- Email: [email protected]
Theory of dust cloud motions in a nonuniform magnetoplasma
1 July 2002
Physics Letters A 299 (2002) 258–261 www.elsevier.com/locate/pla
Theory of dust cloud motions in a nonuniform magnetoplasma P.K. Shukla 1 Institut für Theoretische Physik IV, Fakultät für Physik und Astronomie, Ruhr-Universität Bochum, D-44780 Bochum, Germany Received 2 March 2002; accepted 19 March 2002 Communicated by V.M. Agranovich
Abstract A self-consistent theory for dust cloud motions in a partially ionized nonuniform magnetoplasma is presented, taking into account collisional interactions between the plasma particles, pressure gradient and electromagnetic forces. It is found that dust cloud motions in the radial and azimuthal directions (with respect to the direction of a vertical DC magnetic field) occur owing to the combined effects of dust–ion and dust–neutral collisions as well as due to forces associated with a radial electric field and the gradients of the plasma and magnetic pressures. The relevance of our investigation to laboratory and astrophysical plasmas is pointed out. 2002 Elsevier Science B.V. All rights reserved. PACS: 52.20.-j; 52.25.Zb; 98.38.Dq
About eight years ago, Fujiyama et al. [1] experimentally studied the dynamics of silicon particles in DC and AC silane plasmas in the presence of a modulated magnetic field that is perpendicular to the discharge electric field. The experimental results demonstrated dynamical particle transport due to the balance among electrostatic force, ion drag force and force of gravity on the particles [2]. The removal of dust particles from the processing plasma due to the poloidal ion drag force was also considered by Nunomura et al. [3]. Recently, Sato et al. [4,5] conducted a series of experiments to study the dynamics and control of fine-
E-mail address: [email protected] (P.K. Shukla). 1 Also at the Department of Plasma Physics, Umeå University,
SE-90187 Umeå, Sweden, and the Centre for Interdisciplinary Plasma Science at the Max-Planck Institut für Plasma Physik und Extraterrestrische Physik, Postfach 1312, D-85741 Garching, Germany.
particle clouds in a moderate pressure (75–300 mtorr) partially ionized DC discharge plasma in the presence of a vertical (or axial) magnetic field whose strength varies between 400 gauss to 40 kilogauss. In the experiments of Sato et al. [4,5] the plasma density is in the range 107 –108 cm−3 and the electron temperature is of order of a few eV. The ion temperature is Ti ∼ 0.1 eV and the neutrals and dust grains are at room temperature. Fine dust particles used are spherical with a diameter ∼ 10 µm, mass of order 6 × 10−10 g, and are strongly correlated, as typical dust charge and dust density are qd ∼ 3 × 104e and nd ∼ 103 – 104 cm−3 , respectively, where e is the magnitude of the electron charge. The ion-neutral collision mean free path, of order of a fraction of a mm, is much smaller than the dust cloud size (∼ 1 cm diameter). Sato et al. [4,5] observed rotations of the cloud of negatively charged dust particles perpendicular to the magnetic field. The angular frequency of the dust
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 4 4 3 - 7
P.K. Shukla / Physics Letters A 299 (2002) 258–261
cloud rotation is of the order of 0.1 rad/s, and it depends on the magnetic field strength and rf power in the plasma. On the other hand, Konopka et al. [6] observed rigid and differential dust particle rotations in a nonuniform magnetic field. Several authors [7–9] presented theoretical models for understanding the rotational, vibrational, and spinning motions of charged dust particles in the plasma sheath. Tskhakaya and Shukla [7] examined vibration, rotation and bouncing motions of charged dust particles in inhomogeneous and oscillating electric fields in the plasma. Ishihara and Sato [8] suggested that sheared ion flows in the sheath produce selfrotation of a dust particle, while the external magnetic field causes the dust cyclotron motion. On the other hand, Kaw et al. [9] suggested that the rotation of dust clouds arises spontaneously in the direction of the ion rotation; the latter, in turn, is determined by a competition between the E × B and diamagnetic rotations of the ions in a partially ionized magnetoplasma, where E is the sheath electric field, B = zˆ B0 is the external magnetic field, zˆ is the unit vector along the vertical direction, and B0 is the magnetic field strength. Although the model of Kaw et al. [9] lend support to the experimental observations [4,5] involving rigid dust particle rotation, it appears that there does not exist a self-consistent theory for rigid and differential dust cloud rotations including drag forces (associated with ion–dust and dust–neutral collisions) as well as an electrostatic force caused by the radial sheath electric field and the Jd × B force, where Jd is the dust current density. In this Letter, we develop such a theory for complex dust cloud motions in a partially ionized nonuniform dusty magnetoplasma. We consider the dynamics of negatively charged dust particles in a nonuniform plasma sheath containing an electric field E = rˆ Er and a nonuniform magnetic field B = zˆ B0 (r). The plasma also contains a nonuniform pressure gradient ∂p/∂r, where p = pe + pi ≡ ne Te + ni Ti and nj (Tj ) is the density (temperature) of the particle species j (j equals e for the electrons, i for ions, d for dust grains, and n for neutrals). When the collision frequency νej is much smaller than the electron gyrofrequency ωce = eB0 /me c, we obtain from the inertialess electron momentum equation 0 = −ne eE − ∇pe − ne e
ve × B, c
(1)
259
where me is the electron mass, c is the speed of light in vacuum, ve is the electron velocity, and |ve · ∇| ωce . Eq. (1) shows that electrons acquire an azimuthal drift speed c c ∂pe , Er − (2) B0 ene B0 ∂r due to the radial electric field Er and the electron pressure gradient ∂pe /∂r. Here the subscripts θ and r stand for the azimuthal and radial components, respectively. The ions are coupled with electrons through the radial electric field. In a partially ionized dusty plasma, the ion–neutral collision frequency νin is much larger than the ion–dust collision frequency. Hence, in the steady state, we have from the ion momentum equation vi ρi νin vi = ni eE − ∇pi + ρi νid vd + ni e × B, (3) c where ρi = ni mi is the ion mass density, mi is the ion mass, νid is the ion–dust collision frequency, vi is the ion velocity, and |vi · ∇|, (ne me /ni mi )νei νin . The neutrals are assumed to be stationary. Eliminating E from (1) and (3) we obtain B02 ρi νin vi = Zd nd eE − ∇ p + 8π 1 (B · ∇)B + ρi νid vd − Jd × B, + (4) 4π c where we have used the quasi-neutrality condition veθ = −
ni = ne + Zd nd ,
(5)
and Ampérè’s law 4π J. (6) c Here J = e(ni vi − ne ve − Zd nd vd ) ≡ Ji + Je + Jd is sum of the ion, electron, and dust current densities, Zd is the number of charges residing on the dust particle surface, and vd is the dust particle velocity. It turns out that electrostatic and collisional interactions between ions, electrons and dust particles are nontrivial, as the ion velocity is now controlled by numerous forces in the right-hand side of Eq. (4). The radial and azimuthal components of the ion velocity are determined from ∇ ×B=
vir =
νid Zd nd eEr 1 ∂P + − vdr ni mi νin ρi νin ∂r νdn Zd nd ωci + vdθ , ni νin
(7)
260
P.K. Shukla / Physics Letters A 299 (2002) 258–261
and viθ =
and νid Zd nd ωci vdθ − vdr , νin ni νin
(8)
where P = p + B02 /8π is the sum of the plasma and magnetic field pressures, and ωci = eB0 /mi c is the ion gyrofrequency. The expressions (7) and (8) show that the radial and azimuthal motions of ions are driven by forces involving the radial electric field, gradients of the plasma and magnetic pressures, the ion–dust drag, and the coupling of the dust current density with the external magnetic field. These expressions differ significantly from (3a) and (3b) of Ref. [9], since the latter does not include the coupling of ions with electrons and charged dust particles in the ion momentum equation. In a partially ionized plasma, the dust–neutral collision frequency is much larger than the dust gyrofrequency ωcd = Zd eB0 /md c, where md is the dust mass. Hence, the steady state equation of motion for the dust cloud is 0 = −Zd eE − md νdn vd − md νdi (vd − vi ),
(9)
where the dust particle is assumed to be cold. Since νdi νdn , we obtain from Eq. (9) vdr = −
Zd e νdi Er + vir , md νdn νdn
(10)
and vdθ =
νdi viθ . νdn
(11)
Eqs. (10) and (11) reveal that the radial motion of dust clouds is caused by the radial electric force and the dust–ion drag force, while the azimuthal dust rotation arises from dust–ion collisions in the presence of the azimuthal ion speed that is driven by the Jd × B force. Thus, the physics of the azimuthal dust rotation is quite different from what has been proposed in Ref. [9]. Combining Eqs. (7), (8), (10) and (11), we readily obtain nd νdi mi Zd eEr − vdr ≈ ni νin md mi νdn Zd nd νdi ωci νdi 1 ∂P + − vdθ , (12) νdn ρi νin ∂r ni νdn νin
Zd α nd νdi mi eEr − 2 1 + α ni νin md mi νdn α ∂P νdi , 1 + α2 + νdn ρi νin ∂r
vdθ = −
(13)
where α=
Zd nd νdi ωci . ni νdn νin
(14)
Since νdi (mi /md )νdn , the azimuthal dust speed in the absence of the pressure and magnetic field gradients takes the form vdθ =
α Zd nd νdi e ∂φ , 1 + α 2 ni νdn mi νin ∂r
(15)
where Er = −∂φ/∂r and φ is the sheath potential. We see from Eq. (15) that the azimuthal dust cloud speed increases with the increase√of B0 and attains a maximum value at ωci /νin ∼ (1/ 2 )ni νdn /Zd nd νdi . Furthermore, the dust rotation speed vdθ is inversely proportional to the ion number density and the dust– neutral collision frequency. For rigid dust cloud rotation we can set vθ ∼ rω, and determine the constant angular frequency ω of the dust cloud. To summarize, we have presented a theory for motions of charged dust clouds in a partially ionized nonuniform magnetoplasma. It is found that interactions between dust clouds, electrons and ions are nontrivial in that the dust particle motions are controlled by the combined effects of a radial electric force, the Jd × B and collisional drag forces. Our results show that the azimuthal dust rotation (with respect to a vertical magnetic field) is not due to the cross-field (E × B) drift, as recognized by Sato et al. [4,5]. The azimuthal dust rotation speed is linearly proportional to α when 2 ω /ν ν 2 ). α 1, and it scales as (Zd nd /ni )2 (νdi ci dn in Thus, the rotation speed increases with the increase of B0 and Zd nd . These features are consistent with the experimental observations [4,5]. Furthermore, the radial motion of dust clouds is caused by the radial electric field and gradients of the plasma and magnetic field pressures [6]. In closing, we mention that the results of the present investigation should also be useful in understanding the dynamics of charged dust clouds in partially ionized astrophysical environments [10].
P.K. Shukla / Physics Letters A 299 (2002) 258–261
Acknowledgement This work was partially supported by the European Commission (Brussels, Belgium) through contract No. HPRN-CT-2000-00140 for carrying out the task of the Human Potential Research Training Networks “Complex Plasmas: The Science of Laboratory Colloidal Plasmas and Mesospheric Charged Aerosols”.
References [1] H. Fujiyama, H. Kawasaki, S.C. Yang, Jpn. J. Appl. Phys. 33 (1994) 4216.
261
[2] P.K. Shukla, A.A. Mamun, Introduction to Dusty Plasma Physics, Institute of Physics Publications, Bristol, 2002. [3] S. Nunomura, N. Ohno, S. Takamura, Jpn. J. Appl. Phys. 36 (1997) 877. [4] N. Sato, G. Uchida, R. Ozaki, in: Y. Nakamura, T. Yokota, P.K. Shukla (Eds.), Frontiers in Dusty Plasmas, Elsevier, Amsterdam, 2000, p. 329. [5] N. Sato, G. Uchida, T. Kaneko, Phys. Plasmas 8 (2001) 1786. [6] U. Konopka, D. Samsonov, A.V. Ivlev, Phys. Rev. E 61 (2000) 1890. [7] D.D. Tskhakaya, P.K. Shukla, Phys. Lett. A 279 (2001) 456. [8] O. Ishihara, N. Sato, IEEE Trans. Plasma Sci. 29 (2001) 179. [9] P.K. Kaw, K. Nishikawa, N. Sato, Phys. Plasmas 9 (2002) 387. [10] H. Kamaya, R. Nishi, Astrophys. J. 543 (2000) 257.
Physics Letters A 299 (2002) 258–261 www.elsevier.com/locate/pla
Theory of dust cloud motions in a nonuniform magnetoplasma P.K. Shukla 1 Institut für Theoretische Physik IV, Fakultät für Physik und Astronomie, Ruhr-Universität Bochum, D-44780 Bochum, Germany Received 2 March 2002; accepted 19 March 2002 Communicated by V.M. Agranovich
Abstract A self-consistent theory for dust cloud motions in a partially ionized nonuniform magnetoplasma is presented, taking into account collisional interactions between the plasma particles, pressure gradient and electromagnetic forces. It is found that dust cloud motions in the radial and azimuthal directions (with respect to the direction of a vertical DC magnetic field) occur owing to the combined effects of dust–ion and dust–neutral collisions as well as due to forces associated with a radial electric field and the gradients of the plasma and magnetic pressures. The relevance of our investigation to laboratory and astrophysical plasmas is pointed out. 2002 Elsevier Science B.V. All rights reserved. PACS: 52.20.-j; 52.25.Zb; 98.38.Dq
About eight years ago, Fujiyama et al. [1] experimentally studied the dynamics of silicon particles in DC and AC silane plasmas in the presence of a modulated magnetic field that is perpendicular to the discharge electric field. The experimental results demonstrated dynamical particle transport due to the balance among electrostatic force, ion drag force and force of gravity on the particles [2]. The removal of dust particles from the processing plasma due to the poloidal ion drag force was also considered by Nunomura et al. [3]. Recently, Sato et al. [4,5] conducted a series of experiments to study the dynamics and control of fine-
E-mail address: [email protected] (P.K. Shukla). 1 Also at the Department of Plasma Physics, Umeå University,
SE-90187 Umeå, Sweden, and the Centre for Interdisciplinary Plasma Science at the Max-Planck Institut für Plasma Physik und Extraterrestrische Physik, Postfach 1312, D-85741 Garching, Germany.
particle clouds in a moderate pressure (75–300 mtorr) partially ionized DC discharge plasma in the presence of a vertical (or axial) magnetic field whose strength varies between 400 gauss to 40 kilogauss. In the experiments of Sato et al. [4,5] the plasma density is in the range 107 –108 cm−3 and the electron temperature is of order of a few eV. The ion temperature is Ti ∼ 0.1 eV and the neutrals and dust grains are at room temperature. Fine dust particles used are spherical with a diameter ∼ 10 µm, mass of order 6 × 10−10 g, and are strongly correlated, as typical dust charge and dust density are qd ∼ 3 × 104e and nd ∼ 103 – 104 cm−3 , respectively, where e is the magnitude of the electron charge. The ion-neutral collision mean free path, of order of a fraction of a mm, is much smaller than the dust cloud size (∼ 1 cm diameter). Sato et al. [4,5] observed rotations of the cloud of negatively charged dust particles perpendicular to the magnetic field. The angular frequency of the dust
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 4 4 3 - 7
P.K. Shukla / Physics Letters A 299 (2002) 258–261
cloud rotation is of the order of 0.1 rad/s, and it depends on the magnetic field strength and rf power in the plasma. On the other hand, Konopka et al. [6] observed rigid and differential dust particle rotations in a nonuniform magnetic field. Several authors [7–9] presented theoretical models for understanding the rotational, vibrational, and spinning motions of charged dust particles in the plasma sheath. Tskhakaya and Shukla [7] examined vibration, rotation and bouncing motions of charged dust particles in inhomogeneous and oscillating electric fields in the plasma. Ishihara and Sato [8] suggested that sheared ion flows in the sheath produce selfrotation of a dust particle, while the external magnetic field causes the dust cyclotron motion. On the other hand, Kaw et al. [9] suggested that the rotation of dust clouds arises spontaneously in the direction of the ion rotation; the latter, in turn, is determined by a competition between the E × B and diamagnetic rotations of the ions in a partially ionized magnetoplasma, where E is the sheath electric field, B = zˆ B0 is the external magnetic field, zˆ is the unit vector along the vertical direction, and B0 is the magnetic field strength. Although the model of Kaw et al. [9] lend support to the experimental observations [4,5] involving rigid dust particle rotation, it appears that there does not exist a self-consistent theory for rigid and differential dust cloud rotations including drag forces (associated with ion–dust and dust–neutral collisions) as well as an electrostatic force caused by the radial sheath electric field and the Jd × B force, where Jd is the dust current density. In this Letter, we develop such a theory for complex dust cloud motions in a partially ionized nonuniform dusty magnetoplasma. We consider the dynamics of negatively charged dust particles in a nonuniform plasma sheath containing an electric field E = rˆ Er and a nonuniform magnetic field B = zˆ B0 (r). The plasma also contains a nonuniform pressure gradient ∂p/∂r, where p = pe + pi ≡ ne Te + ni Ti and nj (Tj ) is the density (temperature) of the particle species j (j equals e for the electrons, i for ions, d for dust grains, and n for neutrals). When the collision frequency νej is much smaller than the electron gyrofrequency ωce = eB0 /me c, we obtain from the inertialess electron momentum equation 0 = −ne eE − ∇pe − ne e
ve × B, c
(1)
259
where me is the electron mass, c is the speed of light in vacuum, ve is the electron velocity, and |ve · ∇| ωce . Eq. (1) shows that electrons acquire an azimuthal drift speed c c ∂pe , Er − (2) B0 ene B0 ∂r due to the radial electric field Er and the electron pressure gradient ∂pe /∂r. Here the subscripts θ and r stand for the azimuthal and radial components, respectively. The ions are coupled with electrons through the radial electric field. In a partially ionized dusty plasma, the ion–neutral collision frequency νin is much larger than the ion–dust collision frequency. Hence, in the steady state, we have from the ion momentum equation vi ρi νin vi = ni eE − ∇pi + ρi νid vd + ni e × B, (3) c where ρi = ni mi is the ion mass density, mi is the ion mass, νid is the ion–dust collision frequency, vi is the ion velocity, and |vi · ∇|, (ne me /ni mi )νei νin . The neutrals are assumed to be stationary. Eliminating E from (1) and (3) we obtain B02 ρi νin vi = Zd nd eE − ∇ p + 8π 1 (B · ∇)B + ρi νid vd − Jd × B, + (4) 4π c where we have used the quasi-neutrality condition veθ = −
ni = ne + Zd nd ,
(5)
and Ampérè’s law 4π J. (6) c Here J = e(ni vi − ne ve − Zd nd vd ) ≡ Ji + Je + Jd is sum of the ion, electron, and dust current densities, Zd is the number of charges residing on the dust particle surface, and vd is the dust particle velocity. It turns out that electrostatic and collisional interactions between ions, electrons and dust particles are nontrivial, as the ion velocity is now controlled by numerous forces in the right-hand side of Eq. (4). The radial and azimuthal components of the ion velocity are determined from ∇ ×B=
vir =
νid Zd nd eEr 1 ∂P + − vdr ni mi νin ρi νin ∂r νdn Zd nd ωci + vdθ , ni νin
(7)
260
P.K. Shukla / Physics Letters A 299 (2002) 258–261
and viθ =
and νid Zd nd ωci vdθ − vdr , νin ni νin
(8)
where P = p + B02 /8π is the sum of the plasma and magnetic field pressures, and ωci = eB0 /mi c is the ion gyrofrequency. The expressions (7) and (8) show that the radial and azimuthal motions of ions are driven by forces involving the radial electric field, gradients of the plasma and magnetic pressures, the ion–dust drag, and the coupling of the dust current density with the external magnetic field. These expressions differ significantly from (3a) and (3b) of Ref. [9], since the latter does not include the coupling of ions with electrons and charged dust particles in the ion momentum equation. In a partially ionized plasma, the dust–neutral collision frequency is much larger than the dust gyrofrequency ωcd = Zd eB0 /md c, where md is the dust mass. Hence, the steady state equation of motion for the dust cloud is 0 = −Zd eE − md νdn vd − md νdi (vd − vi ),
(9)
where the dust particle is assumed to be cold. Since νdi νdn , we obtain from Eq. (9) vdr = −
Zd e νdi Er + vir , md νdn νdn
(10)
and vdθ =
νdi viθ . νdn
(11)
Eqs. (10) and (11) reveal that the radial motion of dust clouds is caused by the radial electric force and the dust–ion drag force, while the azimuthal dust rotation arises from dust–ion collisions in the presence of the azimuthal ion speed that is driven by the Jd × B force. Thus, the physics of the azimuthal dust rotation is quite different from what has been proposed in Ref. [9]. Combining Eqs. (7), (8), (10) and (11), we readily obtain nd νdi mi Zd eEr − vdr ≈ ni νin md mi νdn Zd nd νdi ωci νdi 1 ∂P + − vdθ , (12) νdn ρi νin ∂r ni νdn νin
Zd α nd νdi mi eEr − 2 1 + α ni νin md mi νdn α ∂P νdi , 1 + α2 + νdn ρi νin ∂r
vdθ = −
(13)
where α=
Zd nd νdi ωci . ni νdn νin
(14)
Since νdi (mi /md )νdn , the azimuthal dust speed in the absence of the pressure and magnetic field gradients takes the form vdθ =
α Zd nd νdi e ∂φ , 1 + α 2 ni νdn mi νin ∂r
(15)
where Er = −∂φ/∂r and φ is the sheath potential. We see from Eq. (15) that the azimuthal dust cloud speed increases with the increase√of B0 and attains a maximum value at ωci /νin ∼ (1/ 2 )ni νdn /Zd nd νdi . Furthermore, the dust rotation speed vdθ is inversely proportional to the ion number density and the dust– neutral collision frequency. For rigid dust cloud rotation we can set vθ ∼ rω, and determine the constant angular frequency ω of the dust cloud. To summarize, we have presented a theory for motions of charged dust clouds in a partially ionized nonuniform magnetoplasma. It is found that interactions between dust clouds, electrons and ions are nontrivial in that the dust particle motions are controlled by the combined effects of a radial electric force, the Jd × B and collisional drag forces. Our results show that the azimuthal dust rotation (with respect to a vertical magnetic field) is not due to the cross-field (E × B) drift, as recognized by Sato et al. [4,5]. The azimuthal dust rotation speed is linearly proportional to α when 2 ω /ν ν 2 ). α 1, and it scales as (Zd nd /ni )2 (νdi ci dn in Thus, the rotation speed increases with the increase of B0 and Zd nd . These features are consistent with the experimental observations [4,5]. Furthermore, the radial motion of dust clouds is caused by the radial electric field and gradients of the plasma and magnetic field pressures [6]. In closing, we mention that the results of the present investigation should also be useful in understanding the dynamics of charged dust clouds in partially ionized astrophysical environments [10].
P.K. Shukla / Physics Letters A 299 (2002) 258–261
Acknowledgement This work was partially supported by the European Commission (Brussels, Belgium) through contract No. HPRN-CT-2000-00140 for carrying out the task of the Human Potential Research Training Networks “Complex Plasmas: The Science of Laboratory Colloidal Plasmas and Mesospheric Charged Aerosols”.
References [1] H. Fujiyama, H. Kawasaki, S.C. Yang, Jpn. J. Appl. Phys. 33 (1994) 4216.
261
[2] P.K. Shukla, A.A. Mamun, Introduction to Dusty Plasma Physics, Institute of Physics Publications, Bristol, 2002. [3] S. Nunomura, N. Ohno, S. Takamura, Jpn. J. Appl. Phys. 36 (1997) 877. [4] N. Sato, G. Uchida, R. Ozaki, in: Y. Nakamura, T. Yokota, P.K. Shukla (Eds.), Frontiers in Dusty Plasmas, Elsevier, Amsterdam, 2000, p. 329. [5] N. Sato, G. Uchida, T. Kaneko, Phys. Plasmas 8 (2001) 1786. [6] U. Konopka, D. Samsonov, A.V. Ivlev, Phys. Rev. E 61 (2000) 1890. [7] D.D. Tskhakaya, P.K. Shukla, Phys. Lett. A 279 (2001) 456. [8] O. Ishihara, N. Sato, IEEE Trans. Plasma Sci. 29 (2001) 179. [9] P.K. Kaw, K. Nishikawa, N. Sato, Phys. Plasmas 9 (2002) 387. [10] H. Kamaya, R. Nishi, Astrophys. J. 543 (2000) 257.