Theory of dust and dust-void structures in the presence of the ion diffusion

A dust void is a dust-free region inside the dust cloud that often develops for conditions relevant to plasma processing discharges and complex plasma experiments. A distinctive feature of the void is a sharp boundary between the dust and dust-free regions; this is manifested especially clear when dissipation in the plasma is small and discontinuity of the dust number density appear. Here, the structure of the dust void boundary and the distribution of the dust and plasma parameters in the dust structure bordering the void is analyzed taking into account effects of dissipation due to the ion diffusion on plasma neutrals. The sharp boundary between the dust and void regions exists also in the presence of the ion diffusion; however, only derivatives of the dust density, dust charge, electron density and electric field are discontinuous at the void boundaries, while the functions themselves as well as derivatives of the ion drift velocity and the ion density are continuous. Numerical calculations demonstrate various sorts of diffusive dust void structures; the possibility of singularities in the balance equations caused by the diffusion process inside the dust structures is investigated. These singularities can be responsible for a new type of shocklike structures. Other structures are typically self-organized to eliminate the singularities. Numerical computations in this case demonstrate a set of thin dust layers separated by high density thin dust clouds similar to the multiple-layer dust structures observed in the laboratory and in the upper ionosphere. The possibility for existence of a few equilibrium positions of the void boundary is discussed.

Figure 1

Sketch of a void (left) and its converse (right). In the presence of the plasma ionization, a positive space potential develops, creating an outward ambipolar electric field that drives ions outward, applying an outward ion drag force, which can maintain a void. In the absence of the ionization, for example in a space plasma where plasma is generated far away, the dust cloud forms in the complementary shape of a spheroid, with its boundary sustained by an inward ion drag force driven by an inward electric field. Reprinted from Goree et al. [3].

Figure 2

Sketch of the one-dimensional simulation region. A dust cloud fills all space except for a void region of full width $2x_v$ with the center at $x=0$.

Figure 3

Distribution of the main dimensionless parameters in the void and the dust regions in the case when the ion diffusion is neglected. Here, $\tau =0.05$, $x_i=2$, $d=0.2$, $n_0=2$, and $n_{0e}=0.98$. At the dust boundary, the parameter $P$ as well as its derivatives are discontinuous. At the end of the structure, where $P=0$, the derivative of $P$ is also discontinuous.

Figure 4

Distribution of the main dimensionless parameters in the void and the dust regions in the case when the ion diffusion is taken into account. Here, the normalized ionization length is the same $x_i=2$ in the void as well as in the dust regions. Note the nonphysical domain of the negative values of $P$.

Figure 5

The same as in Fig. 2 but for the normalized ionization length $x_i=2$ in the void region and $x_i=4$ in the dust region. Note the nonphysical domain of the negative values of $P$.

Figure 6

The same as in Fig. 2 but for the normalized ionization length $x_i=2$ in the void region and $x_i=1$ in the dust region. The parameter $P$ is everywhere positive. The size of the dust region is of the order or less than $\tau$, i.e., it is of the order of the ion-neutral mean free path.

Figure 7

Distributions of the dimensionless parameters in the void and dust regions for the (normalized) critical ion density in the center $n_0=1.065$.

Figure 8

Distributions of the dimensionless parameters in the void and dust regions with the continuous change of the direction of ion flow. The normalized ionization rate is $x_i=2$ both in the dust and in the void regions.

Figure 9

Parameter $P$ in the dust region for $x_i=2$ in the void region and $x_i=8$ in the dust region. The dust void (size $x_v=0.091$) is to the left (not shown). Other parameters are $z_{0d}=2.9$, $P_{0}d=0.3<P_{cr,1}=0.525$, $P_{cr,2}=0.386$, $n_{e0d}=2.77$, and $n_{0d}=3$.

Figure 10

Shocklike structure in the dust region with the change of the direction of the ion flow velocity. The normalized ionization rate $x_i=2$ in the void and in the dust regions; $t=0.05$. The unperturbed normalized density $n_0=1$ shown in the figure are $n-n_0$ and $z-z_0$. The normalized ion flow velocity $u$ jumps from 0.2 to 0.44, the normalized ion density $n-n_0$ jumps from 1.38 to 0.97, the normalized dust charge $z-z_0$ jumps from 2.8 to 2.9, the parameter $P$ jumps from 0.34 to 0.36, and the electron density and the electric field have only jump of derivatives.

Figure 11

Behavior of the main dimensionless parameters behind the dust-charge diffusive shock.

Figure 12

The main dimensionless parameters in the vicinity of the dust-charge diffusive shock front.