Dust coupling parameter of radio-frequency-discharge complex plasma under microgravity conditions

Oscillation of particles in a dust crystal formed in a low-pressure radio-frequency gas discharge under microgravity conditions is studied. Analysis of experimental data obtained in our previous study shows that the oscillations are highly isotropic and nearly homogeneous in the bulk of a dust crystal; oscillations of the neighboring particles are significantly correlated. We demonstrate that the standard deviation of the particle radius vector along with the local particle number density fully define the coupling parameter of the particle subsystem. The latter proves to be of the order of 100, which is two orders of magnitude lower than the coupling parameter estimated for the Brownian diffusion of particles with the gas temperature. This means significant kinetic overheating of particles under stationary conditions. A theoretical interpretation of the large amplitude of oscillation implies the increase of particle charge fluctuations in the dust crystal. The theoretical estimates are based on the ionization equation of state for the complex plasma and the equation for the plasma perturbation evolution. They are shown to match the results of experimental data processing. Estimated order of magnitude of the coupling parameter accounts for the existence of the solid-liquid phase transition observed for similar systems in experiments.

Figure 1

Distribution of the absolute error involved in the proposed method of particle coordinate determination over the surface area of a pixel. The distance $dr$ between the center of a true Gaussian beam and the determined position of a particle is color coded. The beam amplitude is (a) $A=100$ and (b) $A=50$; for both cases, the width is $w=1.5\text{px}$ and the recognition brightness threshold is $I_{\mathrm{th}}=20$.

Figure 2

View of the particle trajectories convoluted into clews obtained by superposition of successive frames. For different trajectories, the trace length varies from 5 to 185 frames. Shown are (a) a view from the high-resolution camera and (b) its enlarged fragment. The origin of the coordinate system coincides with the void center, which finds itself almost at the discharge chamber axis in the middle of the space between the electrodes. The particle diameter is $2.55\mu \text{m}$ and the argon pressure is $10\text{Pa}$.

Figure 3

Standard deviation (a) $\delta x$ for the coordinate $x$ and (b) $\delta z$, for $z$ as a function of the minimum trace length (minimum number of successive frames available for the observation of an individual particle). Lines show the results of averaging over all particle trajectories within the $x$- and $z$-coordinate range shown in Fig. 1 (field of view of the high-resolution camera); the particle diameter and the gas pressure are the same as in this figure. Blue and red lines indicate the results obtained with and without the trend removal. The particle diameter and the gas pressure are the same as in Fig. 2.

Figure 4

Three-dimensional standard deviations of particles from its equilibrium positions vs the coordinate $z$ for different minimum trace lengths (color coded). Each point indicates averaging over the trace length of an individual particle. The range of coordinate $x$ corresponds to Fig. 2; the particle diameter and the gas pressure are the same as in this figure. For each particle, the trend was removed.

Figure 5

Standard deviations $\delta x$ and $\delta z$ vs (a) the coordinate $x$ and (b) $z$. Dots show averaging for the individual particles, for which the trace length exceeds 40 frames. The range of particle coordinates $z$ in (a) is the same as the range of the $Z$ axis in (b) and the range of particle coordinates $x$ in (b) is the same as the range of the $X$ axis in (a). Lines show fitting the experimental results by the quadratic polynomials. Red lines and dots indicate $\delta x$ and blue, $\delta z$. The particle diameter and the gas pressure are the same as in Fig. 2.

Figure 6

Probability to find a particle shifted from its equilibrium position (a) in the $x$ direction and (b) in the $z$ direction. Corresponding shifts are $\mathrm{\Delta }x$ and $\mathrm{\Delta }z$, respectively. The minimum trace length is 40 frames; averaging is performed over all particles in the range shown in Fig. 2; the particle diameter and the gas pressure are the same as in this figure.

Figure 7

Correlation coefficient $\kappa$ vs the interparticle distance $r$. Dots show averaged experimental results. Averaging was performed over all pairs of particles at the distance within the intervals of 15-$\mu \text{m}$ width. Line indicates the curve fit by exponential. The minimum trace length is 60 frames, the particle coordinate range corresponds to Fig. 2, and the particle diameter and the gas pressure are the same as in this figure.

Figure 8

Three-dimensional standard deviation of a particle from its equilibrium position vs the coordinate $z$. Dots represent the results of experimental data processing and line shows the calculations using formula (3). The gas pressure, the particle diameter, and the coordinate range are the same as in Fig. 2.

Figure 9

Same as Fig. 8. The particle diameter is $3.4\mu \text{m}$ and the argon pressure is $11\text{Pa}$.

Figure 10

Same as Fig. 8. The particle diameter is $3.4\mu \text{m}$ and the argon pressure is $20.5\text{Pa}$.

Figure 11

Coupling parameter for the subsystem of dust particles vs the relative distance from the void boundary along the discharge axis ($z_0$ is equal to the minimum $z$ for corresponding dust cloud; see Figs. 8–10). Dots represent the results of data processing for different experiments: circles, for $2a=3.4\mu \text{m}$ and $p=20.5\text{Pa}$; squares, for $2a=3.4\mu \text{m}$ and $p=11\text{Pa}$; and triangles, for $2a=2.55\mu \text{m}$ and $p=10\text{Pa}$. Dashed and dash-and-dot line indicate the lower and upper bounds from the theory [formula (4)], respectively, for the above-mentioned experimental conditions.