2 Experimental set-up

The experiments are carried out in complex plasma experimental device (CPED) (Kaur et al. Reference Kaur, Bose, Chattopadhyay, Ghosh and Saxena2016) and its schematic is shown in figure 1. An argon plasma is produced by applying a steady potential difference between two coaxially arranged electrodes separated by 4 cm. The upper electrode serves as an anode, while the lower electrode serves as a cathode, as shown in figure 2(a). A metallic ring of inner diameter 6.2 cm and outer diameter 8.2 cm is placed coaxially on the cathode. The height of the ring above the cathode surface is 0.3 cm. More details on the experimental set-up can be found in earlier publications of Kaur et al. (Reference Kaur, Bose, Chattopadhyay, Sharma, Ghosh and Saxena2015a , Reference Kaur, Bose, Chattopadhyay, Ghosh and Saxena2016).

Plasma is produced in CPED over a wide operating range. The discharge current can be varied from 1 to 50 mA. A maximum potential difference of 1 kV can be applied in between the two electrodes. The experiments are usually carried out at high pressures, $P>100~\text{Pa}$ . For the experiments reported in this paper, the discharge current is varied from 15 to 20 mA and the discharge voltage is changed from 400 to 460 V. Pressure is held constant at 167 Pa.

Figure 1. Schematic of complex plasma experimental device (side view). The camera has been aligned along the $x$ -axis; the laser sheet is pointed in the $y$ -direction and $z$ represents the vertical direction.

To study the dust vortices, mono-disperse melamine formaldehyde particles of diameter $6.48\pm 0.08~\unicode[STIX]{x03BC}\text{m}$ are placed on the lower electrode inside the ring. The cathode surface within the metal ring is completely covered by dust particles, as shown in figure 2(a). The levitated dust particles are illuminated by a vertical laser sheet of thickness 0.2 cm. The laser sheet is obtained using a 532 nm, 100 mW laser followed by a cylindrical lens. The laser light scattered by the dust particles is recorded using an sCMOS camera through one of the axial ports of CPED. The sCMOS camera has a maximum resolution of $2560~\text{pixel}\times 2160~\text{pixel}$ . For further details on the camera and optical arrangements, refer to Kaur et al. (Reference Kaur, Bose, Chattopadhyay, Sharma, Ghosh and Saxena2015a , Reference Kaur, Bose, Chattopadhyay, Ghosh and Saxena2016).

Figure 2. (a) Schematic of the electrode arrangement and a coordinate representation of dust torus. Here, $z$ represents the direction opposite to gravity, $r$ represents radial direction, $\unicode[STIX]{x1D719}$ represents the toroidal/azimuthal direction and $\unicode[STIX]{x1D703}$ represents the poloidal direction (direction of rotation of dust particles). (b) Electrode arrangement and coordinate representation for Langmuir probe measurement.

An ultra-thin single Langmuir probe of diameter 0.125 mm and length of 6 mm is used for characterization of the plasma (Kaur et al. Reference Kaur, Bose, Chattopadhyay, Ghosh and Saxena2015c ; Bose et al. Reference Bose, Kaur, Chattopadhyay, Ghosh, Saxena and Pal2017). The electron temperature is determined from the electron retarding exponential region of the I–V characteristics, while the ion density is determined from the ion current. Ion–neutral collisions occur in the probe sheath for probe measurements at 167 Pa neutral pressure and plasma density ${\sim}10^{9}~\text{cm}^{-3}$ . Collision of ions with neutrals in the sheath surrounding the probe in the OML regime $D_{\unicode[STIX]{x1D706}}=r_{p}/\unicode[STIX]{x1D706}_{D}\lesssim 3$ affect the ion collection by the probe in two ways (Schulz & Brown Reference Schulz and Brown1955; Bose et al. Reference Bose, Kaur, Chattopadhyay, Ghosh, Saxena and Pal2017). Orbital destruction of the ions in the sheath results in an increase in the ion current collected by the probe. Elastic scattering of the ions in the probe sheath leads to a decrease in the probe ion current. These two opposing effects are taken into consideration simultaneously by the ‘modified TALBOT and CHOU model’ (Tichý et al. Reference Tichý, S̃ícha, David and David1994). According to this theory, the ion current collected by the Langmuir probe is expressed as $I_{\text{ion}}=\unicode[STIX]{x1D6FE}_{1}\unicode[STIX]{x1D6FE}_{2}I_{L}$ , where, $\unicode[STIX]{x1D6FE}_{1}$ accounts for the increase in ion current due to orbital destruction (Zakrzewski & Kopiczynski Reference Zakrzewski and Kopiczynski1974) and $\unicode[STIX]{x1D6FE}_{2}$ accounts for the decrease in ion current due to scattering of ions (Talbot & Chou Reference Talbot and Chou1969); $I_{L}$ is the collisionless Laframboise current (Laframboise Reference Laframboise1966). The method of determination of ion density from the ion current using the ‘modified TALBOT and CHOU model’ is well described by Bose et al. (Reference Bose, Kaur, Chattopadhyay, Ghosh, Saxena and Pal2017).

The Langmuir probe measurements are carried out in the absence of dust particles using a separate cathode. This cathode is identical to that used in experiments in the presence of dust in all respects, except that the dust on the cathode surface is replaced by a ceramic disc, such that the area of the cathode covered by the dielectric (dust particles or ceramic disc) and the exposed metal remains the same. The thickness of the cathode used for plasma production in the absence of dust during Langmuir probe measurements is slightly smaller on the inner side of the metal ring than the cathode used in the presence of dust. This is done to ensure that the cumulative thickness of the cathode does not increase on placing the ceramic disc on the cathode surface during Langmuir probe measurements, as shown in figure 2(b). This approach allows high quality probe measurements to be made in a dusty plasma without probe contamination by the dust particles.

The effects of dust on plasma parameters have been measured by other researchers by varying the dust density in laboratory experiments. The results of Adhikary et al. (Reference Adhikary, Bailung, Pal, Chutia and Nakamura2007) shows that a dust density of ${\sim}10^{3}~\text{cm}^{-3}$ will not alter the background plasma parameters for a plasma density of ${\sim}10^{9}~\text{cm}^{-3}$ . In our experiments, the dust density is ${\sim}10^{3}~\text{cm}^{-3}$ and the plasma density is ${\sim}10^{9}~\text{cm}^{-3}$ . Therefore, the plasma parameters will not be affected by the presence of dust particles in the experiments reported in this paper.

3 Experimental results and discussion

Figure 3. Evolution of the dust structure at 167 Pa neutral argon pressure with increase in discharge current. Cross-section of the dust torus at (a) 15 mA, and (b) 20 mA discharge currents. Superposition of twenty consecutive images of the dust vortex acquired at (c) 15 mA, and (d) 20 mA discharge current. The yellow arrows in (c,d) indicate the direction of rotation of the dust particles. The dust vortices shown here correspond to a vortex on the left poloidal pane of the torus in figure 2. The brightness and contrast of the image has been adjusted for better viewing. Please refer to the supplementary material for videos of the rotating dust particles available at https://doi.org/10.1017/S0022377819000011.

A poloidally rotating dust structure with toroidal symmetry localized above a metallic ring placed on the cathode surface is observed in an unmagnetized glow discharge plasma (Kaur et al. Reference Kaur, Bose, Chattopadhyay, Sharma, Ghosh and Saxena2015a ,Reference Kaur, Bose, Chattopadhyay, Sharma, Ghosh, Saxena and Thomas b ). The experiments are carried out at high neutral pressure. Initially, at 167 Pa pressure and 15 mA discharge current, a dust torus with a void at the centre is observed. The vertical cross-section illuminated by the laser sheet shows a filled poloidally rotating structure, as shown in figure 3(a,c). Upon increasing the discharge current to 20 mA, the size of the rotating dust structure increases, as shown in figure 3(b,d).

Figure 4. The spatial variation of the velocity of the rotating dust particles at 20 mA discharge current. The two-dimensional velocity profiles are an ensemble average of 849 flow fields (850 frames) in the PIV analysis with an interrogation window size of $32\times 32$ with $16\times 16$ overlap. Velocity vectors show the direction of rotation of dust particles; the colour bar represents the value of dust velocity in $\text{mm}~\text{s}^{-1}$ .

Particle image velocimetry (PIV) analysis is performed on the still images using openPIV software to determine the velocity of the dust particles in a two-dimensional $r$ – $z$ plane (Taylor et al. Reference Taylor, Gurka, Kopp and Liberzon2010; Williams Reference Williams2016). A total of 850 still images of the rotating dust structure are recorded with a sCMOS camera at a resolution $1392\times 1040$ pixel at 142.5 frames per second. For PIV analysis, the interrogation window size is taken as $32\times 32$ with $16\times 16$ overlapping. PIV analysis provides 849 flow fields. These are averaged to determine the mean velocity profile, which represents the velocity profile for the majority of flow fields.

Figure 4, represents the two-dimensional velocity profile of the dust particles at 20 mA discharge current. Peak poloidal velocity ( $v_{\unicode[STIX]{x1D703}}$ ) of $45~\text{mm}~\text{s}^{-1}$ is observed from the two-dimensional velocity profile in figure 4 at 20 mA discharge current. However, the velocity measurements from the PIV analysis near the bottom of the rotating dust structure at (150, 50), as shown in figure 4, is an underestimation. This is because the fast moving dust particles move a considerably larger distance in a single exposure time, and hence appear as lines near the bottom of the rotating structure in the still images, as seen in figure 3(b). The velocity profiles using PIV analysis are best obtained when the particle moves a small distance, typically less than 30 % across the interrogation box (Thomas Jr Reference Thomas1999). At the bottom location, the dust velocity is determined by taking the ratio of the measured track length of the dust particle in a single exposure time and the exposure time. Using this method, the dust velocity is found to be ${\sim}74~\text{mm}~\text{s}^{-1}$ .

We have measured plasma parameters at different horizontal and vertical locations using a Langmuir probe for the two discharge currents of 15 mA and 20 mA. Radial density and floating potential profiles measured at different heights for a background neutral pressure of 167 Pa and discharge current 15 mA are shown in figures 5(a) and 5(b), respectively. Similarly, the density and potential profiles measured at a discharge current of 20 mA are shown in figures 6(a) and 6(b), respectively. A sharp radial density gradient is observed at the location of the dust vortex, in agreement with our previous observations for dust filled vortices (Kaur et al. Reference Kaur, Bose, Chattopadhyay, Sharma, Ghosh and Saxena2015a ,Reference Kaur, Bose, Chattopadhyay, Sharma, Ghosh, Saxena and Thomas b ). This implies that the momentum transferred to the dust particles by the ion flux from the anode to cathode varies radially. A greater momentum will be transferred by the ions to the dust particles where the density is high near the metal ring kept on the cathode surface, while the ions near the centre of the cathode in the low density region will exert a smaller force on the dust particles.

Figure 5. (a) Radial variation of density and, (b) potential profiles measured at a height of 0.9 cm, 1.3 cm and 1.7 cm above the cathode surface. The $r$ – $z$ plane intersecting the dust torus on the left side from the center of the cathode (marked as 0 in figure 2(b)) is sampled by the Langmuir probe. The measurements are carried out at 167 Pa pressure and 15 mA discharge current.

Figure 6. (a) Radial variation of density and, (b) potential profiles measured at a height of 0.7 cm, 0.9 cm, 1.1 cm and 1.7 cm above the cathode surface. The $r$ – $z$ plane intersecting the dust torus on the left side from the center of the cathode (marked as 0 in figure 2(b)) is sampled by the Langmuir probe. The measurements are carried at 167 Pa pressure and 20 mA discharge current.

Figure 7. Radial variation of electron temperature measured at a height of 0.9 cm and 1.7 cm above the cathode at (a) 15 mA and (b) 20 mA discharge currents. The measurements are carried out using a Langmuir probe. The $r$ – $z$ plane intersecting the dust torus on the left side from the center of the cathode (marked as 0 in figure 2(b)) is sampled by the Langmuir probe.

The electron temperature is approximately constant in the region of formation of dust vortex. The radial variation of electron temperature for two representative heights at 15 mA and 20 mA is shown in figures 7(a) and 7(b), respectively. Since, the variation of electron temperature is negligible, the spatial variation of the floating potential is considered to follow the spatial variation of the plasma potential.

A local minimum in the radial profile of the floating potential, and hence, in the plasma potential, is observed at lower heights from the cathode surface as shown in figures 6(a) and 5(a). The local minimum is found to be located near the inner edge of the metal ring kept on the cathode surface coinciding with the location of the dust vortex. The observed local minimum in the radial profile of the floating/plasma potential implies that there is a radial electric field directed towards the minimum. The local minimum will attract the positively charged ions, while it will repel the negatively charged dust particles. The ions moving towards the local minimum will tend to drag negatively charged dust particles along with them, while the Coulomb force will repel the dust particles from the minimum. The presence of a strong monotonic radial density gradient at the location of the local potential minimum implies that the ion drag force is asymmetric about the potential minimum. Thus, it is clear that the forces of electrical origin acting on the charged dust particles have a spatial variation depending on the local values of ion density, electric field, etc.

The effect of forces of electrical origin on the dust vortex is explored by analysing the rotating dust structure with a dust free region at the centre observed at 20 mA discharge current, and 167 Pa neutral pressure, as shown in figure 3(b,d).

The electrical force experienced by the charged dust particle is a function of the charge of the dust particle and the potential gradient in dc glow discharge in addition to other plasma parameters. The charge is calculated by assuming the dust particle to be a spherical condenser and using the fundamental relation, $Q_{d}=C_{d}\unicode[STIX]{x1D6F7}_{d}$ , where, $Q_{d}$ , $C_{d}$ and $\unicode[STIX]{x1D6F7}_{d}$ are the charge, capacitance and surface potential of the dust particle. The surface potential of the dust particle is determined from the balance of the ion and electron flux incident on the dust particle. The experiments are carried out at 167 Pa neutral pressure. At this high pressure, the ion neutral mean free path is $12~\unicode[STIX]{x03BC}\text{m}$ , which is less than the plasma shielding length for $10^{9}~\text{cm}^{-3}$ ion density and 1.3 eV electron temperature. However, the electron mean free path, which is two orders larger than the ion mean free path, is also larger than the plasma shielding length. Collisional ion flux incident on the dust particles is calculated using the interpolation formula given by Khrapak & Morfill (Reference Khrapak and Morfill2008), while the OML theory, is used to estimate collisionless electron flux. Using the above mentioned formulation, the dust charge is determined to be $4616\text{e}^{-}$ for $10^{9}~\text{cm}^{-3}$ ion density, 1.3 eV electron temperature, 167 Pa neutral pressure and $6.48~\unicode[STIX]{x03BC}\text{m}$ dust diameter.

The Coulomb force ( $\boldsymbol{F}_{E}=Q_{d}\boldsymbol{E}$ ) acting on the dust particles, as shown in figure 8(b,d), is determined by multiplying the charge acquired by the dust particles with the local electric field. The value of the electric field is determined from the spatial variation of the floating potential, $V_{f}$ . The vertical electric field at $z=z_{0}$ and $r=r_{0}$ is determined using the formula, $E_{z}(r_{0},z_{0})=-\{V(r_{0},z_{2})-V(r_{0},z_{1})\}/(z_{2}-z_{1})$ , where $z_{2}=z_{0}+0.2~\text{cm}$ and $z_{1}=z_{0}-0.2~\text{cm}$ . And the radial electric field at $z=z_{0}$ and $r=r_{0}$ is determined using a similar formula $E_{r}(r_{0},z_{0})=-\{V(r_{2},z_{0})-V(r_{1},z_{0})\}/(r_{2}-r_{1})$ , where $r_{2}=r_{0}+0.25~\text{cm}$ and $r_{1}=r_{0}-0.25~\text{cm}$ . For example, using the above mentioned method the radial electric field at $r=3.5~\text{cm}$ and $z=0.9~\text{cm}$ is determined to be $\boldsymbol{E}_{r}(r=3.5~\text{cm},z=0.9~\text{cm})=-(232.1-229.8)/0.5=-4.6~\text{V}~\text{cm}^{-1}\hat{r}$ . Therefore, for a dust charge of $4616\text{e}^{-}$ and electric field $=-4.6~\text{V}~\text{cm}^{-1}$ , the Coulomb force turns out to be ${\sim}3.4\times 10^{-13}~\text{N}$ .

The ion flow results in transfer of momentum from ions to negatively charged dust particles. The drift velocity of the ions due to the presence of an electric field is $\boldsymbol{v}_{\text{Di}}=\unicode[STIX]{x1D707}\boldsymbol{E}$ . Mobility is determined using the formula $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D707}_{0}\{1+a(E/P)\}^{-1/2}$ , where $E$ is the electric field in $\text{V}~\text{cm}^{-1}$ , $P$ is the pressure in torr and ‘ $\unicode[STIX]{x1D707}_{0}$ ’ and ‘ $a$ ’ are constants (Frost Reference Frost1957). For argon discharge, ‘ $\unicode[STIX]{x1D707}_{0}$ ’ is equal to $1460~\text{cm}^{2}~\text{V}^{-1}~\text{s}^{-1}$ and ‘ $a$ ’ is equal to $0.0264~\text{torr}~\text{cm}~\text{V}^{-1}$ . At 167 Pa neutral pressure, the ion neutral collisions occur rather frequently, the ion drag force for subthermal ion flow with $l_{i}\leqslant \unicode[STIX]{x1D6FD}_{T}\unicode[STIX]{x1D706}$ is given by Ivlev et al. (Reference Ivlev, Khrapak, Zhdanov, Morfill and Joyce2004, Reference Ivlev, Zhdanov, Khrapak and Morfill2005), Fortov & Morfill (Reference Fortov and Morfill2010) and Kaur et al. (Reference Kaur, Bose, Chattopadhyay, Sharma, Ghosh and Saxena2015a ) as

(3.1) $$\begin{eqnarray}\displaystyle F_{\text{id}}\simeq \frac{1}{6}\left(\frac{T_{i}}{e}\right)^{2}\left(\frac{\unicode[STIX]{x1D706}}{l_{i}}\right)\unicode[STIX]{x1D6FD}_{T}^{2}M_{T}, & & \displaystyle\end{eqnarray}$$

where $l_{i}$ is the ion mean free path, $M_{T}(=v_{\text{Di}}/v_{\text{Ti}})$ is the thermal Mach number, $\unicode[STIX]{x1D6FD}_{T}(=\mathit{e}^{2}Z/\unicode[STIX]{x1D706}T_{i})$ is the thermal scattering parameter and $\unicode[STIX]{x1D706}$ is the effective screening length. The ion drag force, $F_{\text{id}}$ , calculated using (3.1) for ion density $10^{9}~\text{cm}^{-3}$ , electron temperature $T_{e}=1.3~\text{eV}$ , ion temperature $T_{i}=0.03~\text{eV}$ , electric field $E=4.6~\text{V}~\text{cm}^{-1}$ and at neutral pressure $167~\text{Pa}$ is ${\sim}5\times 10^{-13}~\text{N}$ . The value of the electric field used in the calculation of the ion drag force is determined from the radial variation of the floating potential.

The neutral frictional force experienced by the dust particle as it wades through the stationary neutral background is estimated using the formula (Epstein Reference Epstein1924; Liu et al. Reference Liu, Goree, Nosenko and Boufendi2003)

(3.2) $$\begin{eqnarray}\displaystyle F_{\text{nd}}=\unicode[STIX]{x1D6FF}\frac{4\unicode[STIX]{x03C0}}{3}n_{n}m_{d}r_{d}^{2}\bar{c}v_{d}, & & \displaystyle\end{eqnarray}$$

where $m_{d}$ , $n_{n}$ , $r_{d}$ , $\bar{c}$ and $v_{d}$ are the dust mass, neutral density, radius of the dust particle, mean thermal speed of gas atom and dust velocity, respectively. In our experiment $r_{d}=3.24~\unicode[STIX]{x03BC}\text{m}$ and $m_{d}=2.15\times 10^{-13}~\text{kg}$ . At $167~\text{Pa}$ neutral pressure and neutral temperature, $T_{n}\approx 0.03~\text{eV}$ , the neutral density is estimated to be $3.48\times 10^{16}~\text{cm}^{-3}$ . Therefore, the corresponding neutral frictional force acting on a dust particle traversing with a speed $5~\text{mm}~\text{s}^{-1}$ for $\unicode[STIX]{x1D6FF}=1$ is ${\sim}2\times 10^{-13}~\text{N}$ .

Figure 8. (a) Superposition of multiple consecutive images of the dust vortex acquired at 20 mA discharge current and 167 Pa neutral pressure. The brightness and contrast of the image has been adjusted for better viewing. The direction of dust rotation is indicated by the curved yellow arrow and the coordinate system is defined in the upper right corner of the image. (b,c) Show a radial profile of the radial components of the ion drag and Coulomb force, respectively, at a height of 1.5 cm above the cathode surface. A radial profile of the vertical component of the ion drag force and the Coulomb force at a height of 0.9 cm are shown in (d,e), respectively, and the same for their radial components are plotted in (f,g), respectively.

Among all the forces, gravity is one of the major forces. The downward directed gravitational force acting on the spherical melamine formaldehyde particle of mass $2.15\times 10^{-13}~\text{kg}$ is $2.1\times 10^{-12}~\text{N}$ .

The Coulomb and ion drag forces acting near the top of the vortex at a height of 1.5 cm, spanning the radial extent covered by the rotating dust particles, are shown in figure 8(b,c). The direction and the magnitude of these forces near the base of the vortex at a height of 0.9 cm are shown in figure 8(d–f). These Coulomb and ion drag forces are determined as described previously in this paper by making use of local parameters like density, electric field, etc. These calculations are done to understand the relative importance of the forces of electrical origin on the trajectory of dust particles.

The radial variation of the ion drag and Coulomb force acting on dust particles at a height of $z=1.5~\text{cm}$ are shown in figures 8(b) and 8(c), respectively. The radially outward ion drag force ( $F_{\text{id}_{r}}\hat{r}$ ) is greater than the radially inward Coulomb force ( $F_{\text{E}_{r}}\hat{r}$ ). The direction of the effective electrical force, i.e. $F_{\text{id}_{r}}\hat{r}-F_{\text{E}_{r}}\hat{r}$ is consistent with the direction of motion of the dust particles.

The downward directed ion drag forces and the upward directed Coulomb forces at $z=0.9~\text{cm}$ are shown in figure 8(d,e). The downward directed ion drag force clearly dominates the Coulomb force near the metal ring, but away from the inner edge of the ring at $r=2~\text{cm}$ the upward directed Coulomb force dominates over the downward directed ion drag force. However, the upward directed Coulomb force at $r=2~\text{cm}$ is insufficient to balance the force of gravity acting on the dust particles. The radial component of the ion drag and Coulomb force at $z=0.9~\text{cm}$ height are shown figure 8(f,g). The radially inward ion drag force ( $-F_{\text{id}_{r}}\hat{r}$ ) dominates over the outward directed Coulomb force ( $F_{\text{E}_{r}}\;\hat{r}$ ) at $r=4~\text{cm}$ , while the radially outward Coulomb force becomes comparable to the oppositely directed ion drag force at $r=2.5~\text{cm}$ .

Therefore, we can describe the motion of the dust particles in the vortex using the following descriptions. The dust particles approach the cathode sheath edge under the combined action of the downward directed ion drag and gravity near the metal ring. The downward directed dust particles near the edge of the metal ring experience a radially inward ion drag force (refer figure 8 f,g), which imparts momentum in the radially inward direction to the dust particles. Thus, the dust particles approaching the sheath gains momentum both in the radial $(-\hat{r})$ and in the vertically downward direction $(-\hat{z})$ at the inner edge of the metal ring. The downward approaching dust particles enter the sheath under the action of a dominating ion drag force $(-F_{\text{id}_{z}}\hat{z}-F_{\text{id}_{r}}\hat{r})$ , these dust particles are ultimately reflected back by the Coulomb force exerted by the strong sheath electric field and gain a momentum in the upward direction. By the time the dust particles gain a momentum in the upward direction, they have already moved radially inwards due to their radial momentum. The dust particles rise above the sheath surface at a radially inward location (away from the inner edge of the metallic ring) making an angle with the vertical. The radial momentum of the dust particles emerging from the sheath in the $-\hat{r}$ direction is opposed by the neutral frictional force leading to a reduction in the radial momentum in the $-\hat{r}$ direction. The vertically upward directed momentum of the dust particles decreases as they rise above the sheath edge. The downward directed gravity, neutral friction and ion drag force impede the vertically upward momentum of the dust particles and finally cause the dust particles to fall towards the cathode surface.

Thus, it can be concluded that the confinement of the dust particles in the region above the metal ring is due to the cumulative action of the ion drag force, Coulomb force, neutral friction and gravitational force. The rotation of the dust particles is driven by the combined action of the ion drag force and Coulomb force acting on the dust particles.

A comparison of radial ion density profiles (refer to figures 5 a and 6 a) for two values of discharge current shows that the ion density increases when the discharge current is increased from 15 to 20 mA. The ion densities in the region where the rotating dust particles approach the cathode surface at $z=0.9~\text{cm}$ and $r=3.5~\text{cm}$ are $7.6\times 10^{8}~\text{cm}^{-3}$ and $9\times 10^{8}~\text{cm}^{-3}$ at 15 mA and 20 mA discharge current, respectively. This implies that the ion flux towards the cathode will impart a greater momentum to the dust particles at higher discharge current. This will cause the dust particles to penetrate further into the sheath region resulting in a stronger upward momentum from the cathode sheath field, causing an increase in the size of the vortex. A closer look at figure 3(a,b) does show that the distance between the bottom edge of the vortex and the metal ring on the cathode indeed decreases with increase in discharge current. At 15 mA and 20 mA discharge currents, the bottoms of the rotating vortex are 4 mm and 3.6 mm, respectively, from the metallic ring on the cathode.

4 Summary

The crucial role played by the spatial inhomogeneity of floating/plasma potential in confining and aiding the rotation of dust particles in a dc glow discharge plasma is presented in this paper. Dust vortex with a void in the centre is produced in a parallel plate dc glow discharge dusty plasma with a modified cathode geometry. A local minimum in the radial profile of the floating potential is observed at lower heights from the cathode surface at the location of formation of the dust vortex. The floating potential is considered to follow the plasma potential because the electron temperature is found to be constant. The local minimum of the floating potential attracts the positively charged ions but repels the negatively charged dust particles. The radial ion density profile exhibits a strong gradient in the region of formation of the dust vortex, resulting in an asymmetric radial variation of the ion density about the local potential minimum. As a result, the ion drag force dominates the Coulomb force at one side of the potential minimum, while the Coulomb force dominates over the ion drag on the other side of the potential minimum. A delicate balance between these two oppositely directed forces is found to play a crucial role in controlling the trajectory of the rotating dust particles.

Another crucial aspect of the dust vortex that this paper addresses is the reason for confinement of the rotating dust particles in the region above the metal ring on the cathode surface. We have shown by logical arguments and by determining the magnitude and direction of the relevant forces that the rotating dust particles are confined by the cumulative action of ion drag force, Coulomb force, neutral friction and force of gravity.

The evolution of the dust vortex is studied by increasing the discharge current from 15 to 20 mA. The local minimum of the potential profile is found to coincide with the location of dust vortex for both values of the discharge current. However, the size of the dust vortex and the void at the centre of the vortex is found to increase with discharge current.