Melting transition of two-dimensional complex plasma crystal in the DC glow discharge

We now analyze change in the structure of the particle cloud with lowering pressure. In order to get a better understanding of melting transition of the dust crystal in DC plasma, we performed variety of diagnostic test on the experimental data in the form of calculating the pair correlation function, defect measurement, calculation of local bond order ($\Psi_{6}$) and estimating the dust temperature. These parameters gives the information of global as well local structural properties of the cloud. Below we provide a detailed description of the results of our analysis. We kept the discharge voltage constant at 310 volt while reducing the background neutral pressure from 15 to 11 pascal in the unit of 1 pascal. The pair correlation function which, delineate the global system properties in terms of degree of long range order of the distribution of particles, is plotted in Fig 5. Figure  5(a) presents the g(r) vs interparticle distance (r) at pressure of 14 pascal. The nature of the correlation function shows the existence of long range ordering over a distance upto 2 mm with a very pronounced peak which is indicative of the system being in the crystalline state. This can also be confirmed from the splitting of the second peak. With the decrease in pressure, the peaks become shorter and flattered, showing the increasing disorderness of the cloud, and also the number of peaks or the length upto which ordering is observable gets decreased. Fig 5(c) shows the pair correlation function at P = 12 Pa and illustrates that the phase state of the particle changes significantly and it tends towards the liquid state with the primary peak followed by a fast descending second or third peak. At a pressure of 11 Pascal, the system becomes almost a liquid like that can be clearly seen in Fig 5(d), where only a small hump appears after a primary peak in the correlation function. So these pair correlation function indicate a transition from ordered solid structure to the fluid at lowering pressure. It is quite interesting to note that the stability of the crystalline structure in DC plasma maintained only upto a certain range of background neutral pressure and a transition to disordered state occurs at small variation of discharge parameters unlike the rf plasma where a stable crystal can be sustained upto a long range of pressure variation.

In order to get a deep insight on melting dynamics of the structure it is important to understand the variation in structural dynamics of the particle cloud locally. To do that we have calculated the local bond order parameter and the average defect fraction. These quantities are defined locally and measure the structural properties of the lattice directly at the respective particle positions christina_2007 ; christina_thesis . A bond order parameter, $\Psi_{6}$ examine the lattice in terms of the local orientational order of particles christina_thesis ; coudel_2018 . For a given particle k, $\Psi_{6}$ is defined as

$$\Psi_{6}(k)=\frac{1}{N}\left|\sum_{n=1}^{N}e^{6i\theta_{kn}}\right|,$$

(1)

where N is the number of nearest neighbors and $\theta_{kn}$ is the angle of the bond between the kth and nth particles with respect to the x-axis. For an ideal hexagonal structure the bond order parameter is maximum ($\Psi_{6}=1$) hence it is a good measure of the deformation of a cell from the perfect hexagon. Lattice sites with nearest neighbor bond angles deviating from 60^∘ will decrease $\Psi_{6}$. The same accounts for lattice sites with a number of nearest neighbors other than six. For calculating the bond order parameter, we first identify the particle locations and find out the nearest neighbors of all the particles using Delaunay triangulation. After that for every particle k we calculate a vector going to its nearest neighbours n and then calculated the angle of the vector kn with respect to the x-asis. A voronoi map of the particle cloud which is color coded according to $\Psi_{6}$ is shown in Fig 6. Fig 6(a) presents the voronoi map corresponding to the pressure of 15 pascal. As we can see from the figure that the structure shows a high orientation ordering as most of the voronoi cell around the particles exhibits perfect hexagonal structure (colored by blue). Those voronoi cells which exhibit very low ordering, as visible in the right side of the cloud, cause defects. The number of cell having lower bond order is seen to be increased as we decrease the pressure to 13 pascal (Fig 6(b)) however the change is not very significant. At a pressure of 11 pascal the cloud looses its orientational ordering completely and almost all the cells manifest a lower bond order as can be seen in Fig 6(c). The variation in average bond order parameter, $\Psi_{6}$ with pressure is plotted in Fig 7. The mean $\Psi_{6}$ which is obtained by calculating the $\Psi_{6}(k)$ for each unit cell and then taking the average over all the cell gives the information of overall orientational ordering of the system. The local order of the particle cloud is $\sim$ 0.87 at 15 pascal which is almost constant upto 13 pascal. It decreases to 0.48 when we reduce the pressure to 12 pascal. The minimum value of $\Psi_{6}$ is comes out to be 0.4 for 11 pascal which is close to the values reported for the melting dynamics in rf plasmas.christina_thesis

We further parameterize the structure by calculating the defect. Generally, a defect causes the disorganization of particles in a crystal lattice. Most common defect in two dimensional system are point defects or disclinations which arises due to vacancy or an interstitial. A hexagonal lattice with typically six nearest neighbors to each lattice site predominantly encounter two types of point defects namely ‘five-folded’ or ‘seven folded’ lattice sites. Other kind of defects exist in a system are “free dislocation”, which is an additional row of dust particles in an ideal lattice that can be viewed as a pair of a 5-fold and 7-fold disclination, and dislocation pairs (quarter of 5-fold and 7-fold disclination). A variety of theories has been hitherto reported for rf plasma that describe the melting transition of dust crystal based on the formation of these defects melzer_crystal_1996 ; schweigert1 ; christina_thesis . The KTHNY theory melzer_crystal_1996 ; christina_thesis describe the melting of the crystal lattice by two continuous phase transition in which firstly the dislocation pairs unbind into free dislocations followed by the stage where dislocations (pair of a 5-fold and 7-fold disclination) breaks into free disclinations. The first stage is called as hexatic phase where positional order is destroyed but orientation order still can be found in the system. In the second stage the orientation order also disturbed and the liquid state is reached. In our case, we have performed Delauney triangulation to find out the position of each particle and locate the nearest neighbor of corresponding particles. We then find out the position and count the particle having five and seven nearest neighbors respectively. The defect fraction can be calculated by taking the ratio of number of disclinations (5 or 7 folds ) to the total number of particles in a frame.

Fig 8 shows the defect plots at various pressures. The red circle and black square in the figures mark the locations of 5 and 7-fold defects. At pressure of 15 pascal (see Fig 8(a)) we can see the dislocation pairs but mostly free dislocations can be seen. Some chainlike arrangements of the defects also appear. The defect fraction corresponding to five and seven nearest neighbors are calculated as 4.2$\%$ and 2.9$\%$ respectively. There is no significant variation found upto 13 pascal with the defect fractions of 6.3$\%$ (five nearest neigbours) and 3.2 $\%$ (seven nearest neigbours). A notable change in the defect fraction is observed as the pressure is further reduced to 12 Pascal and the defect arrange more and more in chains as visible from Fig 8(c). The defect fraction in the case is calculated as 30.6$\%$ and 20.6$\%$ respectively. At 11 Pascal almost all the defect sites corresponds to free disclination (Fig 8(d)) and defect fraction for 5 fold is calculated as 38$\%$ whereas 29$\%$ for 7- fold. The observed melting transition does not exactly follow the KTHNY theory of melting.

For the experimental measurement of the kinetic temperature a velocity distribution function in two orthogonal direction has been calculated. For a better statistics, a video sequence of 100 frames has been chosen where a particle tracking velocimetry (PTV) analysis package, DAVIS 8 davis is used to construct the velocity vector fields. Fig 9 shows the velocity distribution of our experimental data along the x- and y-directions for discharge parameters of V${}_{d}=310$ volt and P = 14 Pascal. The distribution function are found to be maxwellian as can be seen from the fitted maxwellian function (solid line) on the experimental data points in the figures. The kinetic dust temperature is calculated from the width of the measured distribution by using the formula $E=\frac{1}{2}m\langle v^{2}_{x,y}\rangle=\frac{1}{2}k_{B}T_{x,y}$. Fig 10 shows the measured kinetic temperature of the dust particles with background neutral pressure. It can be seen from the figure that at lower pressure the temperature is higher. It is important to be mentioned that our measured temperature is twice the order of magnitude less than previously reported temperature during melting transition in rf plasma. A rapid fall in the temperature is observed with increasing pressure and above 13 pascal it decays slowly and then saturates at $\sim$ 0.035 eV which is close to the room temperature. The decrease in temperature at increasing pressure can be directly attributed to the collisional cooling of the dust particles. We investigated mechanisms that could be responsible for the heating of the dust particle. Vaulina et al. vaulina_charging predicts the dust charge fluctuation is one of the main mechanisms to heat up the dust particles. The dust charge fluctuation is essentially depends on two parameters; particle charge and their mass. In order to estimate the dust charge, we have followed the procedure outlined in the paper by Khrapak et al.khrapak_charging that allows estimating the charge based on its dependence on the ratio of the charge density of the particles to that of the ions, defined as Havnes parameter havnes

$$P=\frac{aT_{e}}{e^{2}}\frac{n_{p}}{n_{0}}\cong 695aT_{e}\frac{n_{p}}{n_{0}},$$

(2)

where a is the particle radius (in $\mu$m), T_e is the electron temperature (in eV) and n_p and n₀ is the particle and plasma densities, respectively. When P $>1$, the charge and floating potential are significantly diminished, while for P $\ll$ 1 the charge and floating potentials approach the values for an isolated particle melzer_book ; vaulina_charging . Knowing the information of P, one can estimate the dimensionless charge ($z=e^{2}Z_{d}/4\pi\epsilon_{0}ak_{B}T_{e}$)khrapak_charging and hence the corresponding charge residing on the particle, Q. For our range of pressure 15-11 Pascal and the corresponding measured parameters T_e = (2.8-4.3) eV, n₀ = (1.5-0.8)$\times 10^{15}$/m³, n_p = (2.45-1.35)$\times 10^{10}$/m³, the dimensionless charge is estimated as z$\sim$2.25 and the value of charge varies from 1833-2984 (in the unit of e). This shows an increase in the charge at lower pressure. Therefore at lower pressure the charge fluctuation would be higher and hence particle get heated. This can lead to melting of the dust particle. In addition to the charge fluctuation, the streaming ions in also play a very important role in heating up the particles in dc glow discharge plasma. The ion streaming increases with decreasing pressure. However, the asymmetry of the electrodes where cathode tray are larger than the anode and the dust cloud are formed far away from the anode (not facing the anode) has helped in reducing the heating effects associated with ion streaming. The ion path has been further influence by the metal confinement strips which helps ion streaming away from the region between the strips where dust crystal formed. A detailed investigation of such a behavior presented by Hari et al.hari_pop_2018 . Thus with the help of present electrodes configuration and an additional confining strips we are able to get an optimal condition of discharge parameter where we observed a stable crystal formation whereas a small variation in discharge parameters lead to melting unlike to rf plasma where ion streaming is not so strong and hence a large variation in discharge parameter breaks the crystalline structure into fluid-like state.