The formation of a self-consistent crystalline structure is a well-known phenomenon in complex plasmas. In most experiments, the pressure and rf power are the main controlling parameters in determining the phase of the system. We have studied the effect of the externally applied magnetic field on the configuration of plasma crystals, suspended in the sheath of a radio-frequency discharge using the Magnetized Dusty Plasma Experiment device. Experiments are performed at a fixed pressure and rf power where a crystalline structure is formed within a confining ring. The magnetic field is then increased from 0 to 1.28 T. We report on the breakdown of the crystalline structure with the increasing magnetic field. The magnetic field affects the dynamics of the plasma particles and first leads to a rotation of the crystal. At a higher magnetic field, there is a radial variation (shear) in the angular velocity of the moving particles which we believe to lead to the melting of the crystal. This melting is confirmed by evaluating the variation of the pair correlation function as a function of magnetic field.

“Dusty” or “complex” plasmas consist of the usual combination of electrons, ions, and neutral atoms with the addition of charged, dust particulates of size ranging from tens of nanometers to tens of micrometers. In the plasma environment, these heavy dust particles can acquire a negative charge of the order of Zd104105 elementary charges due to the collection of more electrons than ions, which adds much richness to the collective dynamics of the system. Dusty plasmas occur in a variety of natural situations such as planetary rings, comet tails, interplanetary media, and interstellar clouds.1,2 Although the dust particles are highly charged, their mass is much larger than that of the electrons or ions, md1015 to 1013 kg. As a result, the dust particles have a much smaller charge to mass ratio than electrons and ions, which makes the time scales of dust dynamics comparatively much longer. Additionally, the larger sized and highly massive particles possess a very low thermal velocity and hence allow them to be visualized using laser illumination and high-speed cameras. Thus, dusty plasma offers an excellent medium for studying various phenomena at single particle and fluid levels with remarkable temporal and spatial resolutions.

In a typical laboratory experiment, the dust cloud is levitated near the sheath boundary by balancing the electrostatic force due to the sheath electric field and the gravity. The highly charged particles interact with each other via a strong electrostatic potential that may exceed the thermal energy, and the plasma becomes strongly coupled. The strength of coupling can be determined by a coupling parameter Γ, which is the ratio of the interparticle coulomb potential energy to the dust thermal energy (Γ=Q2/(4πϵ0aKBTd)). Several authors previously revealed that above a critical value of coupling parameter Γc=171,3,4 the dust can freeze into perfect crystals, whereas in the regime of 1Γ<Γc, the system exhibits a strongly coupled fluid state. Another parameter is the coulomb screening constant (κ), which is the ratio of interparticle spacing and the screening length. κ is sometimes used as an effective Coulomb coupling parameter,5 Γeff, which is diminished by the effect of shielding. Because it is possible to experimentally control the Coulomb coupling parameter, dusty plasmas have been used to study the phenomenon of phase transition and transport properties of the strongly coupled system.

A large number of experimental studies in the past have been devoted to the formation of the dust crystal and its melting dynamics in a rf plasma. Some of the pioneering experiments in this area have been reported by Chu et al.,6 Thomas et al.,4 and Hayashi et al.7 Chu et al.6 demonstrated the formation of the coulomb crystal and liquid in a rf produced strongly coupled dusty plasma for 10 micron SiO2 particles. They observed the hexagonal, fcc, and bcc crystal structures and solids with coexisting different crystal structures with propagating boundaries at lower rf powers and identified the transition to the more disordered liquid state by increasing the rf power. Hayashi et al.7 demonstrated the formation of coulomb crystals as a result of growth of carbon particles in a methane plasma. They have shown the dependency of the phase transition phenomenon over the particle diameter and Wigner-Seitz radius of the structure. Thomas et al.4 investigated the plasma crystal and its phase transition by changing various controlling parameters including plasma density, temperature, neutral gas, and particle size. Later, various theoretical and experimental research studies on the dust crystal and phase transition phenomena have been reported by several authors.8–11 In all these investigations, it has been concluded that the structural configuration of particle cloud is mainly controlled by the pressure, rf power, and micro-particle size.

In addition to the plasma crystal studies, over the last decade, a series of research focusing the effect of the externally applied magnetic field over the particle dynamics have been reported. It has been well-documented that dusty plasma phenomena become even more complicated in the presence of the external magnetic field. Yaroshenko et al.12 studied various mutual dust-dust interactions in complex plasmas, including the forces due to induced magnetic and electric moments of the grains. Konopka et al.13 have observed the rotation of plasma crystals under the influence of the constant vertical magnetic field (0.014 T) at various discharge conditions in a rf plasma. The estimated shear stress in their experiment has been used to calculate the shear elastic modulus of the dust crystal. Plasma rotation due to azimuthal E × B ion drift in a dc glow discharge plasma has been observed by Uchida et al.,14,15 while the deflection of single-particle trajectory in an electron cyclotron resonance plasma has been examined by Nonumura et al.16 

Some recent theoretical works have shown that the phase transition phenomenon can also be controlled by varying magnetic field strengths.17–20 The presence of the magnetic field alters the internal energy of the particulates in plasma and hence affects the points of phase transition from the solid state to the liquid state. Baruah et al.19 have adopted the modified interaction potential into their molecular dynamic simulation and found that the crystalline behavior changes by changing the magnetic field strength and system turns into fluid or gaseous states above a very high magnetic field (above 1 T). However, the direct effect of the magnetic field on the dust grain dynamics has not been taken into account. Mahmuda et al.20 have reported that the self-diffusion is a criterion for melting of the dust crystal in the presence of the magnetic field. A detailed investigation of crystalline behavior and phase transition of dusty plasma in the presence of the magnetic field is important for the point of understanding of fundamental physics as well as for production and control of the properties of the dust crystal. To understand these effects in depth, a dedicated experimental verification is indeed needed.

In this paper, we have experimentally investigated the effect of the externally applied magnetic field on the configuration of plasma crystals in the sheath of an rf discharge argon plasma. The experiment has been performed in the Magnetized Dusty Plasma Experiment (MDPX) device with mono-disperse melamine formaldehyde particles. The plasma crystal has been formed within a circular ring at a particular discharge condition, and the melting of the crystal has been observed at a higher magnetic field (at 1 T). The melting of the crystals is confirmed by plotting the pair correlation function, and the cause of melting is qualitatively explained by radial variation of the angular velocity measurement of the moving particles.

This paper is organized as follows. In Sec. II, we present the experimental set-up in detail. In Sec. III, we discuss the experimental results on the formation of the dust crystal and its melting dynamics. A brief concluding remark is made in Sec. IV.

The Magnetized Dusty Plasma Experiment (MDPX) device is a recently commissioned, multi-user, high magnetic field experimental platform that is operating at Auburn University. The MDPX device consists of two main components: the superconducting magnet and the primary plasma chamber. The hardware components of the MDPX device are described extensively in previous papers.21–23 For the studies described in this paper, the MDPX device is operated in its vertical configuration with the magnetic field aligned parallel to gravity. All four superconducting coils are energized at the same current to produce a uniform magnetic field (with ΔB/B<1%) at the center of the experimental volume where the plasma and dusty plasma are generated.

An octagonal vacuum chamber with a 355 mm circular inner diameter is used as the plasma source for the MDPX device. The basic configuration for the plasma generation uses a pair of 342 mm diameter electrodes to produce a capacitively coupled, glow discharge argon plasma using a 13.56 MHz radio frequency source connected to a matching network. The lower electrode is powered, while the upper electrode is electrically grounded. The upper electrode has a 146 mm circular hole to allow viewing of the plasma and the dust particles from the top. The lower electrode has a 152 mm wide × 3.2 mm circular depression to aid in the confinement of the dust particles. For this experiment, in the center of the depression, a copper ring that has a 50 mm outer diameter, 41 mm inner diameter, and 1.5 mm thickness is placed to further restrict the size of the particle cloud formed by the suspended particles.

In this experiment, mono-disperse melamine formaldehyde particles of diameter 7.17±0.08μm are used as the dust component. The particles are introduced using a radial “shaker” that is placed into the center of the plasma discharge and tapped to release a few hundred particles. The shaker is then pulled radially outward to avoid disturbing the plasma. The particles are illuminated using a 633 nm, red laser diode that is expanded into a thin, horizontally aligned sheet. The particles are observed to form a 2 layer suspension in the sheath of the plasma. The particles are viewed from the top port of the vacuum chamber using a USB3-based, 2048 × 2048 pixel, Ximea model xiQ camera that can be operated up to 90 frames per second (fps) although for these experiments frame rates of 10 to 60 fps were sufficient to perform these studies. Combinations of particle tracking velocimetry (PTV) and particle image velocimetry (PIV) techniques were used to measure the particle dynamics.24–26 Figure 1(a) shows a schematic drawing of the interior arrangement of the plasma source, and Fig. 1(b) shows the photograph of a plasma crystal that is suspended above the ring in the MDPX plasma chamber.

FIG. 1.

(a) A schematic drawing of the interior of the MDPX plasma chamber and (b) a photograph of a plasma crystal suspended in the plasma.

FIG. 1.

(a) A schematic drawing of the interior of the MDPX plasma chamber and (b) a photograph of a plasma crystal suspended in the plasma.

Close modal

A Langmuir probe has been used to measure the plasma conditions for B = 0 T and without the presence of the dust particles which yields an electron temperature of Te2.5 to 3.5 eV and a plasma density of n 0.5 to 3.0×1015 m3. It is noted that once the magnetic field is energized, the probe traces become severely distorted and are effectively unreliable. So, for the measurements described in this paper, it was critical to find an operating regime in which the plasma and dust particles could remain stable and without distortion over the entire range of magnetic field settings.

Based on the visual observation, we have found that as we set the rf power of 3.5 W at 221±0.5 mTorr (29.4 ± 0.1 Pa) argon pressure, the particle cloud settled into a crystalline structure within the confining ring. For example, Fig. 2 shows an inverted image of particle cloud forming a crystal. From the Voronoi diagram in Fig. 3, it can be clearly seen that the particle cloud is arranged into a nearly hexagonal crystalline structure, although there is some disordering observed in the left region of the cloud. We believe that this may be due to the formation of another layer below the first one. However, most of the crystal points do not show any deviation; this indicates that particles are almost aligned in the vertical direction also. The structure is almost stationary with a small thermal fluctuation around the equilibrium position that we have checked from the overlapped image of 150 consecutive frames. We kept these discharge parameters constant while varying the external magnetic field. Figure 4(a) shows the inverted image of particle configuration at B=0.512 T, and the corresponding Voronoi diagram is shown in Fig. 4(d). It can be clearly seen that system ordering gets changed and tends to the disordered structure. It becomes even more disordered with a further increase in the magnetic field [shown in Fig. 4(b) and associated Voronoi diagram in Fig. 4(e)] and achieves almost a new phase state when we reached the magnetic field of B=1 T. The Voronoi diagram of the particle cloud at B=1 T [Fig. 4(f)] depicts the disordered structure which is nearly a liquid state.

FIG. 2.

The real picture of (a) coulomb crystal and its (b) zoomed view, formed at B = 0, P = 221 mTorr, and rf power = 3.5 W.

FIG. 2.

The real picture of (a) coulomb crystal and its (b) zoomed view, formed at B = 0, P = 221 mTorr, and rf power = 3.5 W.

Close modal
FIG. 3.

Voronoi diagram of the particle location at B = 0 which is shown in Fig. 2.

FIG. 3.

Voronoi diagram of the particle location at B = 0 which is shown in Fig. 2.

Close modal
FIG. 4.

Inverted image of Particle configurations and corresponding Voronoi diagram with the changing magnetic field; (a) and (d) correspond to the structural configuration at B = 0.512 T, (b) and (f) for B = 0.896 T and, (c) and (f) are the inverted snapshot and corresponding Voronoi plot at B = 1 T, delineating the liquid state.

FIG. 4.

Inverted image of Particle configurations and corresponding Voronoi diagram with the changing magnetic field; (a) and (d) correspond to the structural configuration at B = 0.512 T, (b) and (f) for B = 0.896 T and, (c) and (f) are the inverted snapshot and corresponding Voronoi plot at B = 1 T, delineating the liquid state.

Close modal

For further verification of the change in the structure of the particle cloud with increasing magnetic field, we have plotted the pair correlation function, g(r), of the particle cloud in Fig. 5 in a manner that is consistent with previous studies of phase transitions in complex plasmas.9,27,28 Here, g(r) is plotted for each magnetic field setting from B = 0 to 1.153 T. It has been calculated by directly measuring the average distance between particles where 250 frames have been used for averaging. It is important to mention that the discharge parameter is kept fixed at an initial rf power of 3.5 W and an initial pressure of 221 mTorr where we have an observed perfect crystal as shown in Fig. 2. Figure 5(a) presents the g(r) vs interparticle distance (d) at zero magnetic field. As we can see from the figure, the nature of the correlation function shows the existence of long range ordering between the particles with a very pronounced peak which is indicative of the system being in the crystalline state. With the increase in the magnetic field, the peaks become shorter and flattered, showing the increasing disorderness of the cloud, and also the number of peaks gets decreased. Figure 5(c) shows the pair correlation function at B=0.512 T 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. We found that above 1 T of magnetic field, the system becomes almost a liquid like that can be clearly seen in Fig. 5(e), where only a small hump appears in the correlation function.

FIG. 5.

The pair correlation function g(r) of the particle clouds with the changing magnetic field (a) 0 T, (b) 0.256 T, (c) 0.512 T, (d) 0.896 T, and (e) 1.153 T.

FIG. 5.

The pair correlation function g(r) of the particle clouds with the changing magnetic field (a) 0 T, (b) 0.256 T, (c) 0.512 T, (d) 0.896 T, and (e) 1.153 T.

Close modal

In order to diagnose the cause of melting, we have inspected the other dynamical changes that occurred in the particle dynamics in the influence of the magnetic field. It is hitherto reported by several authors13–15,25,31 that adding a vertical magnetic field causes rotation of particle cloud as a whole due to an azimuthal E × B ion drift. Therefore, we have also characterized the effective particle cloud rotation with the changing magnetic field. Figure 6 shows the velocity vector field along with the magnitude of the azimuthal velocity component (vθ) of a particle cloud rotating in the presence of the magnetic field. The particle image velocimetry (PIV) analysis package, DAVIS 8,29 is used to construct these velocity vector fields. It is found that the rotation velocity increases with the increase in the magnetic field. By performing the PIV technique, it is easy to calculate the radial variation of angular velocity as a function of changing magnetic field. To do this, we have to first determine the rotation center of velocity magnitude by fitting it with 2 D elliptic contours. We observe a slight shift in the rotation center as we increase the magnetic field strength which represents that the crystals are not exactly symmetric about the center of the confinement ring. However, the shift is not much significant as can be seen from Fig. 7. After determining the rotation center, we first compare the velocity component in radial and azimuthal directions to get an idea about their contribution in the calculation of velocity magnitude of the particle cloud. Figure 8 shows the vθ and vr components for a higher value of the magnetic field (at 1 T). It can be seen from the figure that the vr contribution is very small and it is almost close to zero. This gives us a confidence that the magnitude of the velocity is primarily the azimuthal velocity. We then calculate the angular velocity of each particle from the measured azimuthal displacement between consecutive frames and its distance from the rotation center. The data have been binned at the interval of 50 pixels (1 mm) in the radial direction and averaged in each bin so as to reduce the uncertainties in the calculated values.

FIG. 6.

Velocity vector field along with the magnitude of the velocity component vθ at (a) 0.512 T, (b) 0.896 T, and (c) 1 T.

FIG. 6.

Velocity vector field along with the magnitude of the velocity component vθ at (a) 0.512 T, (b) 0.896 T, and (c) 1 T.

Close modal
FIG. 7.

Shift in the center of rotation with the magnetic field from B = 0.256–1.28 T.

FIG. 7.

Shift in the center of rotation with the magnetic field from B = 0.256–1.28 T.

Close modal
FIG. 8.

Comparison of radial and azimuthal velocity components for B = 1.0 T.

FIG. 8.

Comparison of radial and azimuthal velocity components for B = 1.0 T.

Close modal

The radial dependence of the particle angular velocity Ω with varying magnetic fields from 0.256 to 1.28 T is shown in Fig. 9. In Fig. 9(a), the angular velocity is shown as a function of the radial position up to B = 0.896 T. For these magnetic fields, two regions in the angular velocity profile are observed. In Region I, which extends from the center (r = 0 mm) to r = 16 mm, the particles are observed to have a nearly constant angular velocity, perhaps with a slight decay beginning near r = 10 mm. In general, in Region II, the motion is characterized by a nearly solid body rotation of the particle cloud. Beyond r = 18 mm, there is then a sharp decrease in the angular velocity, suggesting the presence of a sheared flow.

FIG. 9.

Angular velocity Ω of the particle cloud rotating in the magnetic field vs distance from the center of rotation. (a) The angular velocity profiles for B = 0.256 to 0.896 T. (b) The angular profiles for B = 1.0 to B = 1.27 T. The inset figure in (a) shows the variation of angular velocity in Region I with the magnetic field from 0.256 to 1.28 T.

FIG. 9.

Angular velocity Ω of the particle cloud rotating in the magnetic field vs distance from the center of rotation. (a) The angular velocity profiles for B = 0.256 to 0.896 T. (b) The angular profiles for B = 1.0 to B = 1.27 T. The inset figure in (a) shows the variation of angular velocity in Region I with the magnetic field from 0.256 to 1.28 T.

Close modal

In Fig. 9(b), the radial angular velocities profiles are shown for B 1 T. It is first noted that the profile at B = 1.0 T has the same spatial structure as the data presented in Fig. 9(a), with two regions of particle motion. However, with the increasing magnetic field, we believe that it is reasonable to reclassify the particle motion into three regions. The portion of the cloud labeled as Region I appears to shrink to a radius, r = 8 mm. The cloud now has a more distinct, separate rotation between r = 8 and r = 15 mm which is different from the region r 8 mm. Finally, there is a third region with a gradual decrease in the angular velocity beyond r = 15 mm.

The inset in Fig. 9(a) shows a measurement of the average angular velocity in the part of the cloud identified in Region I (from r = 1 mm to r = 8 mm) for all of the measurements. In this region, the cloud rotation is nearly uniform and can be easily measured and compared across the different magnetic field settings. This plot shows the gradual increase in the average rotation velocity as a function of magnetic field up through B = 1 T and then a general plateau in the velocity. This observation is similar to the results reported by Kaw et al.,30 for an experiment up to B = 1 T.

Therefore, we interpret the melting of the plasma crystal with the increasing magnetic field in the following manner. As the magnetic field is increased, the particle cloud shifted towards the confining ring. The radial electric field near the ring leads to an azimuthal ion E × B drift, which couples to the dust grains and leads to their rotation as described by Uchida et al.14,15 and Ishihara and Sato.31 Alternatively, the azimuthal ion drift can couple to the neutrals which, in turn, couples to the dust leading to a rotation as described by Carstensen et al.32 In this experiment, the measurements presented in Figs. 8 and 9 show that a differential rotation is established between the inner and outer regions of the plasma crystal. This discontinuity in the angular velocity appears to provide the crystal with a breaking point that allows melting to occur with the increasing magnetic field. Above B = 1 T, it is shown that the crystal can no longer retain its long-range order and the particles make the transition to a liquid-like state.

In summary, this paper reports on an experimental observation of the melting of a plasma crystal as a function of magnetic field. These experiments, performed using the Magnetized Dusty Plasma Experiment (MDPX) facility, studied the spatial ordering and structure of monodisperse melamine-formaldehyde dust particles that were held in a capacitively coupled, argon rf glow discharge at a fixed pressure and fixed rf power while varying the magnetic field. The measurements show that at a lower magnetic field (B < 0.5 T), there is an initial rigid rotation of the plasma crystal. With the increasing magnetic field, the outer portion of the crystal develops a differential rotation that eventually leads to heating and melting of the crystal. It is noted that various theories have considered that changes in the particle interaction potential may lead to crystal melting in a magnetic field. While these mechanisms may still play a contributing factor to the melting process, the experiments reported here show that the induced rotation is the dominant mechanism.

The authors would like to thank Professor Uwe Konopka for providing us with a source of monodisperse microparticles and his advice when carrying out these experiments. This work was supported by grants from the National Science Foundation, PHY-1613087, and the U.S. Department of Energy, DE-SC0016330 and DE-SC0010485.

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