From transport to disorder: Thermodynamic properties of finite dust clouds

The quantities entropy and diffusion are measured for two- and three-dimensional (3D) dust clusters in the fluid state. Entropy and diffusion are predicted to be closely linked via unstable modes. The method of instantaneous normal modes is applied for various laser-heated clusters to determine these unstable modes and the corresponding diffusive properties. The configurational entropy is measured for 2D and 3D clusters from structural rearrangements. The entropy shows a threshold behavior at a critical temperature for the 2D clusters, allowing us to estimate a configurational melting temperature. Further, the entropic disorder increases for larger clusters. Finally, the predicted relation between entropy and unstable modes has been confirmed from our experiments for 2D systems, whereas 3D systems do not show such a clear correlation.

Figure 1

(a) Scheme of the experimental 3D setup. The particles confined in the glass box are illuminated by two Nd:YAG lasers from two sides and manipulated by two diode lasers from opposing directions. The three orthogonal high-speed complementary metal-oxide semiconductor (CMOS) cameras allow for a full 3D dynamic measurement. (b)–(e) Yukawa balls consisting of (a) 33, (b) 48, (c) 49, and (d) 60 particles in the $\rho$-$z$ plane with $\rho =\sqrt{x^2+y^2}$. Some of the clusters are slightly elongated along the $z$ axis.

Figure 2

(a) Scheme of the experimental 2D setup. The particles are trapped in a shallow spherical depression in the sheath of the discharge. A top view camera allows us to track the particle movement in the ($x,y$) plane. The four-axis laser-heating systems is used heat the particle ensembles. Also shown are 2D dust clusters in the ($x,y$) plane consisting of (b) $N=20$ and (c) $N=27$ particles. The trajectories of laser-heated clusters are shown over a time span of 10 s.

Figure 3

(a) The INM spectra for the $N=60$ particle cluster for three different temperatures. By convention, the unstable density of states ${\rho }_u\left(\omega \right)$ resulting from imaginary eigenfrequencies is plotted as ${\rho }_u\left(\left|\omega \right|\right)$ on the negative frequency axis ($\omega \to -\left|\omega \right|$). The INM spectra shown here are plotted with a constant offset. (b) The corresponding trajectories of the cluster particles are shown over a time span of 2 s.

Figure 4

Hopping frequency ${\tau }_h^{-1}$ as a function of the temperature $T$ for various dust clusters. The hopping frequency increases slowly as the dust cluster temperature increases.

Figure 5

(a) Diffusion coefficients $D\left(T\right)$ as a function of temperature for Yukawa balls of different size. The dashed line corresponds to a linear fit to the diffusion coefficients for all clusters. (b) Normalized diffusion coefficients $D^{*}$ as a function of the normalized temperature $T^{*}$ for the experimental $N=60$ cluster (triangles), a simulated $N=60$ cluster (open circles), and the extended Yukawa one-component plasma (YOCP) (solid line) of Ref. [7]. Dashed and dotted lines represent linear fits to the measured and simulated data of the finite clusters, respectively.

Figure 6

(a) Trajectories of the particles in an $N=19$, 2D particle cluster over a time span of 250 s. (b)–(d) Observed cluster configurations together with the shell occupation numbers and their probabilities. Dashed lines indicate borders between different shells.

Figure 7

Configurational entropy as a function of the temperature for selected (a) 2D and (b) 3D clusters. Note the different temperature scaling in (a) and (b).

Figure 8

Configurational entropy $S_c$ as a function of the particle number $N$ for finite dust clusters. The symbols represent experimental values for (a) 2D and (b) 3D dust clusters and the lines are results from simulations of Ref. [5] for different values of $\kappa$. In (b), the (blue) circles represent unheated clusters and (red) squares mark laser-heated clusters.

Figure 9

Logarithm of the unstable mode fraction $f_u$ as a function of the configurational entropy $S_c$ for 2D dust clusters (closed circles) and 3D clusters (open triangles). The solid line represents a linear fit to the 2D data. As representatives, the 2D clusters $N=20$ (closed squares) and $N=27$ (open squares) are highlighted. The data of the simulated $N=60$ cluster are added as closed upward triangles.