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Introduction to Dusty Plasmas

Introduction to Dusty Plasmas
Introduction to Dusty Plasmas André Melzer Institute of Physics, ErnstMoritzArndtUniversität Greifswald Germany Extended Lecture Notes see: www5.physik.unigreifswald.deDusty (Complex, Colloidal) Plasmas Dynamics and transport in plasmas Coulomb crystallization of trapped particles Dusty Plasmas = M Miic cr ro os sc co op piic c p pa ar rt tiic clle es s iin n a a gaseous plasma environment Selwyn 1991 … Etching Deposition and… AstrophysicsDusty Plasmas in Astrophysics Comet HaleBopp Saturn ringsDusty Plasmas in Astrophysics Saturn rings 1981 2005Dusty Plasmas in the Atmosphere Noctilucent cloudsDusty Plasmas in Technology Plasma etchingDusty Plasmas in Technology Selwyn 1991 „Killer particle“Dusty Plasmas in the LaboratoryDusty Plasmas under MicrogravityContents of Lecture • Dust charging • Forces • Strongly coupled systems, particleparticle and particleplasma interaction • Waves • Finite systems and normal modes For an extended introduction see: www5.physik.unigreifswald.deCharging plasma Q Q 0 0 ions electrons In typical discharges: Particle will be charges negatively due to higher mobility of electronsOML currents Probe theory of Langmuir and MottSmith 1929 Ion current     8kT eΦ Φ Φ Φ 2 i   I = = = = π π π πa n e 1 − − − − i i     π π π πm kT i  i  increased collection cross section geometry thermal velocities density reduction Boltzmann factor   8kT eΦ Φ Φ Φ 2 e I = = π πa n e exp  = = π π e e electron     π π π πm kT e  i  and Electron current ion currentsParticles as floating probes I (Φ Φ Φ Φ ) = = = = 0 ∑ ∑ ∑ ∑ q fl q  eΦ Φ  eΦ Φ Φ Φ  Φ Φ    T m n fl fl With OML collection e i e     1− − − − = = = = exp       currents only kT T m n kT i i e i  e    Values of eΦ Φ Φ Φ /kT for n =n fl e e i T /T 1 10 20 100 e i 2.50 1.91 1.70 1.24 H Φ Φ Φ Φ ≈ ≈ ≈ ≈ − − − −2kT /e fl e 3.05 2.39 2.16 1.65 He 3.99 3.24 2.99 2.41 ArLimitations of OML description • collisionless (ion) trajectories none of that is • Maxwellian velocity distribution met in „real“ discharges • isotropic Often: ion drift velocity u (much) larger than i ion thermal velocity v SHEATH th,i e ex xa ac ct t   8kT eΦ Φ Φ Φ 2 i solution I = = π πa n e 1 − −  = = π π − − i i   π π π πm kT i  i    approximation   2eΦ Φ Φ Φ 2 I = = = = π π π πa n eu 1 − − − −  i i i 2     m u  i i The capacitance model Particle as a spherical capacitor Q = = C Φ Φ = = Φ Φ fl of capacitance C Capacitance in vacuum C = = = = 4π π π πε ε ε ε a 0 For a particle of a=1µm: 700 e per Volt With the ruleofthumb approximation: Φ Φ Φ Φ=2kT /e e Q = = = = 1400a T μ μ μ μm e,eVOther charging currents Photoelectron emission Particle can become (UV radiation) positively charged electrons hν ν ν ν 2 2 S Se ec co on nd da ar ry y e elle ec ct tr ro on n e em miis ss siio on n II = = = = = = = = π π π π π π π πa a e eμ μ μ μ μ μ μ μ Γ Γ Γ Γ Γ Γ Γ Γ Φ Φ Φ Φ Φ Φ Φ Φ 0 0 ν ν ν ν fl ν ν ν ν 2 I = = = = π π π πa eμ μ μ μ Γ Γ Γ Γ exp(− − − −eΦ Φ Φ Φ /kT ) Φ Φ Φ Φ 0 ν ν ν ν fl p fl ν ν ν ν electrons   E E e e   δ δ δ δ(E ) = = = = 7.4δ δ δ δ exp − − − − e m   E E m m  Charging time scale kT 1 i τ τ τ τ = = = = RC τ τ = = 4π πε ε a τ τ = = π πε ε i 0 2 e π π π πa en v i th,i Time constant for charging of a capacitor 1/I C U Smaller particles 1 τ τ τ τ ∝ ∝ ∝ ∝ i are charged slower a 1 µs Plasma timeSummary Charging • Micrometer sized particles carry 3 4 10 to 10 elementary charges • Charging time: microseconds: Charge in dynamical equilibrium • • C Ch ha ar rg ge e t to o m ma as ss s r ra at tiio o Q Q/ /m m e ex xt tr re em me elly y s sm ma allll: : slow timescalesForces on dust particles • Gravity • Electric field force • Thermophoresis • Ion Drag • Neutral DragGravity r r v 4 3 F = = = = mg = = = = π π π πρ ρ ρ ρa g 3 What else needs to be said Electric Force r r r F = = = = QE = = = = 4π π π πε ε ε ε aΦ Φ Φ Φ E 0 Also for a charged particle with (symmetric) shielding cloud dust r r F = QE shielding Q cloudDrag Forces „streaming“ species A dust p pa ar rt tiic clle e v dt rel Force = momentum transfer x Α Α Α Α x density x velocity of incident particlesNeutral Drag „streaming“ species A p dust Δ Δ Δ Δp particle p‘ v v d dt t r r 4 2 NB: F ≅ ≅ ≅ ≅ − − − −δ δ δ δ π π π πa m v n v n th,n n rel Stokes friction 3 F a 8 p F = = = = − − − −mβ β β βx β β β β = = = = − − − −δ δ δ δ (aλ λ λ λ) π π π π ρ ρ ρ ρv a th,nIon Drag 2 components: 1. Collection Force 2. Coulomb Force 1. Collection Force r r   2 2e eΦ Φ 2   F = πa 1− n u m u i i i i 2   mu  i  cross section as for charging2. Coulomb force r F = σ n u m u coul i i i i p‘ Coulomb scattering cross section 2 σ σ σ σ = = = = π π π πb ln Λ Λ Λ Λ coul π π π π /2 Δ Δ Δ Δ Δ Δ Δ Δp p p p impact parameter for 90° collisions Qe b = = = = π π π π /2 2 4π π π πε ε ε ε mu 0 i i Coulomb logarithm   b max   ln Λ Λ Λ Λ = = = = ln   b  min Thermophoresis Force due to a temperature gradient in the neutral gas ∇T 2 r r 16 a k „hot“ „cold“ n F = = = = − − − − π π π π ∇ ∇ ∇ ∇T 15 v th,n F Fo or rc ce e t to ow wa ar rd ds s c co olld de es st t p po oiin nt t FComparison of forcesTrapping (Laboratory) Electrode F F F F E th g ion sheath P Plla as sm ma a sheath F F F F E th g ion Electrode Trapping in the plasma sheathTrapping (Microgravity and nanometric particles) Electrode F ion F F E th P Plla as sm ma a F F E F th ion sl. 9 Electrode Trapping in the plasma volumeParticle trapping in the laboratory ′′′′ Q E 2 0 ω ω ω ω = = = = 0 m sheath edge Plasma E V 0 E E QE (z ) particle (z) E 0 z 0 mg V(z) z electrode zResonance method Force balance Q(z )E(z ) = = = = mg 0 0 Equation of motion mz + + + + mβ β β βz + + + + Q(z)E(z) = = = = F ext Assumption: constant charge Q(z) = = Q = = 0 ′′′′′′′′ E E( (z z) ) = = = = = = = = E E( (z z ) ) + + + + + + + + E E ( (z z − − − − − − − − z z ) ) Linear electric field 0 0 1 1 2 2 2 ′′ ′′ Potential well mω ω (z − − z ) = = Q E (z − − z ) ω ω − − = = − − 0 0 0 0 2 2 ′′ ′′ Q E 2 0 Resonance frequency ω ω ω ω = = = = 0 mLinear ResonancesCharge measurementSummary Forces • Laboratory: electric field force + gravity: Trapping in the sheath • Microgravity: electric field force + ion drag: T Tr ra ap pp piin ng g iin n t th he e p plla as sm ma a v vo ollu um me e ( (v vo oiid d) ) • Weakly damped particle dynamicsStrongly coupled systems One Component Plasma: (Wigner 1938, Brush et al. 1966) Coulomb Thermal energy energy Solid Phase Γ Γ Γ Γ Γ Γ Γ Γ = = = = 168 c Γ Γ Γ Γ Γ Γ Γ Γ Fluid Phase cYukawa systems Robbins et al. 1988 2   Q r φ φ(r) = = exp − −  φ φ = = − −   4π π π πε ε ε ε r λ λ λ λ 0  D  b b κ κ κ κ = = = = κ κ κ κ = = = = λ λ λ λ λ λ λ λ D Screening strength Crucial parametersPlasma crystals cf: Chu et al. 1994, Thomas et al. 1994, Hayashi et al. 1994Interaction • Horizontal Interaction repulsive Yukawa (DebyeHückel) type (Konopka 2000) • Vertical Interaction attractive forces originVertical Order Attractive forces in the sheath sheath edge Plasma iio on ns s E E 0 0 E QE(z ) particle (z) E 0 z 0 g m electrode zVertcal order: Simulations 1. Ion focus: Attraction 2. supersonic ion flowtion: Nonreciprocal forces, Only the lower particle experiences attractionVertical Order: Experiment 1. Particle: 3.47 μm 2. Particle: 4.18 μm Vertical: force balance Horizontal: free motionVertical Order: Experiment (2) The lower particle experiences attractionVertical Order: Experiment (3) The upper particle does not experience attractionSummary Crystallization • Strongly coupled systems with Yukawa interaction • Attractive forces in the sheath due to ion flow (ion focus, ion wake field)Waves in strongly coupled dust: Dust lattice waves Compressional and Shear Waves in a 2D lattice Transverse Wave modeDust lattice waves: theoretical treatment (n1) b n b (n+1) b (n+2) b x x x x n−1 n n+1 n+2 lliin ne ea ar r c ch ha aiin n w wiit th h m mx x = = = = k k( (x x − − − − 2 2x x + + + + x x ) ) = = = = − − − − + + + + n n+ + + +1 n n− − − −1 spring constant k x = Aexp(inqb − iωt) n 2 iqb −iqb − ω m = k(e + e − 2) = 2k(cosqb −1) k qb 2 2 ω = 4 sin dispersion relation of a linear chain m 2Dust lattice waves: theoretical treatment (n1) a n a (n+1) a (n+2) a x x x x n−1 n n+1 n+2 2 2 2 2   d d V V Q Q r r s sp pr riin ng g c co on ns st ta an nt t f fr ro om m   k k = = V V( (r r) ) = = exp exp − − 2   Yukawa potential dr 4πε r λ 0  D  r=b 2 Q 2 ( ) k = exp(− κ ) 2 + 2κ + κ 3 4πε b 0 2 Q qb dispersion relation w.   2 2 2 ( )( ) ω = exp − κ 2 + 2κ + κ sin   3 Yukawa potential πmε b 2   02D dust lattice waves • many (infinite) neighbors • 2D hexagonal structure • compressional and shear mode qb qbDust lattice waves: 1DDust lattice waves : 1DDust lattice waves : 1D Re q ξ ξ ξ ξ ∝ ∝ ∝ ∝ exp(iqx − − − − iω ω ω ωt) Dispersion ω ω ω ω(q) Driven wave: ω ω ω ω real q q c co om mp plle ex x k qb   2 2 Im q ω ω ω ω + + + + iβ β β βω ω ω ω = = = = 4 sin   m 2   2 ∂ ∂ Φ Φ(x) ∂ ∂ Φ Φ k = = = = = = = = 2 ∂ ∂ ∂ ∂x 2 2 Z e 2 exp(− − − −κ κ κ κ)(2 + + + + 2κ κ κ κ + + + + κ κ κ κ ) 3 4π π π πε ε ε ε b 0Dust lattice waves : 1D Variation of the Re q screening strength Im qDust lattice waves: shear mode Laser pulse Nunomura 2001Dust lattice waves: shear mode Nunomura 2002 wave number frequency frequencyDust lattice waves: transverse mode z n−1 z n z z n+1 n+2 2    Q r r electrostatic force between F(r) = = = = exp− − − − 1+ + + +  2 2    4 4π π π π π π π πε ε ε ε ε ε ε ε r r λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ p pa ar rt tiic clle es s 0  D  D     2 Δ Δ Δ Δz 2 2 r = = a + + Δ Δz ≈ ≈ a + + = = + + Δ Δ ≈ ≈ + + 2a vertical force Δ Δ Δ Δz F (r) = = = = F(r) z component r 2 Q F (n.n.) = = = = exp( ( ( (− − − − κ κ κ κ ) ) ) )( ( ( (1+ + + + κ κ κ κ ) ) ) )(z − − − − z ) z n+ + + +1 n 3 4π π π πε ε ε ε a 0Waves in weakly coupled dust: Dustacoustic waves Complete analog to ionacoustic waves: Ions Dust Electrons Electrons and ions ω ω ω ω ω ω ω ω ω ω ω ω ,ω ω ω ω pd pi pe Dispersion for cold dust, ions DA velocitySummary Waves • Dust lattice waves (strongly coupled system) • Longitudinal (compressional) waves • Shear (transverse) waves • • O Ou ut t o of f p plla an ne e ( (t tr ra an ns sv ve er rs se e) ) w wa av ve es s • Determination of screening strength and interaction potentialDusty plasmas: unique properties • High Particle Charge Z: additional charge carrier in plasma • High Particle Charge Z: strong coupling  crystallization • High Particle Mass m: slow dynamics  video microscopy • Particle surface a: novel type of forces for plasmas • gravity • ion drag • thermophoresis • novel type of waves: dust lattice waves dust acoustic waves
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