Moment fluid equations for ions in weakly ionized plasma

A one-dimensional fluid model for ions in weakly ionized plasma is proposed. The model differs from the existing ones in two aspects. First, a more accurate approximation of the collision terms in the fluid equations is suggested. For this purpose, the results obtained using the Monte Carlo kinetic model of the ion swarm experiments are considered. Second, the ion energy equation is taken into account. The fluid equations are closed using a simple model of the ion velocity distribution function. The accuracy of the fluid model is examined by comparing with the results of particle-in-cell Monte Carlo simulations. In particular, several test problems are considered using a parallel plate model of the capacitively coupled radio-frequency discharge. It is shown that the results obtained using the proposed fluid model are in good agreement with those obtained from the simulations over a wide range of discharge conditions. An approximation of the ion velocity distribution function for the problem under consideration is also discussed.

Figure 1

(a) The gas-dynamic pressure as a function of the ion drift velocity (a logarithmic scale is used for both axes). Circles show the results obtained from the MC simulations for helium (blue) and argon (green). The solid line shows the approximation given by Eq. (1). (b) The effective collision frequency as a function of the ion drift velocity. Circles show the results obtained from the MC simulations for helium (blue) and argon (green). Solid lines show the approximations given by Eq. (2). Squares and crosses show the results obtained using the experimental data of Ref. [21] and data of Phelps [22], respectively. Dotted lines show the approximations proposed in Refs. [19, 37].

Figure 2

The effective collision frequency for argon (left) and helium (right) at low and moderate drift velocities. Squares show the results obtained using the experimental data of Ref. [21]. Solid lines show the approximations proposed in Refs. [19, 37] and given by Eq. (2). Circles show the results of our MC simulations. Triangles show the results obtained for argon using the data of MC simulations presented in Ref. [24]. The simulations of Ref. [24] were performed using the approximation of Phelps [33] both for the hard sphere and charge exchange cross sections.

Figure 3

The parallel IDF ($f_{\parallel }$) for drift flows. Circles show the IDF obtained from the MC simulations. Solid lines show the IDF calculated using the model of Ref. [25] (red line) and model (3) (green line). For simplicity, all results are normalized by the maximum value of the IDF obtained from the MC simulations.

Figure 4

Distributions of the ion number density ($n$), mean velocity ($u$), and internal energy ($p$) for different discharge conditions (here $n_0={10}^{15}{\mathrm{m}}^{-3}$ and a logarithmic scale is used for $n$ and $p$). Solid lines show the results obtained using the fluid model described in Sec. 3. Circles show the results obtained from the PIC-MCC simulations. Panel (a) shows the results for the first test problem (argon, $L=2\mathrm{cm}$). The corresponding gas pressures and voltage amplitudes are $13\mathrm{Pa}, 260\mathrm{V}$ (red); $27\mathrm{Pa}, 212\mathrm{V}$ (blue); $39\mathrm{Pa}, 195\mathrm{V}$ (green); $65\mathrm{Pa}, 160\mathrm{V}$(black). Panel (b) shows the results for the second test problem (argon, $L=6.7\mathrm{cm}$). The corresponding gas pressures and voltage amplitudes are $0.3\mathrm{Pa}, 200\mathrm{V}$ (red); $1\mathrm{Pa}, 160\mathrm{V}$ (blue); $6\mathrm{Pa}, 130\mathrm{V}$ (green), $13\mathrm{Pa}, 100\mathrm{V}$ (black). Panel (c) shows the results for the third test problem (helium, $L=6.7\mathrm{cm}$). The corresponding gas pressures and voltage amplitudes are $4\mathrm{Pa}, 450\mathrm{V}$ (red); $7\mathrm{Pa}, 300\mathrm{V}$ (blue); $13\mathrm{Pa}, 200\mathrm{V}$ (green); $40\mathrm{Pa}, 150\mathrm{V}$ (black).

Figure 5

Distributions of different terms in the ion momentum and energy equations (upper and lower rows, respectively). Here $n_0={10}^{15}{\mathrm{m}}^{-3}$. The results are presented for the second test problem (argon, $L=6.7\mathrm{cm}$) at the gas pressures 0.3, 1, $6\mathrm{Pa}$ and third test problem (helium, $L=6.7\mathrm{cm}$) at the gas pressure of $4\mathrm{Pa}$.

Figure 6

The parallel IDF ($f_{\parallel }$) at different positions in the discharge. The results are presented for the second test problem (argon, $L=6.7\mathrm{cm}$) at the gas pressures 0.3, 1, $6\mathrm{Pa}$ and the third test problem (helium, $L=6.7\mathrm{cm}$) at the gas pressures 4 and $13\mathrm{Pa}$. Blue lines show the IDF obtained from the PIC-MCC simulations. Red lines show the IDF for drift flows calculated using the model of Ref. [25]. Green lines show the IDF given by the model (3). For simplicity, all results are normalized by the maximum value of the IDF obtained from the PIC-MCC simulations.

Figure 7

Distributions of the surface potential (${\phi }_p$) and the ion drag force ($F_d$) calculated for a single dust particle of radius $3.4\mu \mathrm{m}$ at different positions in the discharge. The gas is argon, the discharge length is $3\mathrm{cm}$, the voltage amplitude is $50\mathrm{V}$, and the gas pressures are 20, 10, and $5\mathrm{Pa}$ (from up to down, respectively). Note that the dust potential is negative (${\phi }_p<0$) for the considered conditions. The results were obtained using different models of the IDF. Circles correspond to the IDF taken from the PIC-MCC simulations, red lines correspond to the IDF for drift flows, and green lines correspond to the IDF approximated as a shifted Maxwell distribution with the gas temperature.