Dust density waves: ion flows and finite temperature effects

Transcription

1 Dust density waves: ion flows and finite temperature effects Edward Thomas, Jr. Physics Department, Auburn University This work is supported by National Science Foundation and the US Department of Energy through the NSF/DOE Partnership on Basic Plasma Science and Engineering Diagnostics and Simulations of Dusty Plasmas II 1

2 Outline Dust density waves (DDW) examples derivation of the dust acoustic wave effect of ion flows Experiments with DDWs self-excited waves driven waves diagnostics Outstanding issues Summary 2

3 Dust Density Waves Dust density waves refers to the general class of low frequency, often selfexcited waves in a dusty plasma that are characterized by a modulation of the dust number density. 1 cm λ ~ 2 mm Left: DC glow discharge experiment (Auburn) Above: RF microgravity experiment (Kiel) 1 cm DDW Experiments Issues Summary 3

4 Dust Density Waves and Dust Acoustic Waves Dust density waves resemble dust acoustic waves, but are also affected by ion drifts. A. Piel, et. al., [PRL (2006)] From: R. L. Merlino, Univ. of Iowa A. Barkan, et al., PoP (1995) From: V. Fortov, et al., PoP (2000) 4

5 Dust acoustic waves (DAW) - 1 The original derivation of the DAW was given in N. N. Rao, et al., [PSS, (1990)]. The dispersion relation is derived starting with the one-dimensional continuity and momentum equations for the dust component of the plasma: n d t + x n v ( dv d ) = 0 n d d t (continuity) + n d v d v d x = n dq d m d (momentum) ϕ x The system is closed using Poisson s equation and zero-order quasi-neutrality: 2 ϕ x 2 = e ε 0 ( n i n e Z d n d ) n i0 = n e0 + Z d n d0 (Poisson s) (quasi-neutrality) 5

6 Dust acoustic waves (DAW) - 2 The electrons and ions are assumed to obey a Boltzmann distribution: n i = n i 0 exp eφ T i n e = n e0 exp eφ Assuming plane wave solutions, a~a 0 + a 1 e i(kx-ωt) the system of equations is solved to obtain the dispersion relation: T e ω 2 = T k 2 2 i C D ( 1+ k 2 2 λ ) ; where, C D = D 1+ T i λ 2 D = λ 2 2 De + λ Di and ε = n d n i m d T e εz 2 1 εz ( ) 1 2 For many experiments: T i << T e Long wavelength limit: kλ D << 1 6

7 Dust acoustic waves (DAW) - 3 Comparison of the linearized and complete Rao DAW models. Note the correction at small wavelengths (large k s). Linear approximation : ω = kc D Model parameters: Z = 4600 r d = 1.5 µm ρ = 2.0 g/cm 3 n i0 = 1 x 10 8 cm -3 n d0 = 1.35 x 10 4 cm -3 T i = ev T e = 2.5 ev C D = 1.98 cm/s T i /T e = 0.01 ε = 1.35 x

8 Dust density waves (DDW) - 1 In the most general description, the continuity and momentum equations are solved for all three plasma species (α = e, i, d). We allow for: o a pressure term o an electric field, E, which in zero-order - gives rise to drifts o collisions with background neutrals n α t + ( x n α u α ) = 0 u m α n α α t + u α u α x n + k B T α α x n α q α E = m α n α ν αn u α We solve for the zeroth- and first-order terms assuming plane waves: ~e i(kx-ωt) 8

9 Dust density waves (DDW) - 2 The resulting fluid dispersion relation contains the effects of ion drift, thermal effects, and collisions. 1= 2 ω pi ( ) k 2 V + ω pe 2 ti Ω e ( Ω e + iν en ) k 2 V + ω pd 2 te Ω d ( Ω d + iν dn ) k 2 2 V td Ω i Ω i + iν in Where : Ω α = ω ku α 0, ω pα = n 2 α q 2 α ε 0 m, Vtα = k BT α α m α 1 2 A number of authors have studied various forms of the dispersion relation: Kaw and Singh, PRL (1997), Mamun and Shukla, PoP (2000), Merlino and D Angelo, PoP (2005), Piel, et al., PRL (2007), Williams and Thomas, PoP (2008) 9

10 Dust density waves (DDW) - 3 Comparison of the Rao results with the full fluid dispersion relation. The fluid dispersion contains the effects of the ion flow on the waves. Typical experiments: ω ~ rad/s k ~ 2 6 mm -1 f ~ 6 16 Hz λ ~ 1 to 3 mm 10

11 Experiments on DDWs Experiments on DDWs have been ongoing since the earliest days of dusty plasma research. DDWs have been studied in RF and DC glow discharge plasmas, in Q-machine plasmas, in hot filament discharge plasmas, and under microgravity conditions. Two basic classes of experiments are performed: Experiments on self-excited DDWs Experiments on driven DDWs 11

12 DAW/DDW basic properties - 1 Early experiments on DDW/DAW focused on characterizing the basic properties of self-excited waves. The first experimental result was reported by Barkan, et al., PoP, The displacement of single wavefront is recorded using a video camera and a He-Ne laser as the light source. Measurement of the displacement of a wave front giving a velocity of: C D ~ 9 cm/s. 12

13 DAW/DDW basic properties - 2 In another early experiment, measurements of the frequency of DDWs was performed. Here, a photodiode was used to record the fluctuations in the scattered light intensity of a He-Ne laser that illuminated the dust cloud. Dominant f ~ 5.1 Hz From: Prabhakara and Tanna, PoP (1996) 13

14 DDW as a diagnostic for charge - 1 In the long wavelength limit, the phase velocity of the DAW/DDW can be written as: ω k = T i m d εz 2 1 2; ε = n d n i Therefore, it is possible to use the phase velocity of the DDW and the measured dust number density and ion number density to obtain an estimate of the grain charge: q d = -Ze r d = 0.8 µm m d = 6 x kg ε = 2 to 5 x 10-4 T i (est.) = 0.03 ev v phase ~ 12 cm/s slope = 1/v phase Z ~ 1300 From: Thompson, et al., PoP (1997) 14

15 Driven DDWs measuring dispersion relation - 1 A key measurement for DDWs is the determination of the dispersion relation ω(k). Often, this can be challenging with selfexcited waves because of the spread in k-values. That is, a dust cloud is capable of supporting different wavelengths due to variations in the cloud boundaries. Using a driven DDW at a known frequency, it is possible to obtain a map of the dispersion relation. (a) self-excited waves; (b) (g) driven waves: (b) From: Williams, et al, PoP (2008) 19 Hz, (c) 18 Hz, (d) 16 Hz, (e) 14 Hz, (f) 12 Hz, and (g) 10 Hz 15

16 Driven DDWs measuring dispersion relation - 2 y z voltage-controlled, constant current power supply signal generator f ~ 8 to 40 Hz x Anode Tray Experimental setup for driven DDW V anode = 230 V I anode = 3 ma I modulation = 0.1 to 0.4 ma x-y Plane of laser sheet z Motion of translation stage ammeter, I tray dc power supply Experimental approach: vary ω, measure λ, calculate k 16

17 Driven DDWs measuring dispersion relation - 3 The images to the right show the evolution of the wavelength of the DDW as a function of frequency. 10 Hz 14 Hz Within increasing frequency, there is a corresponding decrease in the DDW wavelength. The wavelength is usually determined by a statistical process by fitting the wavelength from the fluctuation in the light intensity. 1 cm 18 Hz self-excited From: Trottenberg, et al, PoP (2008) From Williams, et al., PoP (2008) 17

18 Driven DDWs measuring dispersion relation - 4 With the determination of the wavenumber, k for each applied frequency, the dispersion relation can be plotted. The blue dots represent estimates of the selfexcited DDW (ω,k) recorded at different times during the measurement process. Note what happens as the applied modulation exceeds the self-excited DDW frequency. 18

19 Diagnostics Among the most basic diagnostics for dusty plasmas are Langmuir probes and laser light scattering (LLS). Langmuir probes (single, double, or triple) and emissive probes measure the background plasma parameters. In LLS, a laser sheet is used to scatter light off of the particles and a video camera is used to capture images of the illuminated particles. These two diagnostic systems form the basis for many of the diagnostic approaches used for measuring low temperature laboratory plasmas. 19

20 Diagnostics - Particle Image Velocimetry (PIV) PIV is a two-dimensional, fluid measurement technique originally developed in the mechanical and aerospace engineering communities to investigate flows in fluids. PIV is well-suited to measure transport phenomena in dusty plasmas. In PIV, two laser pulses with a separation time, t laser, illuminate the dust cloud. Lasers, CCD camera, and data acquisition are synchronized so that each laser pulse appears on a single CCD frame. o The time t laser [0.4 µsec t laser 30 msec] can be defined independently of the frame-grabbing rate of the camera. o However, each pair of images are separated by t sep (set by frame grabbing rate of camera). light intensity t laser t sep time 20

21 2-D PIV Stereo-PIV 2D, 2C: 2-dimensions, 2-components 2D,3C: 2-dimensions, 3-components dusty plasma dusty plasma 2 CCD Cameras CCD Camera Lasers Lasers DDW Experiments Issues Summary 21

22 Diagnostics PIV measurements of DDWs - 1 y x z DDW Experiments Issues Summary 22

23 Diagnostics PIV measurements of DDWs - 2 The full PIV hardware is not always necessary to use the PIV analysis techniques. If images can be recorded at a high frame rate (>50 fps), the image quality may be sufficient to use a PIV software package. The movie shows DDWs propagating outward from a heated central region of a plasma crystal. These images were recorded at 60 fps. Free software packages such as MATPIV and URAPIV are available for processing images. From L. Couedel, MPE 23

24 Some outstanding issues with DDWs Significant progress has been made in understanding the properties of dust density waves. The theoretical understanding of these waves has made significant advances in both the development of fluid and kinetic theories Experimenters are also becoming more experienced and adept at carefully designing experiments that can discriminate between the subtle differences among the models. Some examples of outstanding issues: Ion-dust interactions and ion drag forces Dust cloud plasma interfaces and boundary effects Microgravity effects Thermal effects Grain charge variation Collisions 24

25 Boundary effects on DDW In many experiments, the DDWs propagate until reaching the cloud boundary. The mechanisms that dissipate the wave energy or that cause the wave to reflect at the boundary have not been studied extensively. bifurcation From Thomas and Merlino, IEEE TPS (2002) 25

26 Boundary effects on DDW Additionally, the three-dimensional structure of the DDWs evolves throughout the volume of the dust cloud. The images show 6 slices through a cloud from z = 80 to z = 90 mm ( z = 2 mm). From Thomas and Fisher, APS-DPP (2008) DDW Experiments Issues Summary 26

27 Thermal effects on DDW - 1 Finally, it is noted that the fluid dispersion relation is not only dependent upon ion/electron flows, but also upon the temperature of each species. A topic of recent debate is what is the role of the dust kinetic temperature in determining the properties of a dusty plasma? Perhaps a more fundamental question is what is the physical interpretation of the dust kinetic temperature? 1= 2 ω pi ( ) k 2 V + ω pe 2 ti Ω e ( Ω e + iν en ) k 2 V + ω pd 2 te Ω d ( Ω d + iν dn ) k 2 2 V td Ω i Ω i + iν in 2 2 Where : Ω α = ω ku α 0, ω pα = n α q α ε 0 m, Vtα = k B T α α m α

28 Thermal effects on DDW - 2 A recent study by Rosenberg, et al., [PoP, 2008] used a kinetic theory approach to study the possible role of the dust kinetic temperature in DDWs. In this model: T e /T d = 0.05 and T e /T i =

29 Ordered Disordered Ordered Disordered Ordered Disordered Crystal Liquid Ordered Disordered Representative dust kinetic temperature measurements State P (mtorr) T d (ev) Method Reference Weakly-coupled ~50 ~0.1 ~10 ~0.1 ~ ~ ~20 ~2000 ~50 ~300 MKE Melzer, PRE (1996) MKE Quinn, PoP (2000) MKE, NM Ivanov, PoP (2005) MKE Nosenko, PoP (2006) MD Zhakhovskii, JETP Lett. (1997) M,PIV Williams, PoP (2007) Weakly-coupled 160 ~35 DAW Thomas, PoP (2007) No simple conclusion can be drawn about the relevance of T dust or what leads to T dust >> T i, T e. rf plasma / laser excited / dc plasma 29

30 Summary The discovery of the DAW/DDW was an early success for the field of dusty plasmas. Their discovery has led to a number of detailed investigations of the fluid and kinetic properties of these waves. In spite of great progress in understanding the generation and propagation of these waves, a number of questions remain unanswered. This means that many opportunities exist for new researchers to learn more about these waves. And, new phenomena are always being discovered From: S.-H.. Kim, et al., PoP (2008) 30

31 Selected References / Acknowledgements Theory N. N. Rao, et al., Planet. Space Sci., 38, 543 (1990) M. Rosenberg, J. Vac. Sci. Technol. A, 14, 631 (1996) P. Kaw and R. Singh, Phys. Rev. Lett., 79, 423 (1997) A. A. Mamun, et al., Phys. Plasmas, 7, 2329 (2000) R. L. Merlino and N. D Angelo, Phys. Plasmas, 12, (2005) M. Rosenberg, et al., Phys. Plasmas, 15, (2008) Experiments A. Barkan, et al., Phys. Plasmas, 2, 3563 (1995) H. R. Prabhakara and V. L. Tanna, Phys. Plasmas, 2, 3176 (1996) V. E. Fortov, et al., Phys. Plasmas, 7, 1374 (2000) E. Thomas, Jr. and R. L. Merlino, IEEE Trans. Plasma Sci., 29, 152 (2002) T. Trottenberg, et al., Phys. Plasmas, 13, (2006) A. Piel, et. al., Phys. Rev. Lett., (2006) E. Thomas, Jr., et al., Phys. Plasmas, 14, (2007) I. Pilch, et al., Phys. Plasmas, 14, (2007) J. Williams and E. Thomas, Jr., Phys. Plasmas, 15, (2008) S.-H. Kim, et al., Phys. Plasmas, 15, (2008) PIV E. Thomas, Jr., Phys. Plasmas, 6, 2672 (1999) E. Thomas, Jr. and J. Williams, Phys. Plasmas, 13, (2006) Review articles Special issue, Contributions to Plasma Physics, 49, Issues 3 & 4-5, (2009) 31