Introduction to Dusty Plasmas

André Melzer

Institute of Physics,

Ernst-Moritz-Arndt-Universität Greifswald Germany

Extended Lecture Notes see: www5.physik.uni-greifswald.de

Coulomb crystallizationof trapped particles

Dynamics and transport in plasmas

Dusty (Complex, Colloidal) Plasmas

Dusty Plasmas = Microscopic particles in a

Selwyn 1991

Astrophysics… Etching

Microscopic particles in a gaseous plasmaenvironment

Deposition and…

Dusty Plasmas in Astrophysics

Comet Hale-Bopp

Saturn rings

Dusty Plasmas in Astrophysics

Saturn rings 1981

2005

Dusty Plasmas in the Atmosphere

Noctilucent clouds

Dusty Plasmas in Technology

Plasma etching

Dusty Plasmas in Technology

Selwyn 1991

„Killer particle“

Dusty Plasmas in the Laboratory

Dusty Plasmas under Microgravity

Contents of Lecture

• Dust charging

• Forces

• Strongly coupled systems, particle-particle and

particle-plasma interaction

• Waves

• Finite systems and normal modes

For an extended introduction see:www5.physik.uni-greifswald.de

Charging

plasma

Q<0

ions

electrons

Q<0

In typical discharges:Particle will be charges negatively due to higher mobility of electrons

OML currents

ΦΦΦΦ−−−−====

ii

iii

kT

e

m

kTenaI 1

82

ππππππππ

Ion current

Probe theory of Langmuir and Mott-Smith 1929

ΦΦΦΦ====

ie

eee

kT

e

m

kTenaI exp

82

ππππππππ

Electron current

increased collectioncross section

thermal velocitiesgeometry

density reductionBoltzmann factor

electronand

ion currents

Particles as floating probes

∑∑∑∑ ====ΦΦΦΦq

flqI 0)(

ΦΦΦΦ====

ΦΦΦΦ−−−−

e

fl

i

e

e

i

i

e

i

fl

kT

e

n

n

m

m

T

T

kT

eexp1 With OML collection

currents only

Te/Ti 1 10 20 100

H -2.50 -1.91 -1.70 -1.24

He -3.05 -2.39 -2.16 -1.65

Ar -3.99 -3.24 -2.99 -2.41

Values of eΦΦΦΦfl /kTe for ne=ni

ekTefl /2−−−−≈≈≈≈ΦΦΦΦ

Limitations of OML description

Often: ion drift velocity ui (much) larger than ion thermal velocity vth,i SHEATH

exact

• collisionless (ion) trajectories

• Maxwellian velocity distribution

• isotropic

none of that ismet in „real“ discharges

ΦΦΦΦ−−−−====

2

2 21

ii

iiium

eeunaI ππππ

ΦΦΦΦ−−−−====

ii

iii

kT

e

m

kTenaI 1

82

ππππππππ

exact solution

approximation

The capacitance model

aC 04πεπεπεπε====

flCQ ΦΦΦΦ==== Particle as a spherical capacitorof capacitance C

Capacitance in vacuum

For a particle of a=1µm: 700 e per Volt

With the rule-of-thumb approximation: ΦΦΦΦ=2kTe/e

eVe,m1400 TaQ µµµµ====

Other charging currents

hνννν electrons

Photoelectron emission(UV radiation)

02 <<<<ΦΦΦΦΓΓΓΓ==== eaI µµµµππππ Secondary electron emission

Particle can become positively charged

electrons

0)/exp(

0

2

2

>>>>ΦΦΦΦΦΦΦΦ−−−−ΓΓΓΓ====

<<<<ΦΦΦΦΓΓΓΓ====

flpfl

fl

kTeeaI

eaI

νννννννν

νννννννν

µµµµππππ

µµµµππππ

−−−−====

m

e

m

eme

E

E

E

EE exp4.7)( δδδδδδδδ

Secondary electron emission

Charging time scale

ithi

ii

venae

kTa

,20

14

πππππεπεπεπεττττ ==== RC====ττττ

Time constant for

charging of a capacitor

C U 1/I

ai1∝∝∝∝ττττ Smaller particles

are charged slower

Plasma

time

1 µs

Summary Charging

• Micrometer sized particles carry 103 to 104 elementary charges

• Charging time: microseconds: Charge in dynamical equilibrium

• Charge to mass ratio Q/m extremely small: • Charge to mass ratio Q/m extremely small: slow timescales

Forces on dust particles

• Gravity

• Electric field force

• Thermophoresis

• Ion Drag

• Neutral Drag

Gravity

gagmFrvr

3

3

4πρπρπρπρ========

What else needs to be said ?

Electric Force

EaEQFrrr

ΦΦΦΦ======== 04πεπεπεπε

Also for a charged particle with (symmetric) shielding cloud

dust

shieldingcloud

Q

EQFrr

=

Drag Forces

„streaming“ species

dustparticle

A

Force = momentum transfer x ΑΑΑΑ x density x velocity

particle

vrel dt

# of incident particles

Neutral Drag

„streaming“ species

dustparticle

A

v dt

p

p‘

∆∆∆∆p

v dt

relnnth,n2

3

4vnvmaFrr

ππππδδδδ−−−−≅≅≅≅

av

pxmF

nth,

8

ρρρρππππδδδδββββββββ −−−−====−−−−==== &

NB:

Stokes friction

F ~ a

(a>λλλλ)

Ion Drag

2 components:

1. Collection Force

2. Coulomb Force

1. Collection Force

e Φ2r

iiii

i

umunmu

eaF

Φ−=

2

2 21π

r

cross section as for charging

2. Coulomb force

iiii umunF coulσ=r

ΛΛΛΛ==== ln22/coul ππππππππσσσσ b

p

p‘

∆∆∆∆p

Coulomb scattering cross section

20

2/4 iium

Qeb

πεπεπεπεππππ ====

====ΛΛΛΛ

min

maxlnlnb

b

p∆∆∆∆p

impact parameter for 90° collisions

Coulomb logarithm

Thermophoresis

T∇T

v

kaF ∇∇∇∇−−−−====

rr

nth,

n2

15

16ππππ „hot“ „cold“

Force towards coldest point!

Force due to a temperature gradientin the neutral gas

F

Force towards coldest point!

Comparison of forces

Trapping (Laboratory)

Electrode

Plasma

Fth FgFE Fion

sheath

Electrode

Plasma

Fth FgFE Fion

sheath

Trapping in the plasma sheath

Trapping (Microgravity and nanometric particles)

Electrode

Plasma

FEFionFth

Electrode

Plasma

FthFE Fion

Trapping in the plasma volume

sl. 9

Particle trapping in the laboratory

E

E

V 0

sheathedge Plasma

m

EQ ′′′′==== 02

0ωωωω

QE

mg

(z ) E

z

electrode z

E

z

0

0

particle

V(z)

(z)

Resonance method

mgzEzQ ====)()( 00Force balance

ext)()( FzEzQzmzm ====++++++++ &&& ββββ Equation of motion

0)( QzQ ====

)()()( zzEzEzE −−−−′′′′++++====

Assumption: constant charge

)()()( 00 zzEzEzE −−−−′′′′++++==== Linear electric field

200

20

20 )(

2

1)(

2

1zzEQzzm −−−−′′′′====−−−−ωωωω Potential well

m

EQ ′′′′==== 02

0ωωωω Resonance frequency

Linear Resonances

Charge measurement

Summary Forces

• Laboratory: electric field force + gravity: Trapping in the sheath

• Microgravity: electric field force + ion drag:Trapping in the plasma volume (void)Trapping in the plasma volume (void)

• Weakly damped particle dynamics

Strongly coupled systems

One Component Plasma:(Wigner 1938, Brush et al. 1966)

c

c 168

ΓΓΓΓ<<<<ΓΓΓΓ

====ΓΓΓΓ>>>>ΓΓΓΓ Solid Phase

Fluid Phase

Coulomb

energy

Thermal

energy

Yukawa systems

−−−−====

D0

2

exp4

)(λλλλπεπεπεπε

φφφφr

r

Qr

λλλλκκκκ b====

Robbins et al. 1988

Dλλλλκκκκ b====

Screening strength

Crucial parameters

Plasma crystals

cf: Chu et al. 1994, Thomas et al. 1994, Hayashi et al. 1994

Interaction?

• Horizontal Interactionrepulsive Yukawa (Debye-Hückel) type

(Konopka 2000)

• Vertical Interactionattractive forcesorigin?

Vertical Order

E0

sheathedge

Plasmaions

Attractive forces in the sheath?

QE

mg

(z ) E

z

E

electrode z

E0

0

0

particle (z)

ions

Vertcal order: Simulations

1. Ion focus: Attraction2. supersonic ion flowtion:

Non-reciprocal forces, Only the lower particleexperiences attraction

Vertical Order: Experiment

1. Particle: 3.47 µm

2. Particle: 4.18 µm

Vertical: force balance

Horizontal: free motion

Vertical Order: Experiment (2)

The lower particleexperiencesattraction

Vertical Order: Experiment (3)

The upper particledoes not

experienceattraction

Summary 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

Compressionaland Shear Waves in a 2D lattice

TransverseWave mode

Dust lattice waves: theoretical treatment

n b(n-1) b (n+1) b (n+2) b

nx

1+nx

2+nx

1−nx

)2( ++++−−−−==== xxxkxm && linear chain with )2( 11 −−−−++++ ++++−−−−==== nnnn xxxkxm && linear chain with spring constant k

)exp( tiinqbAxn ω−=

2sin4

)1(cos2)2(

22

2

qb

m

k

qbkeekm iqbiqb

=

−=−+=− −

ω

ω

dispersion relation of a linear chain

Dust lattice waves: theoretical treatment

n a(n-1) a (n+1) a (n+2) a

nx

1+nx

2+nx

1−nx

spring constant from

−==

rQrV

Vdk exp)(

22

spring constant fromYukawa potential

dispersion relation w.Yukawa potential

−==

= Dbr

r

r

QrV

dr

Vdk

λπεexp

4)(

0

2

2

2

( )( )2

30

2

22exp4

κκκπε

++−=b

Qk

( )( )

++−=

2sin22exp 22

30

22 qb

bm

Qκκκ

επω

2D dust lattice waves

• many (infinite) neighbors

• 2D hexagonal structure• compressional and shear mode

qb qb

Dust lattice waves: 1D

Dust lattice waves : 1D

)exp( tiiqx ωωωωξξξξ −−−−∝∝∝∝

Driven wave:

ω ω ω ω real

q complex

Re q

Dispersion ωωωω(q)

Dust lattice waves : 1D

q complex

Im q

====++++

2sin4 22 qb

m

kiβωβωβωβωωωωω

)22)(exp(4

)(

2

30

22

2

2

κκκκκκκκκκκκπεπεπεπε

++++++++−−−−

====∂∂∂∂

ΦΦΦΦ∂∂∂∂====

b

eZ

x

xk

Variation of

the

screening

strength

Re q

Dust lattice waves : 1D

Im q

Dust lattice waves: shear mode

Laser pulse

Nunomura 2001

Dust lattice waves: shear modefr

eq

ue

nc

y

Nunomura 2002

wave number

fre

qu

en

cy

Dust lattice waves: transverse mode

nz

1+nz

2+nz

1−nz

++++

−−−−====

rr

r

QrF

λλλλλλλλπεπεπεπε1exp

4)(

2

2

electrostatic force betweenparticles

DDr λλλλλλλλπεπεπεπε4 2

0

a

zazar

2

222 ∆∆∆∆

++++≈≈≈≈∆∆∆∆++++====

(((( )))) (((( )))) )(1exp4

.).(

)()(

130

2

nnz

z

zza

QnnF

r

zrFrF

−−−−++++−−−−====

∆∆∆∆====

++++κκκκκκκκπεπεπεπε

particles

vertical forcecomponent

Waves in weakly coupled dust: Dust-acoustic waves

Complete analog to ion-acoustic waves:

Ions DustElectrons Electrons and ions

pepipd ωωωωωωωωωωωωωωωω ,<<<<<<<<<<<<

DA velocity

Dispersion for cold dust, ions

Summary Waves

• Dust lattice waves (strongly coupled system)

• Longitudinal (compressional) waves• Shear (transverse) waves• Out-of-plane (transverse) waves• Out-of-plane (transverse) waves

• Determination of screening strength and interaction potential

Dusty 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