Irina I. Lisina, Olga S. Vaulina

JIHT RAS

Distribution of the Kinetic temperature   Z / X   Upper / Lower  

J.B. Pieper, J. Goree, PRL (1996)   ТZ / ТX < 1      

A.K. Mukhopadhyay, J. Goree, PRE (2014)   ТZ / ТX < 1   ТUP / ТLOW < 1   

A.A. Samarian, B.W. James, S.V. Vladimirov, and N.F. Cramer, PRE (2001) ТZ / ТX > 1      

A. Aschinger, J. Winter, NJP (2012)   ТZ / ТX > 1   ТUP / ТLOW < 1  

O.S. Vaulina, E.V. Vasilieva, O.F. Petrov and V.E.

Fortov, Phys. Scr. (2011)  ТZ / ТX > 1   ТUP / ТLOW < 1  

A.V. Ivlev, J. Bartnick, M. Heinen, C.-R. Du, V.

Nosenko, and H. Löwen, PRX (2015)  ТZ / ТX < 1   ТUP / ТLOW > 1  

[O. Petrov et al. 2011]

Side view:

Top view:

Z

X

Tdust > Te , Ti , Tn

Microparticles can gain energy from the surrounding plasma

•  if the charge spatially varies [Zhakhovskii V.V., Molotkov V.I., et al. // JETP Lett.,1997] [Zhdanov S.K., Ivlev A.V., Morfill G.E. // Phys. Plasmas, 2005]

•  if the charge randomly varies with time [R.A. Quinn and J. Goree // Phys. Rev. E, 2000] [Vaulina O.S., Khrapak S.A, et al. // Phys. Rev. E., 1999] [A.V. Ivlev, U. Konopka, G. Morfill // Phys. Rev. E, 2000] [G. Norman, V. Stegailov, A. Timofeev // Contrib. Plasma Phys., 2010]

•  if the electric field cause effect of delayed charging

[Nunomura S., Misawa T., et al. // Phys. Rev. Lett., 1999] [Pustylnik M.Y., Ohno N., et al. // Phys. Rev. E, 2006] [ A.A. Samarian, B.W. James, et al. // Phys. Rev. E, 2001]

these theoretical models do not always allow us to explain the increase in

kinetic energy (> 0.5 eV) for the dust particles for common conditions of laboratory experiments

It was for the first time assumed in [Schweigert V.A., et al. // PRL, 1996.] that an increase in dust particles’ energy can be caused by an ion induced instability The authors have also proposed a simple model of anisotropic pair interaction, which is similar to interaction occurs due to ion focusing And numerically have shown that the location and density of the ion cloud is mainly depend on the position of the upstream particle and weakly depend on the position of the downstream one. The simulation confirmed that this mechanism can provide both the conditions for alignment and a dramatic increase in the dust temperature.

ü  only radial component of dust kinetic energy was observed

[V. A. Schweigert, et all, Phys. Rev. E (1996)] [V. A. Schweigert, et all, Phys. Rev. Lett., (1998)] [S. A. Maiorov, S. V. Vladimirov, Phys. Rev. E (2000)] [S. V. Vladimirov, S. A. Maiorov, Phys. Rev. E (2003)] [W.J. Miloch, et all, Phys. Rev. E (2008)] [I.H. Hutchinson, Phys. Rev. E (2012)] [С.А. Майоров, Б.А. Клумов, Кр. сооб. ФИАН, (2013)] [V.V. Zhakhovskii, V.I. Molotkov, et all, JETP Lett., (1997)] [Y. Hayashi, K. Tachibana, J.Vac.Scl.Technol.A (1996)] [C. Killer, et all, Phys. Rev. B (2011)] [A. Aschinger, J. Winter, NJP (2012)] [T.W. Hyde, et all, Phys. Rev. E (2013)] [ A.V. Ivlev, , et all, PRX (2015) ]

Numerical, analytical and experimental studies of spatial distribution of electrostatic potential around a dust particle in an anisotropic plasma flow:

( )

Db

ϕ θϕ **

** [Lisina I.I., Vaulina O.S. // EPL, 103 (2013) 55002]

* [M. Lampe, et all, Physics of plasmas (2000)]

*

W

W

/ ,

/

p

p

q Q

d l

=

=

*

*

q

d

( , ) exp( ) exp( )kj jdkj kj jd

kj p jd p

l lQ ql ll l l l

ϕ κ κ= − + −

Fkj = Qφkj(1) (lkj, ljd).

Fjk ≠ Fkj

[M. Lampe, G. Joyce, G. Ganguli, IEEE Trans. Plasma Sci.(2005)]

2

int2 ( )k j

k jk kfr ext brl l l

j k j

l ld l dlM F l M F Mg Fdt dtl l

ν= −

−= − + + +

−∑ r r

r rr rr rr

r r

Mgr

0 ( )extz zF Q E zβ= +r r r

extrF Q rα=r r ext

rFr

We put N particles with quasi dipol-dipol interaction into a linear trap

int ( )F l Ql

∂ ϕ∂

= −

dpM =r R&&

The statistical approach proposed by Langevin, allows us write the stochastic equations of motion for each particle

Equations of motion

0.15

0.25

0.35

0.45

0.55

0.65

0.75

0.05 0.20 0.35 0.50 q*

d*

κ = 2 κ = 1κ = 0.5

* / 0.1 5frξ ω ν= ÷:

Параметры задачи:

0.16 , / 0.37pq Q d l= − =

0.16 , / 0.25pq Q d l= − = ∑≠ jkjk

kjU;,

UΣ =

The region of stable existence for two different particle

arrangements depending on the parameters of

quasi dipol-dipol interaction

UΣ ≡ Uv <Uh ,

[Hayashi Y., 1999]

[Hayashi et al., 1999]

The criterion for the particle arrangements:

Uh ≅ Uv

Fragments of vertical cross section of five-layer structures (particle tracks in XZ-plain, ∆Y~ 0.3 lp) :

[Lisina I.I., Vaulina O.S. // EPL, 103 (2013) 55002]

0.1

0.2

0.4

0.8

1.6

3.2

6.4

12.8

-0.15 -0.05 0.05 0.15

ϕ (Vz)

ϕ (Vx)

 

 

ξ  =  0.33  ____ξ  =  1  _______ξ  =  3  

0.15 , / 0.2pq Q d l= − =

0.16 , / 0.37pq Q d l= − =

�  particle tracks Kinetic temperature ≥ 0

3 = ( - )2

δK K T

Kz > Kx ~ Ky

0

1

10

-0.2 -0.1 0.0 0.1 0.2

ϕ(Vz);

Vx, Vz, sm/s

ϕ(Vx);

Layered structure

Kz ≥ Kx ~ Ky

= 0.014 eV2T

Chain

Kx =0.14 eV

Kz =0.21 eV

Kz=0.1 eV Kx =0.05eV

= 0.05 eV2T

[Lisina I.I., Vaulina O.S. // EPL (2013)]

[Lisina I.I., Vaulina O.S. // Phys. Scr. (2014)]

а1

z = F12

(1), а2z = F21

(1)

а1x(y)

≈ F12/lp , а2x(y)

≈ F21/ lp

the correlation equations for Brownian force : <Fb1 > = < Fb2 > ≡ 0, <Fb1 Fb2 > = 0, <Fb1 V2 > = <Fb2 V1 > ≡0, <Fb1 ξ2 > = <Fb2 ξ1 > ≡0, <Fb1 ξ1 > = <Fb2 ξ2 > ≡0, <Fb1 V2 > = <Fb2 V1 > ≡0; Fkj

(1) - The first derivative of the Fkj at a distance lp,

- velocities of the 1st and the 2nd particles respectively.

< > denotes time averaging for t→ ∞.

)(1

)(2

)(1

)(1

)(1

)()(

12

)(1

2

)( xzb

xzxzxzxzxzxz

fr

xz

FaQdt

dMvdt

dM +−−−−= ξξξβξξ

)(2

)(1

)(2

)(2

)(2

)()(

22

)(2

2

)( xzb

xzxzxzxzxzxz

fr

xz

FaQdt

dMvdt

dM +−−−−= ξξξβξξ

dtd

V )2(1)2(1

ξ=

09 Bearing that MV1(2)

2 ≡ T1(2) = T+ δT1(2), νfrδT1(2) = νfrT1(2) - <V1(2)Fb1(2)>, and as the particles move along closed trajectories then <ξ1 V1 > = <ξ2 V2 > ≡0, which lets us go to equations describing the additional energy "swap" and the energy redistribution between the particles As well as the energy redistribution on the degrees of freedom When the ratio . In this case, the redistribution of energy between two particles becomes impossible and the system completely destroyed.

( ) ( ) ( ) ( ) 2( ) 1 2 1 2

( ) ( ) 2 ( ) ( ) ( ) 21 2 1 2

0.5 ( )2 0.5( ) ( 2 )

z x z x z x z xz x

z x z x z x z x z xfr

Т Т Т a aТa a a a M

δ δδ

β ν+ −

≡ =+ + + +

2 2

1 1

1 / /1 / /

Т Т Т Т T Т TТ Т Т Т T Т T

δ δδ δ+ + + Δ

≡ =+ + −Δ

0)( )(2

)(1 →+ xzxz aa ∞→Δ TТ xz /)(

z z

x x

Т Т ТТ Т Т

δδ+

≡+

The analytical solutions are illustrated by solid curves. The simulation results are represented by symbols. The dashed lines indicate the value of d* when the system is destroyed due to development of vertical instability.

|q*| = 0.1 (green)

|q*| = 0.3 (red)

χ ≅ 3 β x(y)/β z =3.5

The proposed linear theory is in good agreement with the simulation results, even at high heating (when the kinetic energy K of particles much higher than the thermostat temperature T)

system is destroyed

system is destroyed

The system is completely destroyed when

ω*/νfr=3

0)( 2112 →∂∂+∂∂ zFzF

ω*/νfr

∝ 1/νfr2

2

1 2

1 2

z z z

z z

Т a aТ a aδ ⎛ ⎞−

= ⎜ ⎟+⎝ ⎠

The swap in particle energy is due to the non conservativeness of the dust particle system. Due to results obtained are the strong heating is possible when there are •  attraction forces between dust grains and •  friction forces. If nonreciprocal forces аre the repulsive forces, the kinetic temperatureof particle system can not be increased more than twice: When νfr → 0 the growth of the kinetic energy is restricted due to their electrostatic interactions and random forces : When νfr →∞ , then νfr

—1 →∞ ,02)( →∝∂ −

frxzТ ν 1/ 12 →ТТ

( ) ( ) ( ) ( )( ) 2 21 2 1 2( ) ( ) constantz x z x z x z xz xТ Т a a a aδ ≡ − + =

|q*| =0.275, d*≈0.45

|q*| =0.2, d* ≈0.45

TТ ≈∂

0

1

10

0.1 0.2 0.3 0.4 0.5

d*

Тz/Тx

κ1=κ2≡0

κ1=1.5, κ2=0

κ1=2, κ2=0

thin line: q*= 0.1 bold line: q*= 0.3

O.S. Vaulina et al., Phys. Scr. 2011 A. Aschinger, J. Winter, NJP 2012 A.A. Samarian et al., PRE 2001

J.B. Pieper, J. Goree, PRL 1996 A.K. Mukhopadhyay, J. Goree, PRE 2014

1.0

1.5

2.0

2.5

3.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

d*

Т2z/Т1

z

•  A  simple  Yukawa  model    (par2cle  size  a<<λ)

( ) expYZe lll

ϕλ

⎛ ⎞= −⎜ ⎟

⎝ ⎠

•  Macropar2cle  in  a  bulk  plasma:        

( ) ( ) ( )Y adl l lϕ ϕ ϕ= +

•  Test  par2cle  in  a  plasma  flow:        

q  an  a=rac2ve  part  due  to  ion  focusing  

q                                                 in  a  collisionless  plasma  [Montgomery    et  al.,  PоP(1968),  Kompaneets    et  al.,  PоP(2009)]      

q                                                 in  a  collisional  regime  [Stenflo    et  al.,    Phys.  Fluids  (1973),  Chaudhuri    et  al.,  PоP(2007)]    

3( )l lϕ −∝2( )l lϕ −∝

U = eZϕ(l)    is  the  poten2al  energy  of  interac2on  between  par2cles                  (pair  approxima2on)  

Lampe    et  al.,    Phys.Plas.(2000)    

q                                                 in  a  collisionless  plasma  [  Allen    (1992),      Khrapak    et  al.  (2001)  ]    

q                                                 in  a  collisional  (weakly)  regime  [  Filippov  et  al.  (2007),    Khrapak    et  al.  (2008)  ]  

q                                                                                                                                                                               with  ioniza2on  sources    [  Filippov  et  al.  (2007)  ]    

2( )l lϕ −∝1( )l lϕ −∝

( ) ( )1 2 2( ) exp expl Ze A l A l lϕ κ κ= − + −⎡ ⎤⎣ ⎦