Simulating the dynamics of complex plasmas

Complex plasmas are low-temperature plasmas that contain micrometer-size particles in addition to the neutral gas particles and the ions and electrons that make up the plasma. The microparticles interact strongly and display a wealth of collective effects. Here we report on linked numerical simulations that reproduce many of the experimental results of complex plasmas. We model a capacitively coupled plasma with a fluid code written for the commercial package comsol. The output of this model is used to calculate forces on microparticles. The microparticles are modeled using the molecular dynamics package lammps, which we extended to include the forces from the plasma. Using this method, we are able to reproduce void formation, the separation of particles of different sizes into layers, lane formation, vortex formation, and other effects.

Figure 1

Modeled geometry, based on the PK-3 Plus chamber [82]. Two parallel electrodes with a radius of 3 cm are separated by a distance of 3 cm. The electrodes are surrounded by spacers 5 mm wide. The nominal sheath width is set to 5 mm. The sheaths are solved analytically, while the bulk plasma is treated with a fluid model. We only model the right half of the plasma chamber midplane and assume cylindrical symmetry.

Figure 2

Electron temperature $T_e$ in the plasma bulk at a gas pressure of 20 Pa and an input current of 20 mA. The gray lines in the top and bottom indicate the position of the electrodes and spacers.

Figure 3

Distribution of ion density $n_i$ (and, via quasineutrality, electron density) in the plasma bulk at a pressure of 20 Pa and an input current of 20 mA. The gray lines in the top and bottom indicate the position of the electrodes and spacers.

Figure 4

Ion flux ${\Gamma }_i$ in the plasma bulk as a function of position for a pressure $p=20\mathrm{Pa}$ and an input current $I=20\mathrm{mA}$.

Figure 5

Mean ion density as a function of rf voltage for different pressures. The lines are guides to the eye that connect series of constant pressure. The input currents used for each pressure were $I=5$, 10, 20, 30, and $40\mathrm{mA}$. For the two highest pressures and lowest input current, the voltage falls below 10 V and the plasma model is no longer applicable. That is why we omit these data in the plot. The case shown in Figs. 2 and 3 corresponds to the voltage $V_{\mathrm{rf}}=47.5\mathrm{V}$ and the pressure $p=20\mathrm{Pa}$. In general, the ion density increases with pressure and with voltage.

Figure 6

Magnitude of the ambipolar electric field $E_a$ in the plasma bulk at a pressure of 20 Pa and an input current of 20 mA. The gray lines in the top and bottom indicate the position of the electrodes and spacers.

Figure 7

Formation process of a particle cloud with a superposition of nine frames during the injection process (left) and the equilibrium particle cloud after 20 injections and a time span of $3.5\min$ after the last injection (right). The gas pressure is 20 Pa and the input current is 25 mA. The gray box in the images indicates the dimensions of the simulated plasma bulk: one-half of the vertical midplane of the plasma chamber without sheaths, with dimensions $4.5\times 2{\mathrm{cm}}^2$. The centerline of the plasma chamber corresponds to the left gray line at the edge of the field of view. The color of the microparticles indicates the vertical velocity $v_z$, ranging from blue (light gray) for $-2\mathrm{mm}/\mathrm{s}$ to red (not reached in the figure) for $2\mathrm{mm}/\mathrm{s}$. Purple (dark gray) indicates no movement in the $z$ direction, as for the time $t=0$ ms in the left panel.

Figure 8

Vortices composed of particles 3.4 $\mu$m in diameter in the PK-3 Plus setup under microgravity conditions in argon at a pressure of 10 Pa. The black lines show the streamlines of the particle movement. The vector field of the mean particle velocities is overlaid. The color and length of the vectors indicate the total velocity and their orientation in the direction of movement. The dimensions of the particle cloud are $33\times 16{\mathrm{mm}}^2$. Data courtesy of the PK-3 Plus team [82].

Figure 9

Simulated vortex for a pressure of 10 Pa, 3.4-$\mu$m-diam particles, and an input current of 16 mA. The particle charge was set to 1/2 of that obtained with orbital-motion-limited theory. The dimensions of the particle cloud are $23\times 10{\mathrm{mm}}^2$. The streamlines of the particle movement are shown in black, analogous to Fig. 8. Superimposed over this are the mean particle velocities. The length and the color of the vectors indicate the total velocity and their orientation in the direction of movement. The image was mirrored to the left side of the void to ease comparison with Fig. 8.

Figure 10

Particle cloud made up of three different particle sizes: 6.8-$\mu$m [red (light gray)], 3.4-$\mu$m (white), and 1.55-$\mu$m [blue (dark gray)] diameters at a pressure of 20 Pa and a current of 30 mA. The field of view is 33.9 $\times$ 20.0 mm${}^2$. The particles were injected in five sets, each composed of a bunch of each particle size. Within each set, the largest particles were injected first. After the injections, the cloud was left alone to evolve for 3.5 min, during which it sorted by particle size. Even after this time, the sorting is not complete: There are still some smaller particles mixed into the larger ones further out.

Figure 11

Lane formation in the PK-3 Plus laboratory on board the International Space Station. Microparticles $3.4\mu \mathrm{m}$ in diameter, shown in red (dark gray), move through a cloud of $6.8-\mu \mathrm{m}$-diam particles. The buffer gas is argon at a pressure of $30\mathrm{Pa}$. The crossover from (a) lane formation to (b) phase separation is visible. Courtesy of Du et al. [104] (©IOP Publishing Ltd and Deutsche Physikalische Gesellschaft; published under a CC BY-NC-SA license).

Figure 12

Batch of microparticles 3.4 $\mu$m in diameter (white) in a cloud of particles 6.8 $\mu$m in diameter, moving towards the center of the chamber. In the background, the positions of the small particles are overlaid on the background cloud during their voyage towards the center. The total field of view is $15\times 15{\mathrm{mm}}^2$. The inset (with a field of view of $3.8\times 3.8{\mathrm{mm}}^2$) shows one frame when the microparticles start penetrating into the background cloud, starting to form lanes. The gas pressure is 10 Pa and the input current is 12 mA. In contrast to the other examples shown in this paper, here the sheath width is set to $3\mathrm{mm}$ and the temperature that the microparticles are subjected to from the background gas is $600\mathrm{K}$. The vertical dashed line marks the approximate position where lane formation turns into phase separation.

Figure 13

Mach cone excited by a projectile moving through a cloud of $2.55-\mu \mathrm{m}$-diam particles at a gas pressure of $10\mathrm{Pa}$. The experiment was performed under microgravity in the PK-3 Plus laboratory on board the International Space Station. (a) Difference between two successive frames, making visible the Mach cone as the difference in the particle densities. (b) Superposition of 64 frames that shows the continuation of the trajectory upward. Courtesy of Schwabe et al. [106].

Figure 14

Simulated Mach cone in a cloud of $2.55-\mu \mathrm{m}$-diam particles at a gas pressure of $10\mathrm{Pa}$. The field of view is $33.9\times 22.0{\mathrm{mm}}^2$. A projectile is dragged by an external force horizontally towards the chamber center and upward. The positions of the projectile are shown with white circles that are spaced $10\mathrm{ms}$ apart in time. In the background, a single simulation frame is shown, with the corresponding projectile position as a solid circle. The colors of the background particles indicate the vertical velocity of the microparticles and are scaled between $-3$ [cyan (light gray)] and $3\mathrm{mm}/\mathrm{s}$ [blue (dark gray)]. The Mach cone is clearly visible in the movement of the particles. The image was mirrored to the left for easy comparison with Fig. 13.