Analyzing the liquid state of two-dimensional dust clusters: The instantaneous normal mode approach

The liquid state and the freezing transition of finite two-dimensional dust systems are studied using the instantaneous normal mode (INM) analysis. This technique measures the instantaneous unstable modes of a cluster configuration and relates them to the liquid properties of the system. Here, the INM analysis has been applied to experiments on laser-heated dust clusters. From the experiments, diffusion constants and melting temperatures for clusters of different size have been derived. The INM diffusion constants have been compared to those derived from other standard approaches. The scatter of the diffusion constant retrieved by the INM is smaller than that retrieved by other methods, allowing a more reliable determination of melting temperatures. Moreover, the behavior of double-well and escape modes, which reflect certain topological properties of unstable modes, correlates very well with the behavior of the diffusion constant. Further, the dynamic nature of the unstable modes has been determined as mostly shearlike. Finally, the INM results on the experiments are checked against those from Langevin simulations.

Figure 1

Scheme of the experimental setup. Four heating laser beams are swept by galvanometer mirrors randomly over the dust cluster. The particle motion is recorded by a high-speed camera. See also [11].

Figure 2

Density of states for a cluster with $N=26$ at four different temperatures. The gray area indicates the unstable branch ${\rho }_u$ containing the imaginary values of ${\omega }_{\ell }$, where ${\rho }_u\left(\omega \right)$ is plotted as ${\rho }_u\left(\right|\omega \left|\right)$ on the negative frequency axis ($\omega \to -\left|\omega \right|$).

Figure 3

Diffusion coefficient $D$ derived from INM analysis as a function of cluster temperature. A linear fit to the data $D\left(T\right)$ is also indicated. The insets show the trajectories of the particles in the cluster with $N=26$ at different temperatures.

Figure 4

The ratio ${\rho }_u\left(\right|\omega \left|\right)/{\rho }_s\left(\omega \right)$ as a function of ${\omega }^2$ for $N=26$ together with the best exponential fit at four different temperatures.

Figure 5

Diffusion constants derived from INM as a function of temperature for different values of the screening strength $\kappa$ in Eq. (3).

Figure 6

(a) Mean-squared displacement MSD over time for $N=26$ at a temperature of $T=9270$ K. (b) Corresponding time-dependent diffusion constant $D\left(t\right)$ derived from the experiment (solid line). A model calculation according to Ref. [55] is shown for comparison.

Figure 7

Diffusion constants derived from MSD, VACF, and INM (symbols) as a function of temperature, respectively. The lines indicate best linear fits to the diffusion constants where the dashed, dash-dotted, and dotted lines are the fit to VACF, MSD, and INM values, respectively.

Figure 8

Freezing temperature of dust clusters with different particle number $N$. Shown are temperatures from the experiment and from simulated dust clusters. The upper panel shows the frequency of the least stable mode from Ref. [32].

Figure 9

Ratio of the density of states $f_u/f_s$ for a cluster with $N=26$ at different temperatures. In (a) all INM modes are used, in (b) only DW modes, and in (c) only escape modes.

Figure 10

Plane of mode frequency $\omega$ and compressional part ${\psi }_c$ and shear part ${\psi }_s$, respectively, for the $N=26$ cluster at $T=22190$ K. Darker colors indicate a higher abundance of modes. To guide the eye, the lines indicate the mean values of ${\psi }_c$ (dash-dotted) and ${\psi }_s$ (dashed).