Dust voids due to dust-phase-space vortices in plasmas

PHYSICS OF PLASMAS

VOLUME 11, NUMBER 5

MAY 2004

Dust voids due to dust-phase-space vortices in plasmas A. A. Mamuna) and P. K. Shukla Institut fu¨r Theoretische Physik IV, Ruhr-Universita¨t Bochum, D-44780 Bochum, Germany

共Received 11 November 2003; accepted 27 January 2004; published online 13 April 2004兲 It is shown that a dust void in a dusty plasma can be formed due to the presence of positive plasma potential, which is built up owing to a nonisothermal dust particle distribution arising from negatively charged dust particle trapping. It is found that the dust void shrinks with the increase of the dust charge density, while it enlarges with the increase of the dust temperature. The implications of our results to the possibility for the formation of dust voids in space dusty plasmas, particularly in Saturn’s dusty rings, is discussed. © 2004 American Institute of Physics. 关DOI: 10.1063/1.1688331兴

I. INTRODUCTION

creased in a dust cloud. However, in a void the comparatively higher ionization rate leads to an electric field that is directed outward from the void center. This gives rise to an outward ion flow that exerts an outward ion drag force on the dust particles. Therefore in equilibrium there is a balance of forces on a dust particle: an inward electrostatic force and an outward ion drag force. Using the models based on ion drag force,4,18,19 thermophoretic force,4,14,19 and some other forces,4,19 linear20–23 analyses have been made to predict the formation of dust voids in dusty plasmas. However, computer simulation studies24 show that the ion-neutral collisions increase the gas temperature by a maximum of 1 K, and that the thermophoretic force or the ion drag force cannot explain the appearance of the void. They also imply that the ion drag force model of a dust void only works if one accounts for the enhancement of the ion drag force.25 Since observed voids are of large amplitudes, the force balance models are inappropriate for their formation. Accordingly, nonlinear models26 –28 have been developed to explain the formation of a plasma void in a dusty plasma. Specifically, Mamun et al.26 suggested that a one-dimensional plasma void is formed due to the presence of ion holes associated with a nonisothermal ion distribution that is produced by ion trapping in the negative plasma potential. Jovanovic´ and Shukla27 have shown that cylindrical symmetric dust voids are caused by a large potential drop associated with the ion holes and double layers in a plasma with a model dust Boltzmann density distribution. Avinash et al.28 presented a nonlinear theory for the void formation in a colloidal plasma. Their model is based on the nonlinear evolution of a zero-frequency linear instability which grows in the nonlinear regime to form a void. The dust grains in a dusty plasma could be heated up due to a variety of processes,29 including stochastic dust charge fluctuations, the variations of the plasma parameters, radiative heating, etc. In such a situation, the dust particle distribution can be far away from the Maxwellian. A nonthermal dust particle distribution can also arise due to the dust particle trapping in electrostatic perturbations that are created by two-stream instabilities.3 Accordingly, for a non-Maxwellian dust particle distribution, Schamel et al.30 predicted the existence of a continuum, undamped, ultralow-frequency dust

Dusty plasmas1– 4 have become an outstanding and challenging research area because of their ubiquity, versatile applications, and complexities. The physics of dusty plasmas has not only introduced a great variety of new phenomena associated with waves and instabilities,5–9 but also it has initiated a number of new experimental discoveries. The most important experimental discoveries in dusty plasmas are dust plasma crystals,10–12 dust Mach cones,13 dust voids,14,15 etc. Though the entire plasma volume is usually filled by the dust particles, dust-free regions 共known as dust voids兲 inside the dust cloud have been observed in dust laden plasmas.14,15 Morfill et al.14 observed centimeter-size stable dust voids, which formed without any initial turbulent phase in their microgravity experiment 共where dust particles are already sufficiently large兲. Samsonov and Goree15 observed that as dust particles in a sputtering plasma grew in diameter, the void was developed by a sudden onset of a filamentary mode in which both the ionization rate and the dust number density were modulated. These experimental observations14,15 thus reveal that a dusty plasma is not always composed of a homogeneous distribution of dust particles, but under some conditions, it is accompanied by a dust-free region. Voids have also been observed in colloidal suspensions16 as well as in the Milky Way stretching across the sky.17 A number of theoretical models have been proposed to explain the physics of the void formation. Morfill et al.14 predicted that the thermophoretic force 共a neutral temperature gradient force兲, which is more dominant than all other forces on a dust particle in the absence of gravity, is responsible for the formation of voids. Goree et al.18 explained the dust void formation by the balance of the electrostatic and ion drag forces on a dust particle. The balance of the electrostatic and ion drag forces involves electron depletion 共reduction of the electron number density in a dust cloud due to the absorption on dust particles兲 and electron-impact ionization. The electron-impact ionization rate is therefore dea兲

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Phys. Plasmas, Vol. 11, No. 5, May 2004

A. A. Mamun and P. K. Shukla

FIG. 1. The potential profiles 共upper plot兲 represented by Eq. 共11兲 and the corresponding dust number density profiles 共lower plot兲 for ␴ e ⫽0.1, ␤ ⫽⫺0.9, ␴ d ⫽10, ␣ ⫽0.01, Z d ⫽40 共solid curves兲, Z d ⫽15 共dotted curves兲, and Z d ⫽10 共dashed curves兲.

FIG. 2. The potential profiles 共upper plot兲 represented by Eq. 共11兲 and the corresponding dust number density profiles 共lower plot兲 for ␴ e ⫽0.1, ␤ ⫽⫺0.9, ␴ d ⫽10, Z d ⫽10, ␣ ⫽0.06 共solid curves兲, ␣ ⫽0.03 共dotted curves兲, and ␣ ⫽0.01 共dashed curves兲.

acoustic waves propagating with a phase speed close to the dust thermal speed. They are characterized by the deficit or surplus of the dust particles trapped in the trough of the wave potential. In this paper, we adopt the dust-phase-space vortex model of Schamel et al.30 to show the existence of a dust void in a dusty plasma with Boltzmann electron and ion density distributions. A self-consistent solution of Poisson’s equation reveals the formation of large amplitude rarefactive dust density perturbations associated with a positive plasma potential. The void size decreases 共increases兲 with the increase of the dust charge density 共the dust temperature兲. Thus our approach of understanding a dust void is based on a stationary BGK 共Bernstein-Greene-Kruskal兲 electrostatic equilibrium of the Poisson-Vlasov system of equations in which the distributions of electrons and ions are Maxwellian, while that of charged dust grains are non-Maxwellian in the wave potential. Our approach is therefore different from those models which are based on the balance of different forces4,19 acting on dust grains in a plasma. Our model is relevant to space dusty plasmas, particularly in Saturn’s dusty rings. The manuscript is organized as follows. The governing equations are provided in Sec. II. The properties of small amplitude dust voids are examined in Sec. III. The possibility for the formation of arbitrary amplitude dust voids for space dusty plasma parameters are investigated in Sec. IV. Finally, a brief discussion is presented in Sec. V.

II. GOVERNING EQUATIONS

We consider a one-dimensional, collisionless, unmagnetized dusty plasma composed of electrons and ions following Boltzmann distributions, negatively charged dust particles obeying a trapped/vortex-like distribution. The electrostatic potential distribution associated with the charge-density perturbation is obtained by Poisson’s equation, d 2␾ ⫽ ␣ Z d n d ⫹ 共 1⫺ ␣ Z d 兲 n e ⫺n i , d␰2

共1兲

where n e , n i , and n d are the electron, ion, and dust number densities, normalized by their equilibrium values n e0 , n i0 , and n d0 , respectively, ␾ is the electrostatic plasma potential normalized by k B T i /e (T i is the ion temperature, k B is the Boltzmann constant, and e is the magnitude of the electron charge兲, Z d is the number of electrons residing on the dust grain surface, ␰ is the space variable normalized by ␭ Di ⫽(k B T i /4␲ n i0 e 2 ) 1/2, and ␣ ⫽n d0 /n i0 . The normalized electron and ion number densities (n e and n i ), which follow the Boltzmann distribution, are given by n e ⫽exp共 ␴ e ␾ 兲 ,

共2兲

n i ⫽exp共 ⫺ ␾ 兲 ,

共3兲

where ␴ e ⫽T i /T e . The Boltzmann distribution follows from the balance of the electrostatic and pressure gradient forces.



Dust voids due to dust-phase-space vortices in plasmas

FIG. 3. The potential profiles 共upper plot兲 represented by Eq. 共11兲 and the corresponding dust number density profiles 共lower plot兲 for ␴ e ⫽0.1, ␤ ⫽⫺0.9, Z d ⫽10, ␣ ⫽0.06, ␴ d ⫽60 共solid curves兲, ␴ d ⫽20 共dotted curves兲, and ␴ d ⫽10 共dashed curves兲.

To model the dust distribution in the presence of trapped particles, we employ the Schamel distribution31,32 which solves the dust Vlasov equation.30 Thus we have f d ⫽ f d f ⫹ f dt , where f df⫽

1

冋

册

1 exp ⫺ 共 v 2 ⫺2Z d ␴ d ␾ 兲 , 2 冑2 ␲

共4兲

for 兩 v 兩 ⬎ 冑⫺2Z d ␴ d ␾ , and f dt ⫽

1

冋

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FIG. 4. The direct solutions of Poisson’s equation 关Eq. 共1兲兴 for the potential profiles 共upper plot兲 and the corresponding dust number density profiles 共lower plot兲 for ␴ e ⫽0.1, ␤ ⫽⫺0.3, ␴ d ⫽10, ␣ ⫽0.01, Z d ⫽80 共solid curves兲, Z d ⫽60 共dotted curves兲, and Z d ⫽40 共dashed curves兲.

n d ⫽I 共 Z d ␴ d ␾ 兲 ⫹

2

冑␲ 兩 ␤ 兩

W D 共冑⫺ ␤ Z d ␴ d ␾ 兲 ,

共6兲

where I 共 x 兲 ⫽ 关 1⫺erf共冑x 兲兴 exp共 x 兲 , W D 共 x 兲 ⫽exp共 ⫺x 2 兲

冕

x

0

共7兲共8兲

exp共 y 2 兲 dy.

We note that for Z d ␴ d ␾ ⬍1, Eq. 共6兲 gives

册

1 exp ⫺ ␤ 共 v 2 ⫺2Z d ␴ d ␾ 兲 , 2 冑2 ␲

共5兲

for 兩 v 兩 ⭐ 冑⫺2Z d ␴ d ␾ . Here v is the dust particle speed normalized by the dust thermal speed v Td ⫽(k B T d /m d ) 1/2, ␴ d ⫽T i /T d , and 兩 ␤ 兩 (⫽T d /T dt ), which is the ratio of the free dust temperature T d to the trapped dust temperature T dt , is a parameter determining the number of trapped dust particles. We note that the dust distribution function, as prescribed above, is continuous in velocity space, and satisfies the regularity requirements for an admissible BGK solution.33 It is obvious from Eqs. 共4兲 and 共5兲 that ␤ ⫽1 ( ␤ ⫽0) represents a Maxwellian 共flat topped兲 distribution, whereas ␤ ⬍0 represents a vortex-like excavated trapped dust distribution corresponding to an underpopulation of trapped dust particles. The situation ␤ ⬍0 is of our present interest. Thus integrating the dust distributions over the velocity space, the dust number density n d for ␤ ⬍0 can be expressed as

n d ⯝1⫹Z d ␴ d ␾ ⫺

4 共 1⫺ ␤ 兲 3 冑␲

1 共 Z d ␴ d ␾ 兲 3/2⫹ 共 Z d ␴ d ␾ 兲 2 2

⫹¯ .

共9兲

We are interested in examining the possibility for the formation of a standing dust void in a dusty plasma. III. SMALL AMPLITUDE DUST VOIDS

We first consider small amplitude dust holes ( ␾ Ⰶ1) for which Eqs. 共1兲–共3兲 and 共9兲 reduce to 4 d 2␾ ⯝a ␾ ⫺ b ␾ 3/2, d␰2 3

共10兲

3/2 where a⫽1⫹(1⫺ ␣ Z d ) ␴ e ⫹Z 2d ␣ ␴ d and b⫽Z 5/2 d ␴ d ␣ (1 ⫺ ␤ )/ 冑␲ . Now, using appropriate boundary conditions 共namely, ␾ →0 and d ␾ /d ␰ →0 at ␰ ⫽⫾⬁) the localized solution of Eq. 共10兲 is


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A. A. Mamun and P. K. Shukla

FIG. 5. The direct solutions of Poisson’s equation 关Eq. 共1兲兴 for the potential profiles 共upper plot兲 and the corresponding dust number density profiles 共lower plot兲 for ␴ e ⫽0.1, ␤ ⫽⫺0.3, ␴ d ⫽10, Z d ⫽30, ␣ ⫽0.03 共solid curves兲, ␣ ⫽0.02 共dotted curves兲, and ␣ ⫽0.01 共dashed curves兲.

␾ ⫽⌼ sech4

冉冊

␰ , ⌬

共11兲

where ⌼⫽(15a/16b) 2 and ⌬⫽4/冑a. Equation 共11兲 depicts a positive plasma potential, resulting an electron density hump 共with the amplitude ␴ e ⌼), and an ion density dip 共with the amplitude ⫺⌼). The width of both the electron density hump and the ion density dip is ⌬. The dust particle number density profiles corresponding to the potential structure given by Eq. 共11兲 for different dusty plasma parameters 共viz., Z d , ␣ and ␴ d ) are displayed in Figs. 1–3. Figures 1–3 show that for ␾ Ⰶ1, there exists dust particle number density holes 共dust voids兲. Figure 1 共2兲 implies that the dust voids shrink when the dust grain charge 共dust particle number density兲 is increased. Figure 3 indicates that the dust voids enlarge when the dust temperature is increased. To compare our results shown in Figs. 1–3 with our analytical results, let us approximate the amplitude ⌼ and the width ⌬. Using Z d ␣ ⬍1, ␴ e ⬍1, and Z d ␴ d Ⰷ1, which are valid in most space1– 4 and laboratory9–15 dusty plasma situ⫺1 ⫺2 and ⌬ ations, we have ⌼⯝2.76Z ⫺1 d ␴ d (1⫺ ␤ ) ⫺1 ⫺1/2 ⫺1/2 ␴ d . It is important to note here that our ⯝4Z d ␣ weakly nonlinear theory discussed above is only valid for Z d ␴ d ⌼⬍1, i.e., ⫺ ␤ ⬎0.662. IV. ARBITRARY AMPLITUDE DUST VOIDS

We are now interested in examining the possibility for the formation of arbitrary amplitude standing dust holes

FIG. 6. The direct solutions of Poisson’s equation 关Eq. 共1兲兴 for the potential profiles 共upper plot兲 and the corresponding dust number density profiles 共lower plot兲 for ␴ e ⫽0.1, ␤ ⫽⫺0.3, Z d ⫽30, ␣ ⫽0.01, ␴ d ⫽40 共solid curves兲, ␴ d ⫽20 共dotted curves兲, and ␴ d ⫽10 共dashed curves兲.

共voids兲. We numerically solved Eq. 共1兲 with n e , n i , and n d given by Eqs. 共2兲, 共3兲, and 共6兲, respectively, and obtained the plasma potential structures, and the corresponding dust number density profiles for different values of dust parameters, namely Z d , ␣, ␴ d , and ␴ e . We found that for the same set of dusty plasma parameters, the direct numerical solutions of Poisson’s equation 关Eq. 共1兲兴 gave the same results as we obtained in the small amplitude limit. This means that for usual dusty plasma parameters1– 4,9–15 and for ⫺ ␤ ⬎0.662, the weakly nonlinear theory 共which is valid for ⫺ ␤ ⬎0.662) presented above is sufficient for describing dust holes 共voids兲. However, it is interesting to examine the selfconsistent and direct solutions of Poisson’s equation when the weakly nonlinear theory is no longer valid, i.e., when ⫺ ␤ ⬍0.662. Therefore we obtain direct numerical solutions of Poisson’s equation for ␤ ⫽⫺0.3, and examine the effects of dusty plasma parameters 共viz., Z d , ␣, ␴ d , and ␴ e ) on these numerical solutions. We note that one can numerically solve Eq. 共1兲 either as a boundary-value problem or as an initial-value problem. However, for simplicity, we have solved it as an initial value problem by using the initial values, ␾ (0)⫽0 and ␾ ⬘ (0)⫽10⫺6 , where ␾ ⬘ ⫽d ␾ /d ␰ 关one can take some other values of ␾ ⬘ (0), since the increase 共decrease兲 in its value only shifts the potential or density profile towards 共away from兲 the origin兴. The results are displayed in Figs. 4 –7. Figures 4 –7 show that the self-consistent solutions of Poisson’s equation give rise to the dust density holes 共dust voids兲. Figure 4 共5兲 implies that the dust voids shrink when the dust grain charge 共dust particle number density兲 is



Dust voids due to dust-phase-space vortices in plasmas

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applicable to understand experimental observations of a dust void in plasmas if there are indications of dust grain heating and associated modification of the dust particle distribution function due to quasilinear effect. We therefore propose to conduct new laboratory experiments in which dust grains are of low charge state and are sufficiently heated up due to a variety of processes,29 such as stochastic dust charge fluctuations, the variations of the plasma parameters, radiative heating, etc. ACKNOWLEDGMENTS

The work of A. A. Mamun was supported by the MaxPlanck Institut fu¨r Extraterrestrische Physik at Garching. He thanks Professor Greg Morfill for encouragement. C. K. Goertz, Rev. Geophys. 27, 271 共1989兲. D. A. Mendis and M. Rosenberg, Annu. Rev. Astron. Astrophys. 32, 418 共1994兲. 3 F. Verheest, Waves in Dusty Space Plasmas 共Kluwer, Dordrecht, 2000兲. 4 P. K. Shukla and A. A. Mamun, Introduction to Dusty Plasma Physics 共Institute of Physics Publishing Ltd., Bristol, 2002兲. 5 N. N. Rao, P. K. Shukla, and M. Y. Yu, Planet. Space Sci. 38, 543 共1990兲. 6 P. K. Shukla and V. P. Silin, Phys. Scr. 45, 508 共1992兲. 7 F. Melandsø, Phys. Plasmas 3, 3890 共1996兲. 8 M. Rosenberg, J. Vac. Sci. Technol. A 14, 631 共1996兲. 9 R. L. Merlino, A. Barkan, C. Thompson, and N. D’Angelo, Phys. Plasmas 5, 1607 共1998兲. 10 J. H. Chu and L. I, Phys. Rev. Lett. 72, 4009 共1994兲. 11 H. Thomas, G. E. Morfill, V. Demmel, J. Goree, B. Feuerbacher, and D. Mohlmann, Phys. Rev. Lett. 73, 652 共1994兲. 12 Y. Hayashi and K. Tachibana, Jpn. J. Appl. Phys., Part 1 33, 804 共1994兲. 13 D. Samsonov, J. Goree, Z. W. Ma, A. Bhattacharjee, H. Thomas, and G. E. Morfill, Phys. Rev. Lett. 83, 3649 共1999兲. 14 G. E. Morfill, H. Thomas, U. Konopka, H. Rothermel, M. Zuzic, A. Ivlev, and J. Goree, Phys. Rev. Lett. 83, 1598 共1999兲; H. Thomas, G. E. Morfill, and V. N. Tsytovich, Plasma Phys. Rep. 29, 895 共2003兲. 15 D. Samsonov and J. Goree, Phys. Rev. E 59, 1047 共1999兲. 16 B. V. R. Tata, E. Yamahara, P. V. Rajamani, and N. Ise, Phys. Rev. Lett. 78, 2660 共1997兲. 17 H. J. Rood, Annu. Rev. Astron. Astrophys. 26, 245 共1988兲. 18 J. Goree, G. E. Morfill, V. N. Tsytovich, and S. V. Vladimirov, Phys. Rev. E 59, 7055 共1999兲. 19 T. Nitter, Plasma Sources Sci. Technol. 5, 93 共1996兲. 20 A. V. Ivlev, D. Samsonov, J. Goree, G. E. Morfill, and V. E. Fortov, Phys. Plasmas 6, 741 共1999兲. 21 A. V. Ivlev and G. E. Morfill, Phys. Plasmas 7, 1094 共2000兲. 22 X. Wang, A. Bhattacharjee, S. K. Gou, and J. Goree, Phys. Plasmas 8, 5018 共2001兲. 23 K. Avinash, Phys. Plasmas 8, 351 共2001兲. 24 M. R. Akdim and W. J. Goedheer, Phys. Rev. E 65, 015401 共2001兲. 25 S. A. Khrapak, A. V. Ivlev, G. E. Morfill, and S. K. Zhdanov, Phys. Rev. Lett. 90, 225002 共2003兲. 26 A. A. Mamun, P. K. Shukla, and R. Bingham, Phys. Lett. A 298, 179 共2002兲. 27 D. Jovanovic´ and P. K. Shukla, Phys. Lett. A 308, 369 共2003兲. 28 K. Avinash, A. Bhattacharjee, and S. Hu, Phys. Rev. Lett. 90, 075001 共2002兲. 29 G. Sorasio, R. A. Fonseca, D. P. Resendes, and P. K. Shukla, in Dust Plasma Interaction in Space, edited by P. K. Shukla 共Nova Science Publishers, New York, 2002兲, p. 37. 30 H. Schamel, N. Das, and N. N. Rao, Phys. Plasmas 8, 671 共2001兲. 31 H. Schamel, Plasma Phys. 14, 905 共1972兲. 32 H. Schamel, Phys. Plasmas 7, 4831 共2000兲. 33 I. B. Bernstein, J. M. Greene, and M. D. Kruskal, Phys. Rev. 108, 546 共1957兲. 1

2

FIG. 7. The direct solutions of Poisson’s equation 关Eq. 共1兲兴 for the potential profiles 共upper plot兲 and the corresponding dust number density profiles 共lower plot兲 for ␤ ⫽⫺0.3, Z d ⫽30, ␣ ⫽0.01, ␴ d ⫽10, ␴ e ⫽0.1 共solid curves兲, ␴ e ⫽0.05 共dotted curves兲, and ␴ e ⫽0.01 共dashed curves兲.

increased. Figure 6 indicates that the dust voids enlarge when the dust temperature is increased. Figure 7 shows that the effect of ␴ e is almost insignificant on the void structure. V. DISCUSSION

We have considered a one-dimensional, collisionless, unmagnetized plasma composed of Maxwellian electrons and ions, and non-Maxwellian negatively charged dust particles. The non-Maxwellian distribution of dust particles has been obtained due to their trapping in large amplitude plasma potentials. We have obtained self-consistent solutions of Poisson’s equation for both small and arbitrary amplitudes. It has been shown that the dust void shrinks 共enlarges兲 when the dust charge density 共dust temperature兲 is increased. The present analysis of a dust void uses a Boltzmann ion distribution and a nonisothermal dust particle distribution, contrary to our previous work26 which employed nonisothermal ions and an ensemble of cold stationary dust grains. Our results should be useful for understanding the origin of a dust void in space dusty plasmas, particularly in Saturn’s rings, which are under radiative background and are mainly composed of charged dust grains. We hope that forthcoming observations from the CASSINI mission will lend support to our theoretical prediction. The present theory should also be
