Theory Group

Annual Report 2006

Recrystallization of a 2D Plasma Crystal

C.A. Knapek, D. Samsonov, S. Zhdanov, U. Konopka, G.E. Morfill

A monolayer plasma crystal consisting of micron-sized particles, levitated in the sheath of a radio frequency discharge, was melted by applying a short electric pulse to two parallel wires located at the height of the particles. Structural properties and the particle temperature were examined during the stage of recrystallization. A liquid- like phase was followed by a transient state characterized by energy release and the partial restoring of long range order. Numerical simulations revealed the same regimes of recrystallization as those observed in the experiment.

Phase transitions, solid to liquid and back, of 2-dimensional systems are a subject of great interest. Theory describes a continuous melting transition with the appearance of a so called ''hexatic'' phase, which is characterized by long range orientational order, but no long range positional order, between the solid and liquid state. Plasma crystals provide an ideal opportunity to observe phase transitions of 2D systems at the kinetic level due to the direct visualization of the individual particles and the short restoring time they need to reach equilibrium after a perturbation. A single layer of micron-sized spherical particles arranged in a hexagonal crystalline structure can easily be generated in the sheath of a plasma. An example for such a crystal is shown in Fig. 1.

Fig. 1: A two-dimensional plasma crystal of charged microparticles illuminated by a green laser and levitated above the circular electrode by an electric field. (Picture courtesy of M. Kretschmer)

The crystal can be melted into a gaseous state by application of a strong negative electric pulse. The observation of the following transition from the molten unordered state back to an ordered crystal gives information on the processes happening on the level of the particle motion itself.

An Argon plasma has been ignited in a vacuum chamber by applying a radio frequency power between a horizontal electrode and the grounded chamber walls, and dust particles (melamine-formaldehyde spheres with an diameter of 9.19 µm) have been injected. The particles levitate in a plane above the lower horizontal electrode at the height where downward gravitational and upward electric forces are in balance. They arrange in a 2D hexagonal crystal due to their mutual interactions and the radial confining forces given by the shape of the lower electrode. Two parallel wires were mounted horizontally at both sides of the crystal, approximately at the height of the particles. To these, an electric pulse of -253 V lasting for 0.2 s was applied, which pushed the particles from both wires to the center of the chamber and melted the crystal. The particles were illuminated by a horizontally spread laser sheet from the side. A digital high speed camera recorded the whole process of melting and recrystallization from the top view with a frame rate of 500 frames per second (fps).

Fig. 2: Particle temperature versus time during the recrystallization. The vertical dashed lines mark the regimes I - IV of different temperature decay. The inset shows the temperature decay found in computer simulations. The red solid lines represent exponential fits.

From the images, we extracted particle coordinates by searching for contiguous regions of a few pixels with brightness values above a given threshold, and identified those as particles. These particles were then traced from frame to frame to obtain velocities. For our analyses we used 2300 such particles, located in a chosen region of the field of view.

The mean particle temperature T was derived for each time step from the velocity distributions. The temperature T is highest for the melted state (Fig. 2). Then follows a regime (I) where the dust particles are slowed down mainly by the collision with neutral gas atoms in the plasma. This process, called Epstein damping, implies an exponential decay of T with time, whose slope is given by the plasma conditions. After 1 second of this state, the decrease of T slows down but is still exponential. We found two consecutive regimes (II, III), where some release of energy seems to heat the particles and weakens the effective cooling rate. The temperature reaches its initial value (before melting) after about 4.5 seconds in regime IV. A simulation of a particle system, having the same conditions as in our experiment, yielded the same behaviour (inset in Fig. 2).

Our aim then was to connect the kinetic behaviour of the system to structural properties, being a measure for the order, i.e. the state (e.g. crystalline or fluid) of the system at each time step. The degree of order depends on the positions of particles relative to each other. Here we distinguish between the local structure in the nearest vicinity of a particle, and the global appearance - the continuity of this structure over larger distances. In a liquid for example, particles are distributed randomly over the system on global scales. However on a local scale, they are subject to their mutual interactions, implying a certain interparticle distance. In an ideal 2D plasma, crystal particles are arranged in a continuous hexagonal lattice with a constant interparticle distance and 60^o angles between the nearest neighbour bonds.

The range of order is described by means of the pair- and bond correlation functions. The pair correlation function gives the probability to find a particle at a distance r to another particle. By a fitting process, we obtained the translational correlation length for each image. The larger its value is, the larger is the distance scale on which the system is highly ordered. Similarly, the bond correlation function is a measure for the long range orientational order. It measures the average orientation of nearest neighbour bonds, separated by a distance r in the crystal. The bond correlation function decayed by a power law in the initial state, and exponentially elsewhere. The latter defines an orientational correlation length, comparable to the translational correlation length.

Locally, order can be destroyed by defects. Particles with a number of nearest neighbours differing from the canonical value of six in case of a hexagonal lattice, we call defects. Most common are defects with five or seven neighbours. They often appear in pairs. We calculated their fraction in each frame.

We found that both, defects fraction and correlation lengths, evolve according to power laws proportional (T-T_c)α, where T_c is a critical temperature. We found similar absolute values for the exponent α, ranging from 0.25 - 0.35. This behaviour holds for all regimes of the temperature decay.

Fig. 3: Color-coded maps of the crystal for different regimes of recrystallization. The background color corresponds to the value of the modulus of Ψ (see colorbar at the top). The arrows represent the vector field of arg(Ψ). Defects are marked by red (5-folds) and blue (7-folds) dots. a) Crystal structure before melting; b) liquid-like state immediately after melting (regime I); c),d) consecutive regimes II,III of crystallization; e) metastable regime IV; f) crystallite structure observed in simulations.

While the defect fractions and the translational correlation length return to their initial values, the orientational order shows a different behaviour after melting. The bond correlation function decays exponentially with increasing correlation length, in contrast to the initial power law decay. According to theory, this behaviour would indicate a liquid state. This however, is in conflict with the general appearance of the system, being highly ordered also on larger scales (Fig. 3).

The orientational order is investigated closer by means of the local bond order parameter. It is calculated from the angles between the bonds from one center particle to all its nearest neighbours, which define the unit cell around this particle. The parameter provides information on the closeness of the unit cell to an ideal hexagon, and on the orientation of the cell with respect to an arbitrary chosen fixed axis. Fig. 3 shows colour-coded maps of selected frames. The brighter the colors the closer the cells are to ideal hexagons. The arrows indicate the orientation of the unit cells. Further, defect locations are marked by red (5 neighbours) and blue (7 neighbours) dots. After an unordered state (Fig. 3b), crystallization proceeds first to a system of small ordered 'crystallites' with arbitrary orientations separated by strings of defects. As the system cools down, these crystallites grow and merge with neighbouring regions (Fig. 3 c-d). A metastable state is reached which is characterized by highly ordered adjoined crystalline domains (Fig. 3e). Since the orientation of unit cells changes abruptly across the domain boundaries - even to directions with opposite signs for adjoining domains - long range orientational order cannot be found in the final state of the system, in contrast to the initial state (Fig. 3a). The formation of a crystallite was also observed during the cooling regime in the simulation data (Fig. 3f).

In conclusion, the transition of a liquid-like state of a monolayer of particles to a state of high translational order and low defect fraction was investigated. The exponential temperature decay changes from pure Epstein damping to a slower decay rate until the temperature of the initial state is reached. Defect fraction and translational order are also restored to their initial values. Long range orientational order is not present due to the non-uniform orientation of domains across the crystal. It was found that the crystal slowly returns to a more uniform state after a sufficiently long time (approximately 30 minutes). The regime of slow temperature decay (II and III in Fig. 2 and 3) deserves special attention. The occurring ''heating'', i.e. the slower temperature decay, could have two origins: Either, latent heat released by the dissolution of lattice defects and the tilting of nearest neighbour bonds into the (hexagonal) ground state, or the strong coupling between the particles, which would simulate successively larger (more massive) ''unit cells'' or ''effective'' particles. Finally, a unique classification of the found regimes with respect to the existing theory - especially regarding the hexatic phase - can not be carried out, since not all of the quantities which describe the structural properties exhibit the predicted behaviour.