes significant and (cid:104)| ψ |(cid:105) drops from 0 .
93 to 0 .
74 (black and blue curves). It takesabout 80 ms for the system to restore initial order; thechain of defects is preserved thereafter, as shown by theyellow curve. (a) y * ( ∆ ) −6−4−20246 (b)10 − − E k ( e V ) −10 −5 0 5 10 15x* ( ∆ )0 5 10 15 20E k (eV) FIG. 4: Upstream particle as a moving heat source. Panel (a)shows a map of in-plane kinetic energy of the lattice particlesaveraged from data for 30 consecutive video frames. The up-stream particle is marked by a circle located at (0,0) and ismoving from right to left. Here x ∗ and y ∗ represent the lon-gitudinal and the transverse axes along the trajectory. Panel(b) shows the in-plane kinetic energy E k of the wall particles.The black plus signs are the means of a gauss-fit of the dataalong the evolution and error bars represent the standard de-viation. As an example of the effect of upstream particles ontransport phenomena, we consider heat transport in thelattice. As discussed in the previous paragraph, a movingupstream particle disturbs the lattice and generates localdisorder. During this process, its kinetic energy is trans-ferred to the lattice particles and therefore it acts as amoving heat source. The transferred heat (in-plane par-ticle kinetic energy) is concentrated on the channel wallsfor about 5 interparticle spacings behind the upstreamparticle, see Fig. 4. Then the heat starts to spread outtransversely. The peak value is located slightly behindthe upstream particle and can on average reach a fewtens of eV.The usual analysis of heat transport (based on the ideaof heat diffusion) is clearly not applicable in this situa-tion. The moving heat source can be properly taken intoaccount in the following way. If we only concentrate onthe kinetic energy of the particles in the walls, namelyin the longitudinal direction, we can simplify the heattransport model to a quasi-one-dimensional model [36]with a thermal diffusivity χ = 2 γL + vL , where L isthe inhomogeneity (gradient) length. The RHS of the equation contains two terms, the first one is associatedwith conduction and the second with convection. Solvingthe equation, we obtain L , = (cid:113) ( v γ ) + χ γ ∓ v γ , cor-responding to the inhomogeneity length in front of ( L )and behind ( L ) the moving heat source, respectively.The experimental kinetic energy profile can be well fit-ted by an exponential function E k ∝ exp( −| x | /L f,b ). Forthe inhomogeneity length in front of the upstream par-ticle, L f = 0 . ± .
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