Three-dimensional structure of a string-fluid complex plasma

Three-dimensional structure of complex (dusty) plasmas was investigated under long-term microgravity conditions in the International-Space-Station-based PK-4 facility. The microparticle suspensions were confined in a polarity-switched dc discharge. The experimental results were compared to the results of the molecular dynamics simulations with the interparticle interaction potential represented as a superposition of isotropic Yukawa and anisotropic quadrupole terms. Both simulated and experimental data exhibited qualitatively similar structural features indicating the bulk liquid-like order with the inclusion of solid-like strings aligned with the axial electric field.

PK-4 [20] laboratory is a presently operational complex plasma facility on board the ISS. It is intended for the microgravity investigations of anisotropic fluid phase of complex plasmas. 3D analysis of microparticle suspensions trapped by a polarity-switched discharge in the ground-based PK-4 setup showed the presence of crystalline order [21]. Under the short-term microgravity conditions during the parabolic flights, the suspensions trapped in a PK-4 chamber in a combined rf and dc discharge were investigated. Their 3D analysis revealed coexistence of solid-, fluid-and string-like phases [22].
In this work, we report on the first observation of a 3D structure of a uniform string-fluid complex plasma. Up to now, the observations of the isotropic-to-stringfluid transition in complex plasmas were either incomplete or to a certain extent controversial. Electrorheology in plasmas was discovered in [16], where the structural analysis was performed using the anisotropic scaling indices. Further analysis of the data from the same experiments [13] showed that the external fast oscillating electric field caused crystallization of the microparticle suspension and not the transition from isotropic to string fluid. In parabolic flight experiments of Ref. [23], the microgravity conditions were achieved only for about 20 s in each parabola. Unlike [23], our work was done in an ISSbased facility. This gave us sufficient time to investigate the 3D structure of our microparticle suspensions and to observe its evolution. In the 3D structure of our suspension, we were able to identify the undoubtful signatures of strings. Similarly to Refs. [13,16,23], we supplemented our experimental data with molecular dynamics (MD) simulations. Data from the experiments and simulations were processed using the same techniques.
Experiments were performed in the flight model of PK-4 setup on board the ISS [20]. In PK-4, a dc discharge plasma is produced inside a glass tube of 3 cm diameter. The discharge can be operated in continous dc as well as in polarity switching mode. Typical polarity switching frequency of the order of hundreds Hz is still low enough for the dc discharge to stabilize within half of the switching period, but already high enough for the microparticles not to react on the polarity switching. Maximal current in the dc mode as well as for one of the polarities in the polarity switching mode is 3 mA. Gas pressure can be varied between 0.1 and 2 mbar.
The working area of the tube has a length of about 20 cm. In this area, the microparticles can be illuminated by a laser sheet. The scattered light is observed by the two video cameras.
Melamine formaldehyde microparticles are injected into the plasma chamber outside of the working area and need to be transported before they can be seen by the cameras. In this particular work, a dc trapping technique, in which the microparticles of 3.4 µm diameter are transported by a dc discharge in argon (current 0.5 mA, pressure 0.4 mbar) and trapped by a polarity-switched discharge (current 0.5 mA and frequency 500 Hz), was used.
The trapped microparticle suspension was scanned in the direction perpendicular to the image (Fig. 1a). For this, the cameras and the laser optics were synchronously moved to keep the plane of the laser sheet always in the focus of the cameras. The scanning speed of v scan = 0.9 mm/s results in about 15 µm separation between the consecutive frames along the Y axis. Video data from camera 1 only were used in this work. The field of view with the dimension 600 × 600 pixels centered on the axis of the plasma chamber was used. Axial position of this field of view was chosen in the rightmost part of the camera 1 image, where the illumination laser sheet has minimal thickness.
Scanning allowed to reconstruct the 3D structure of the microparticle suspension. This was done in the following way. First, all the 2D images recorded during the scan were stacked into a 3D image with a voxel volume of 14.18 × 15.00 × 14.25 µm 3 . In this 3D image, the objects consisting of voxels with the intensities above certain threshold were identified. These objects correspond to the trajectories of each microparticle during the scan. 3D position of each microparticle was determined by weighting the intensities of the voxels belonging to the respective 3D object. An example of a reconstructed 3D structure is shown in Fig. 1b. Five scans with the interval of ≈ 2 min between them were performed.
We show the data from the two scans: The first scan was performed right at the beginning of the experiment, whereas the second was performed about 6 min later. Diameter of the suspension decreased from 8.5 mm to 7.2 mm between those two scans, whereas the discharge parameters were kept constant. The average microparticle number density was about 3.4 × 10 4 cm −3 . The microparticle charge according to the measurements in [24] was about 2100 elementary charges.
We compared the experimental observations with the results of MD simulations [13,16,23]. In those simulations, the microparticle interaction potential was a superposition of Yukawa and quadrupole-like terms [16]. The origin of the anisotropy is the fast oscillating electric field of a polarity-switched dc discharge. The quadrupole-like term mimics the polarization of an ion cloud screening the microparticle. The simulated system was embedded in the Langevin thermostat. We used the following parameters: thermal ion Mach number M = 1.2, screening parameter κ = 3.6 and coupling parameter Γ ≃ 10 3 . In addition, we used the data for the Yukawa melt [25] with κ = 3 as a reference isotropic fluid. Estimation based on the dust-free plasma density 2.1 × 10 8 cm −3 for the experimental conditions under consideration yields κ ≈ 3.6. Ion Mach number for the electric field of 2.5 V/cm estimated using the modified Frost formula [26] yields M ≈ 0.33.
Using the 3D microparticle positions, we determined the pair correlation function of the suspension where r, θ and φ are the current spherical coordinates, r ij , θ ij and φ ij are the length, polar angle and azimuthal angle of a vector connecting microparticles i and j, N is the number of microparticles, n d is the macroscopic microparticle number density. About 6 × 10 3 microparticles in the central part of the suspension were used for G(r, θ, φ) calculation. We have also defined a twodimensional distribution G φ (r, θ) = 2π 0 G(r, θ, φ)dφ. The suspension exhibited a jerky drift in the axial direction. The axial velocity distribution of the microparticles measured between the scans can be characterized with the average v drift = 0.2 mm/s and full width at half maximum ∆v fwhm = 0.7 mm/s. The drift weakens the correlations between the microparticles in the scanning direction. We estimate the maximal radius r max , within which the correlations are not affected by the drift, as r max /D = v scan / v 2 drift + (∆v fwhm /2) 2 , where D is the average interparticle distance. For our experimental conditions with D = 0.32 mm, r max = 0.79 mm. For the calculation of G(r, θ, φ), we consider the maximal correlation radius equal to 1 mm. Figs. 2a and 2b show the G φ function for the two scans. In both cases, correlations exhibit three peaks over r cor- responding to D, 2D and 3D. The polar angle correlations exhibit well defined peaks at θ = ± π 2 over three interparticle distances. These peaks reveal strong correlations in the X direction, suggesting string-like character of the suspension. Fig. 2b shows even more complicated arch-like structures at least for two interparticle distances, which may be interpreted as characteristics of interstring correlations. Similar peaks at θ = ± π 2 and a very similar structure in G φ is observed in our MDsimulations (Fig. 2c).
In Fig. 3(a), g(r) and the cumulative RDFs N (< r) = 4πn d r 0 ξ 2 g(ξ)dξ are plotted. The latter function allows to estimate the mean nearest neighbours number N nn in the first coordination shell. Experimental g(r) reveal nearly close packed structures (N nn = 12). Both twopoint correlation functions (see Fig. 3(b)) g(r) and P (α) exhibit liquid-like structures. Analysis of our suspensions with the help of bond orientational order parameters (BOOP) ( [28][29][30]) q 4 and q 6 confirms the liquid-like character of our suspensions (Fig. 4a, c, e). Comparison of the first peaks of g(r) with those for the Yukawa melt allows us to estimate the relative coupling parameter [31] Γ m /Γ ≃ 1.25, where Γ m is the coupling parameter at melting. Fig. 3(c) shows the p(θ) functions for all the four systems under consideration. All of them except for the isotropic Yukawa melt exhibit peaks in the vicinity of θ = ± π 2 . Angular distribution of nearest neighbors exhibits similar behavior. It is presented in Figs. 4(b), 4(d) and 4(f) in a projection onto a sphere. The areas with enhanced density of nearest neighbors clearly concentrate in the polar regions. We also note that RDF and BADF for the two experimental datasets look almost identical, θ/π p(θ)  In panel (a), the cumulative RDFs N (< r) are also plotted with dashed lines, revealing close packing (12 nearest neighbors in the first coordination shell) structures for the microparticle suspension. g(r) and P (α) clearly show liquid-like order, whereas p(θ) reveals string-like features in the axial direction. In panel (c), the range of angles between − π 2 and 0 is omitted due to symmetry.
whereas the PADF, angular distribution of nearest neighbors and G φ (r, θ) show enhancement of the string-like features.
To quantitatively characterize the strings that consti- In plates (b, d, f), the local number density of particles is color-coded. As seen from the BOOP data, the local orientational order clearly reveals liquid-like structure for all considered systems. Unlike BOOP data, the angular distributions demonstrate bond alignment along the X axis for the experimental system and simulated anisotropic system. Yukawa melt is an isotropic system, as expected.
tute our fluid, we calculated the probability distribution of the string-like clusters (SLCs). The calculation was performed in two steps: First, we identified the pairs of neighboring particles, whose bonds are almost parallel to the axis of the discharge tube. The allowed angular deviation from the axis was estimated using the PADFs (Fig. 3(c)). After that, the pairs were connected into SLCs with the help of friend-of-friends algorithm (see e.g. [11]). Fig. 5 shows the probability distributions for the observed microparticle suspensions and the simulated systems as well. Size spectra of SLCs decay exponentially for all the systems under consideration, however, the decay rate in the anisotropic cases decreases significantly with respect to the isotropic Yukawa melt. Also, the second scan, where the strings are much better developed, exhibits smaller decay rate than the first scan. By measuring the distances between the microparticles inside the SLCs and counting their statistics, we were able to calculate the one-dimensional RDF g s (r) of the strings. The result for the second scan is shown in the inset (a) of Fig. 5. The function is normalized to the total number of microparticles belonging to SLCs and their linear density in the strings. We stress that this g s (r) exhibits crystalline order. Therefore, the observed system represents bulk liquid with the inclusion of solidlike strings aligned along the discharge tube axis.
To summarize, we have investigated the 3D structure of the microparticle suspensions in a dc plasma with fast oscillating electric field under microgravity conditions. Although radial distribution, bond angle distribution functions and rotational invariants q 4 and q 6 exhibited fluid order, angular functions derived from the 3D pair correlation function showed clear axial peaks. Decay rate of the size spectrum of the string-like clusters was found to decrease on the increase of the axial peak. The strings exhibited solid-like order. Similar features were shown to appear in the MD simulations on the addition of the quadrupole-like component to the microparticle interaction potential. Finally, we observed for the first time the 3D structure of a string-fluid complex plasma. This opens up the possibilities for novel experiments, e.g., observations of isotropic-to-string-fluid transition, which will be explored using the PK-4 apparatus.
The authors thank Dr. C. Knapek for careful reading of the manuscript. All authors greatly acknowledge the joint ESA-Roscosmos "Experiment Plasmakristall-4" on-board the International Space Station. B. K. was supported by the German Academic Exchange Service (DAAD) with funds from the German Aerospace Center