1. Ionization Equation of State for the Dusty Plasma Including the Effect of Ion–Atom Collisions
D.I. Zhukhovitskii
Joint Institute for High Temperatures, RAS
Scientific-Coordination Section on “Non-Ideal Plasma Physics”
November 19–20, 2018, Presidium RAS, Leninskiy avenue, 32a, Moscow 1

2. Ionization equation of state (IEOS) is a relation between two parameters specific for the charged components of gas discharge plasma containing a cloud of the dust particles. Such parameters are the electron, ion, and particle number density, and the particle charge. A complete set of IEOS's makes it possible to calculate all plasma state parameters provided that a single one is known. This makes IEOS similar to the common equation of state.
[1] D.I. Zhukhovitskii, V.I. Molotkov, and V.E. Fortov, Physics of Plasmas 21,
63701 (2014).
[2] D.I. Zhukhovitskii, Phys. Rev. E 92, (2015).
[3] D.I. Zhukhovitskii, Physics of Plasmas 24, (2017).

3. Typical parameters of dusty plasma in RF low-pressure argon discharge
Typical particle diameter, number density, and charge are 2a = 2.55 mm,
nd = 3x105 cm–3, and Z ~ 103, respectively.
The Wigner–Seitz cell radius is rd = (3/4pnd)1/3 = 93 mm.
The ion Debye screening length rDi ~ 100 mm, the ion mean free path is
la = 200 mm.
Other parameters of the system are pAr = 10 Pa, TAr = 300 K.
The Coulomb coupling parameter for the interparticle interaction is
G = Z2e2/rdTAr ~ 103 .

4. Wigner–Seitz cell model of dusty plasma

5. Screening length in the cell model of a sense dust cloud
The distribution of electric potential in the cell is defined by the Poisson equation
to be solved with the boundary conditions
For the homogeneous charge background (ni – ne)e , the ratio of ion potential energy in the cell to its temperature is
where t = Te/Ti , F = Ze2/aTe . Comparison with the “shifted Coulomb” and the Yukawa potentials –1/x + 3/2 and –exp(–3x/2)/x for x = r/rd << 1 shows that the screening length should be defined as rs = (2/3)rd [rd = (3/4pnd)1/3].
The cell is quasineutral (a/rd << 1
r = rd /la ),

6. Particle potential in the Wigner–Seitz cell
rs = (2/3)rd
can exceed rDi considerably.

7. Particle charge equation
The electron current to the particle is defined by their Boltzmann distribution,
j–/pa2nevTe = exp(–F) , where vTe,i = (8Te,i/pme,i)1/2. The ion current is defined by the atom–ion collisions and the ion production [S.A. Khrapak et al., Phys. Rev. E 85, ]
The collisionless regime (OML approximation) can be considered if la/rs > 10.
The condition j– = j+ yields the particle charge equation

9. Ionization equation of state for the dust cloud
The electric driving force is Fe = –ZendE and the ion drag force is Fid = (3/8)(3/4pnd)1/3nileE . Here, the ion mean free path l is corrected for the ion collisions against the particles,
Thus, the force balance equation yields
The IEOS’s follow from this equation, the particle charge equation, and the quasineutrality equation:

10. Dust particles number density distribution, 2a = 3
Dust particles number density distribution, 2a = 3.4 µm, p = 11 Pa [Naumkin et al., PRE 94, (2016)]

11. Dust particles number density distribution, 2a = 3. 4 µm, p = 20
Dust particles number density distribution, 2a = 3.4 µm, p = 20.5 Pa [Naumkin et al., PRE 94, (2016)]

12. Particle number density as a function of the electron number density at a = 10-4 cm-3
(Atom–ion), this work; (OML), D.I. Zhukhovitskii, Physics of Plasmas 24, (2017).

13. Havnes parameter as a function of the electron number density at a = 10-4 cm-3
H = Znd /ne
= g /(1 – g)
(Atom–ion), this work; (OML), D.I. Zhukhovitskii, Physics of Plasmas 24, (2017).

14. Particle number density as a function of the dust particle diameter at pAr = 10 Pa

15. Particle number density as a function of the dust particle diameter for ne = 3.5x108 cm-3

16. Havnes parameter and the electron number density from experimental data
2a, mm
pAr, Pa
Te, eV
nd, 104 cm-3 H
ne, 108 cm-3
Ref.
1.55 15
3.8
65.2
0.723
5.60
[1]
2.55
26.7
0.537
4.52
9.55 30
4.5
1.97
0.113
3.64
[2] 10
3.5
22.3
0.760
2.89
[3]
28.0
0.818
3.27
[4]
3.4 11
8.01
0.491
1.98
20.5
6.30
0.243
2.73
6.8
3.65
0.202
3.10
[1] S.A. Khrapak, B.A. Klumov, P. Huber, V.I. Molotkov, A.M. Lipaev, V.N. Naumkin, A.V. Ivlev,
H.M. Thomas, M. Schwabe, G.E. Morfill, O.F. Petrov, V.E. Fortov, Y. Malentschenko, and S. Volkov, Phys. Rev. E 85, (2012).
[2] D. Caliebe, O. Arp, and A. Piel, Phys. of Plasmas 18, (2011).
[3] M. Schwabe, K. Jiang, S. Zhdanov, T. Hagl, P. Huber, A.V. Ivlev, A.M. Lipaev, V.I. Molotkov,
V.N. Naumkin, K.R. Sütterlin, H.M. Thomas, V.E. Fortov, G.E. Morfill, A. Skvortsov, and S. Volkov, EPL 96, (2011).
[4] V.N. Naumkin, D.I. Zhukhovitskii, V.I. Molotkov, A.M. Lipaev, V.E. Fortov, H.M. Thomas, P. Huber, and G.E. Morfill, Phys. Rev. E 94, (2016).

17. The interparticle distance difference as a function of argon pressure
ne = ( pAr)x108 cm–3
Experiment: S.A. Khrapak,
B.A. Klumov, P. Huber,
V.I. Molotkov, A.M. Lipaev,
V.N. Naumkin, A.V. Ivlev,
H.M. Thomas, M. Schwabe,
G.E. Morfill, O.F. Petrov,
V.E. Fortov, Y. Malentschenko, and S. Volkov, Phys. Rev. E 85, (2012).

18. Conclusion
In the Wigner–Seitz cell model for dusty plasma, the screening length for the particle charge is equal to 2/3 of the cell radius.
The collisionless regime is realized solely if the ion mean free path is at least ten times larger than the screening length.
Enhancement of the ion current to the particle decreases its charge as compared to the estimation from OML model.
IEOS obtained for the stationary dust cloud provides an interpretation not only of the particle number density dependence on the electron (ion) number density and the particle radius but on the argon pressure as well.
Calculation results are in a reasonable agreement with the experimental data in a wide range of system parameters.
This work was supported by Presidium RAS program No.13 “Condensed Matter and Plasma at High Energy Densities”.

19. Thank you for your attention!
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