PARTICLE SIMULATION ON MULTIPLE DUST LAYERS OF COULOMB CLOUD IN CATHODE SHEATH EDGE K. ASANO, S. NUNOMURA, T. MISAWA, N. OHNO and S. TAKAMURA Department of Energy Engineering and Science, Graduate School of Engineering, Nagoya University, Nagoya 464-8603, Japan Abstract We have observed a Coulomb dust cloud with a funnel shape composed of multiple layers formed in sheath-plasma boundary in a dc glow discharge. In order to make clear an underlying physics for formation of such a Coulomb cloud, we have performed the one- and three-dimensional particle simulations. The one-dimensional shows an effect of the self-gravity on the trapping of the dust particles in the plasmasheath boundary. The three-dimensional particle simulation shows that dust particles compose a Coulomb crystal structure by assuming a moderate friction force. A radial electric force causes multiple layers structure of dust particles. These simulation results are in a good agreement with an experiment ones. 1 Introduction Dusty plasmas, ubiquitous in space as well as in the laboratory, have recently attracted strong interests in experimental and theoretical investigations. In recent experiments, Coulomb crystals formed in dusty plasmas have been observed in a number of devices. In our experiment, three-dimensional volumetric Coulomb crystal-like cloud in a dc glow discharge configuration has been observed [1], which has a very interesting shape of three-dimensional funnel with vertex at bottom as shown in Figure 1. This dust cloud can be trapped in the plasma-sheath boundary area, because the dust cloud is supported against gravity by the electric field at the sheath edge above the cathode. Each dust particle has a large negative charge, then these dust particles couple strongly with each other through the repulsive Coulomb interaction. In the experiment, Coulomb coupling parameter is estimated to be 10-380. The present paper is intended to investigate the formation of dust cloud levitated near the plasma-sheath boundary by using the molecular dynamics (MD) simulation and to compare it with the experimental results. MD simulation is suitable for our purpose because it can calculate all forces directly which act on a dust particle such as the Coulomb interaction among particles. We also consider the change of dust s charge in the sheath region where the charge neutrality of ions and electrons is not satisfied. Three-dimensional funnel-shaped dust clouds can be reproduced in this MD simulation, which is in a good agreement with the experimental result. We have applied various external fields in our simulation to investigate the influence on the structure of the dust cloud. Radial electric force is found to have strong effects on the structural change of the dust cloud.
Fig. 1: (a):the typical three-dimensional funnel-shaped dust clouds with the superposition of a few discrete layers and (b):schematic view of experimental setup. Radius of dust particles is 1.5µm ± 0.25. 2 Simulation Model Figure 2 shows a schematic of the simulation model. The modified Child-Langmuir formula with frequent ion-neutral collisions is employed to determine the potential structure in the cathode sheath. The charge on each particle is determined by the product of its capacitance and the floating voltage. The floating voltage is obtained by the balance between the ion and electron fluxes to the surface of dust particle, in which a strong reduction of electron density due to the sharp increase of potential barrier in the sheath is considered. So in the deep inside of the sheath, the dust particle is charged positively, then the dust particle can not be levitated by the electric field. Fig. 2: (a):model of this simulation in cathode sheath edge and (b):the assumed potential profiles. The space potential in the cathode sheath is given as follows [3], V s (z) = 3 5 ( 3 2 ) 3/5 en se C s (δ z) 5/3, (1) ɛ 0 µ i where δ is the thickness of the cathode sheath, z is the height from the cathode surface, n se is the ion and electron density at the sheath edge, C s is the ion sound velocity, and, µ i is the ionic mobility given as (eλ i /2m i ) 1/2, where m i and λ i are the mass and the mean free path of an ion. When we assume that δ is 26mm, and the bias voltage of the cathode
electrode, V c,is 290V. The electric field can be given from eq.(1) as follows, E s (z) = { 2.12 10 5 (0.026 z) 2/3 [V/m] (z <δ) 0 (z δ). (2) The charge q d on a spherical dust particle is generally given as 4πɛ 0 r d V f, where r d and V f are the radius and the floating voltage of the dust particle, respectively. V f, defined as the potential difference between the surface of the dust particle and the space around the dust, is determined by the balance between the electron and the ion currents flowing into the surface of the dust particle. The electron current is given for the Maxwell-Boltzmann distribution by I e = 4πr 2 d en se 4 ( ) 1/2 ( ) 8kB T e e(vs (z)+v f (z)) exp πm e k B T e. (3) The ion current into a dust is described by ( ) 1/2 I i = πrden 2 kb T e se. (4) m i To simplify, we neglect the collection of ions based on the Coulomb collision with a negatively charged dust particle. Detailed analysis on this effect was discussed in [3]. V f should be given for these currents to balance, that is, I e + I i = 0. From eqs.(3) and (4), V f (z) = 1 ( ) k B T e π m e ln V s (z). (5) 2 e 8 m i Fig. 3: The electric field at the cathode sheath edge and the floating potential of a dust particle given by eq.(1) and eq.(5), respectively. The first term in the right side of eq.(5), shows a floating potential V f0 in a plasma. If an argon plasma is assumed in this simulation, m i is estimated to be 10 26 kg and V f0 is given to -5.1V for T e 1eV. Distribution of V f (z) and E s (z) in the cathode sheath are shown in Figure 3. V f becomes positive at z =23.7mm in the inside of sheath, which means the dust particle charges positively. This result indicates that a dust particle can not be levitated in the deep inside of the cathode sheath. The position where a dust particle charges positively depends on the electron temperature, T e. Equation of motion on a dust particle is described as follows, m d dv d dt = f d kv d, (6) where m d and v d are the mass and the velocity of dust particle. f d is the force acting on a charged dust particle, including gravity force, electric field force, and Coulomb forces,
f Coulombi, induced by other dust particles. It is given as follows, f coulombi = N j=1,i j 1 4πɛ 0 q i q j r 2 ij ( ) ( rij +1 exp r ) ij rij λ d λ d r ij where r ij r i r j and λ d is shielding length. k is friction term acting on a dust particle due to elastic collision with neutrals, given as k = n n v n m n πrd 2 [2], where m n, n n and v n are the mass, the density and the thermal speed of the neutrals, respectively. In this simulation k is estimated to be k 10 12 kg/s with an assumption of m n 10 26 kg, n n 10 23 m 3 and v n 1023m/s whose temperature, T n, is assumed to be 300K. And a collision time τ given by k/m d is calculated to be τ 10 2 s. (7) 3 Simulation Results and Discussion 3.1 One-dimensional simulations on multiple dust layers In order to understand basic characteristics of the levitation of dust particles in the cathode sheath edge, we have performed one-dimensional simulation along z by changing a number of the dust particles which are introduced into the sheath. The dust particle falling into cathode sheath is supported against gravity force by the electric force at the sheath edge given by q d E s. Figure 4 shows the dependence of the levitation positions of dust particles whose radius and weight are 2.0µm and 6.7 10 14 kg, respectively, on the number of levitated dust particles. V f0 is given as 2.9V, which makes a dust particle charge positive at z = 24.6mm. The dust particles in the sheath are supported by electric force and Coulomb force due to dust particles in lower side against downward force due to the gravity and Coulomb forces coming from other dust particles located in upper side. The dust particles above the sheath edge are supported by only Coulomb forces due to the downward dust particles. After the dust particles become stationary, then a new dust particle is added on them to make the column of the dust particles. As the number of levitated dust particle is increasing, this column is gradually moving downward due Fig. 4: One dimensional simulation on multiple dust layers. (a):shielding length, λ d = 0.2mm and (b):λ d = 20mm
to its own weight, so called, self-gravity. As mentioned in Sec. 2, in the deep inside of sheath a dust particle can not be charged enough negatively due to strong reduction of the electron density. So the dust particle being pushed down below the threshold position due to self-gravity, can not be supported and is falling to the cathode electrode. There is a limit to number of the dust particles to levitate at the sheath edge. Figure 4 also shows that number of the levitated dust particles depends on shielding length, λ d of the Coulomb force. The longer λ d, the less number of the levitated dust particles is obtained. But when λ d is longer than 20mm in this simulation, number of the levitated dust particles becomes constant. This is because that λ d = 20mm is enough long compared with the distance between lavitated dust particles which is about 10 2 mm 10 1 mm. 3.2 Three-dimensional simulations on multiple dust layers and Coulomb crystal In this section, we will analyze the three-dimensional structure of the dust cloud in cathode sheath edge. First, dust particles are put randomly within positions of x, y from 1.5mm to 1.5mm and z from 26mm to 26.1mm. The number of them is 250, whose radius and weight are 2.5µm and 1.3 10 13 kg. V f0 is given to be 5.1V, which makes dust particle charge positive at z = 23.7mm We assume that dust particles are under the effect of radial electric field such as E r (r) = Ar 3,r = x 2 + y 2. The dust particles, repulsive each other, are pushed inward and finally crystallize, as shown in Figure 5(a), when the radial electric field is given as E r (r) = 4.9 10 2 r 3 [V/mm]. On the other hand, when we assumed the strong radial electric field E r (r) = 0.8r 3 [V/mm], the structure of the dust particles is changed from the mono layer structure as shown in Figure 5(b). Four layers are clearly observed, and each of them is found to be crystallize individually. This is because that a strong radial electric force makes the repulsive dust particles approach too closely. Furthermore when we use lighter dust particles than those in Figure 5, less layers are observed. It means that the number of layers depends on the radius or weight of the dust particle because the light particle hardly moves to the deep inside of sheath against strong electric field. In the experiment, the dust particles have a dispersion of size distribution. We have employed completely the same-sized dust particles. It may suggest that a few dispersions of size distribution of the dust particle is necessary to make multiple layers, funnel-shaped dust cloud. Or, other kind of radial electric field may be needed. 4 Conclusion Molecular dynamics simulations have been performed to investigate the Coulomb crystal formation of dusty plasma near the plasma-sheath boundary.
Fig. 5: Coulomb cloud in cathode sheath edge in, (a):weak radial electric field, E r = 4.9 10 2 r 3 [V/mm] and (b):strong radial electric field, E r = 0.8r 3 [V/mm]. One-dimensional simulation shows that the column of dust particles supported by electric force at the sheath edge is gradually pushed downward with increase in number of the dust particle because of the effect of the self-gravity. The dust particles at bottom of the column which do not have negative charge, are dropped, which gives the limit to the number of levitated dust particles. Three-dimensional simulations can reproduce the dust cloud with multiple dust layers depending on the radial electric force. Weak radial electric force makes crystal mono layer, while a strong electric force makes multiple dust layers, which are crystallized individually. Number of layers in the sheath edge depends on radius or weight of the dust particle. Acknowledgment The authors wish to thank for Y. Uesugi for discussions. One of us (K.A.) is grateful to D. Nishijima and H. Kojima for making this document. References [1] S. Nunomura, N. Ohno and S. Takamura, Jpn. J. Appl. Phys. 36(1997)L949. [2] S. Nunomura, N. Ohno and S. Takamura, Jpn. J. Appl. Phys. 36(1997)877. [3] S. Nunomura, N. Ohno and S. Takamura, submitted to Phys. Plasmas.