Network analysis of three-dimensional complex plasma clusters in a rotating electric field

Network analysis was used to study the structure and time evolution of driven three-dimensional complex plasma clusters. The clusters were created by suspending micron-size particles in a glass box placed on top of the rf electrode in a capacitively coupled discharge. The particles were highly charged and manipulated by an external electric field that had a constant magnitude and uniformly rotated in the horizontal plane. Depending on the frequency of the applied electric field, the clusters rotated in the direction of the electric field or remained stationary. The positions of all particles were measured using stereoscopic digital in-line holography. The network analysis revealed the interplay between two competing symmetries in the cluster. The rotating cluster was shown to be more cylindrical than the nonrotating cluster. The emergence of vertical strings of particles was also confirmed.

Figure 1

Experimental setup. A 3D cluster of micron-size particles is suspended in the glass box mounted on top of the rf electrode in a capacitively coupled discharge. The particles are charged negatively and are manipulated by applying voltages on the conductive side plates of the box. By shifting sinusoidal signals on the adjacent plates by $\pi /2$, a rotating electric field is created at the position of particles. Expanded laser beams at 532 nm illuminate the particles from two perpendicular directions. The diffracted light is attenuated by neutral density filters (NDF) and registered by CCD cameras A and B. Insets show the interference patterns (a) of the whole cluster and (b) of an individual dust particle.

Figure 2

Dependency of detected communities on resolution parameter $\gamma$. (a) Ratio $\eta$ of intercommunity distance to community size for different values of $\gamma$ for the rotating (triangles) and nonrotating (squares) cluster. The maximum values are plotted as solid symbols. The standard deviations are plotted as error bars and are concealed by the symbols in the case of the nonrotating cluster. (b)–(d) Histograms of the number of particles per community for different values of $\gamma$.

Figure 3

Time evolution of detected strings for resolution parameter $\gamma =4$. (a) Projection of the rotating cluster on the $xy$ plane at $t=0$. Particles which were found to be in the same string are grouped together and have the same color. (b) String affiliation of particles over time. A transition from one string to another appears as a change in color of the line corresponding to the particle. (c),(d) Same for the nonrotating cluster.

Figure 4

Method for generating an unweighted network from the data. (a) 3D plot of the clockwise rotating cluster. The 60th particle is plotted as a blue triangle and the particles $j$ satisfying $|{\rho }_{60}-{\rho }_j|<\epsilon$ are plotted as red squares. (b) The corresponding adjacency matrix. The nodes connected to the 60th node can be read off the 60th column (or line) of the matrix. (c) Representation of the network. The nodes representing the particles have the same markers as in (a).

Figure 5

Comparison of network measures for the clockwise rotating cluster with the measures for null models of different ratios $R$. The histograms (for all particles and time steps) of measures on networks of the clockwise rotating cluster are plotted in black. They are identical in all three rows. The null models are plotted in red (gray) with $R=0$ (top row), 2 (middle row), and $\infty$ (bottom row), where $R$ is the ratio of particles from a perfectly spherical geometry to particles from a cylindrical geometry. First column: local connectivity ${\kappa }_{\nu }$; second column: clustering coefficient $C_{\nu }$; third column: average path length $L_{\nu }$. The differences are marked as hatched areas. The good agreement of all three network measures in the middle row is clearly visible.

Figure 6

Comparison of experimental cluster (top) to null model (bottom). Projections of the clockwise rotating cluster onto the $\rho z$ plane (a) and $xy$ plane (b). The two main components of $A^{\text{cyl}}$ are plotted as triangles (squares) and correspond to the inner (outer) ring regions of the cluster. For the sake of clarity the network edges are only plotted in panel (a). The network is identical to the one in Fig. 4, only the positions of the nodes are chosen differently: While in Fig. 4 a more abstract representation of the network was chosen to show its structure, here the positions correspond to the projections of the particles. (c),(d) The same for the null model whose network measures showed the best agreement with the cluster: The ratio of particles in a spherical arrangement to particles in a cylindrical arrangement is $R=2$.

Figure 7

Ratio $R=n^{\text{sph}}/n^{\text{cyl}}$ vs noise amplitude of null models for $A^{\text{cyl}}$ (a) and $A^{\text{sph}}$ (b). The results are averaged over 40 calculations, and the standard deviations are plotted as error bars. The results in Table 1 correspond to a noise amplitude of $0.15$ mm.

Figure 8

Projection of particle positions on the $\rho z$ plane for clockwise (a) and counterclockwise (b) rotating, and nonrotating (c) clusters. Vertical lines and semicircles are adjusted to the cylindrical structure of the rotating clusters and to the spherical structure of the nonrotating clusters, respectively, in order to guide the eye. Particle positions at each time step correspond to a transparent marker, leading to lines in the event of a string transition and solid structures if the particle stays in the same position throughout the time series.

Figure 9

Radial profiles of the repulsive Yukawa-type interactions of the particles for clockwise (a) and counterclockwise (b) rotating, and nonrotating (c) clusters. The cylindrical projection of the Yukawa forces is plotted vs the cylindrical radius $\rho$. The solid lines are the best-rms-fit parabolas [Eq. (11)] to the force profiles. The insets show the mean values of $A_{\rho }$ in pN/mm and $B_{\rho }$ in pN/mm${}^2$ which were calculated for each time step, with the standard deviations indicated as errors. The vertical lines are the same as in Fig. 8. (d)–(f) The same for the $z$ profiles of the Yukawa forces, here, the few particles suspended below the cluster at $z\simeq -2$ mm (see Fig. 8) were left out. The insets show the parameter $A_z$ in pN/mm of the linear fit to the profiles.