A corona discharge-assisted technique for spreading of a gently deposited dielectric droplet into a uniform thin film over a dry isothermal conductive substrate is proposed. The surface charge was built up over the droplet interface through ion bombardment using a sharp emitter electrode. Interaction of the surface charge density and intense electric field generates an interfacial electrical pressure and leads to a uniform axisymmetric spreading of the droplet in the radial direction. It was shown that the droplet expansion process can be controlled through variation of the ion injection current and/or discharge exposure time. It is also demonstrated that the proposed technique can be analogous to the classical Stefan's squeezing liquid flow between two separated parallel discs. The dynamics of the film spreading owing to the corona discharge can be predicted through a simplified analytical model based on this analogy.
Spreading of a liquid droplet over a solid surface is important for a wide variety of applications, such as coating and deposition processes (Carré et al. 1996; Schiaffino & Sonin 1997; Bergeron et al. 2000), spray cooling of heated surfaces (Bernardin et al. 1997; Frohn & Roth 2000) and thin film evaporation (Mang & Dresel 2007). The active control of the droplet deposition process is still challenging and the current techniques highly depend on the fluid properties and surface condition. The conventional way to increase the spreading ratio of a droplet is to increase the impact velocity. However, high speed impact causes some drawbacks, such as non-uniformity of the resulting film thickness, micro-satellite droplet formation owing to violent breakup and recoiling/rebounding phenomenon. In most of the above listed applications, the droplet spreading process needs to be uniform as a non-uniform film thickness leads to other complications (Bussmann et al. 2000; Mehdizadeh et al. 2004). Moreover, preserving the contact between the resulting liquid film and the solid surface during the spreading process leads to efficient deposition with reduced material wastage and environmental pollution. Therefore, new methods are demanded to enhance the deposition efficiency and mitigate the drawbacks of high-speed droplet impact.
The corona discharge process has been found to be advantageous for several scientific and industrial applications. It creates a partially ionized gaseous medium with free space charge carriers drifting towards a grounded electrode. These free charge carriers can be deposited as surface charge on the targets. Although the corona discharge has been extensively applied to create net motion in gaseous media, little effort has been expended to obtain a net flow in dielectric liquids. Recently, perpendicular corona discharge was applied to a liquid–vapour interface to investigate a new type of interfacial electrohydrodynamic instability, which is the so-called rose-window instability (Vega & Pérez 2003; Vega & García 2006). The dielectric liquid was placed in a container and charge carriers injected through the gaseous phase towards the dielectric interface using a high-voltage corona needle electrode fixed at some height above the liquid surface. The surface charge density accumulates on the interface and creates strong electrical pressure, so that the thickness of the dielectric layer locally reduces. Subsequently, the electric field in the thin film region increases and the surface charge easily migrates to the ground electrode owing to conduction phenomenon. The charge migration mitigates the electrical pressure, and the local thickness of the liquid film increases. The process is resumed by charge accumulation over the interface and the rose-window instability patterns are generated. Although several experimental and theoretical studies have focused on understanding the interaction of corona discharge with the dielectric liquid interface and the induced instabilities (Atten & Moreau 1972; Koulova-Nenova & Atten 1997), no effort can be found in the literature to generate a net pressure gradient through interfacial corona discharge exposure.
In the present work, a new forced spreading technique for a dielectric droplet over a conductive substrate is proposed. In order to generate a strong enough electrical pressure, the droplet interface was exposed to a corona discharge. The interfacial electrical pressure resulting from the ion deposition on the liquid–vapour interface was used to actively control the liquid film spreading process. In order to understand the physics of spreading in the presence of corona discharge, a simplified mathematical model based on theoretical postulations is proposed. In this model, the dynamic expansion of the droplet is correlated to the corona discharge parameters, physical/electrical properties of the dielectric liquid and gaseous medium in which the discharge occurs. The theoretical postulations of the mathematical model were verified through further demonstrative experiments. A comparison between theoretical predictions and measured spreading dynamics is provided.
2. Experimental setup and procedure
A schematic view of the experimental setup is shown in figure 1. In order to develop perpendicular corona discharge, a universal fitting with a precise adjustable mechanism was constructed. This fitting was electrically insulated and connected to a XYZ holder. The position of the substrate with respect to the high voltage electrodes was adjusted using a laser pointer and the XYZ holder.
In order to confirm the pivotal role of the ion injection process owing to corona discharge in the film spreading phenomenon, two sets of experiments were performed with two different electrode configurations: (i) parallel plate geometry and (ii) needle plate geometry. For parallel plate electrode experiments, a mirror-finished stainless steel disc was chosen and all the sharp edges, including the rim of the electrode, were rounded and fully finished to minimize the back corona discharge. For the needle plate experiments, two standard hypodermic needles, 15 and 22 g, were selected as emitter electrodes. The inner and outer diameters for 15 g were 1.37 and 1.83 mm, respectively. For 22 g, the inner diameter was measured to be 0.41 mm and the outer diameter was 0.72 mm. The length of both the needles was identical and chosen to be 40 mm. The tips of the needles were cut off perpendicular to their axis with a laser cutter. No particular irregularity in the needle cross section was found after the preparation process.
The needles were connected to a positive polarity of a DC high-voltage power supply and the substrate was grounded. The unipolar ion injection strength was controlled through variation of the applied voltage using these two emitter electrodes. As the oxidation owing to the ozone generation may affect the sharpness of the electrodes, the V –I characteristics for both needles were monitored. The corona discharge characteristics for these emitter electrodes were found to be stable during the experiments.
The substrate was a mirror-finished stainless steel plate. In the present study, silicone oil was chosen to be the main dielectric working fluid. The dielectric droplet was gently deposited on the grounded substrate using a micro-syringe. A few additional qualitative experiments were performed using glycerine, deionized water and solution of silicone oil with different concentrations of di-2-ethylhexyl sodium sulphosuccinate (AOT).
The initial size of the deposited droplet, Do, was evaluated by converting the droplet weight to its equivalent spherical size. Droplets with different sizes were obtained using the two different hypodermic needles and with various injected volumes. It should be noted that immediately after the droplet deposition, some spontaneous spreading occurs owing to its own weight and capillary effects. Thirty seconds after deposition, the high-voltage power supply was switched on and the droplet evolution was recorded using a digital camera (SONY DCR-SX43) at 30 fps. A typical set of photographic sequential frames for a silicone oil droplet shown in figure 2a were used to plot the dynamic evolution of the film diameter, D(t) (figure 2b). The measurement system converts the number of pixels directly to the dimensions of the spreading film. In order to check the uniformity of expansion, the film diameters were measured at different peripheral angles. The non-uniformity of the measured droplet spreading ratio, β=D(t)/Do, never exceeded 1 per cent. In order to diagnose the onset of any interfacial deformation owing to the ion injection, an additional optical system including a laser beam and a photodiode detector was used. For a fixed incident angle, any changes in the mirror angle simply produce a detectable electric signal. All the experiments were repeated three times to ensure the reproducibility of the experimental data.
3. Results and discussion
In order to prove that interfacial charging is responsible for the spreading phenomenon, we studied the spreading behaviour of a silicone oil droplet in a uniform and strong electric field in the absence of ion injection. Two parallel discs with smooth rounded edges enabled us to achieve a high electric field close to the air breakdown limit with negligible ion injections. For this extreme case, where the principal radii of curvature of the high-voltage electrodes tend to infinity, monitoring the oil droplet shape using a laser diode and photodiode detectors showed no noticeable change in its surface or the contact line. The range of applied voltage was 0–40 kV DC and no measurable electric current was recorded during these experiments. This shows that the Laplacian electric field in the absence of corona discharge yields no changes either in the droplet shape or in the contact line position. Therefore, the electrical pressure generation owing to the permittivity and static field difference across an air–dielectric liquid interface does not influence the deposited droplet shape. However, by introducing a series of mini-grooves on the high-voltage electrode and repeating the experiment, a weak corona discharge was established between the parallel plates and a noticeable electric current in the order of approximately 0.1 μA was measured at the highest possible applied voltage, which was close to the breakdown limit. At these conditions, a minor depression on the droplet interface was detected. These observations suggest that the spreading mechanism is governed by ion deposition on the droplet surface. In order to obtain significant film spreading, one should locally enhance the strength of ion injection to create localized surface charge density and electric field on the droplet interface. The needle-plate configuration was adopted to provide more significant spreading.
The typical evolution of a silicone oil droplet exposed to the corona discharge (V =15 kV applied voltage and total corona current of I=15 μA) is shown in figure 2b. It can be observed that the droplet is squeezed to a pancake shape and spreads axisymmetrically in the radial direction. The expansion process is uniform and the spreading ratio increases monotonically with the discharge exposure time.
It is noticeable that the evolution process involves two distinct regimes: a short and long time ones. In the short-term evolution (for t<10 s), when the ratio of the film diameter to the film thickness is moderate, the spreading proceeds very rapidly, but its rate decreases quickly with time. For the long-term exposure (for t>10 s), as the diameter of the squeezed droplet increases and the liquid film thickness dramatically decreases, the spreading rate becomes significantly slower.
The mechanism of droplet spreading in the presence of unipolar ion injection is rather complicated. A figurative explanation of the spreading mechanism is shown in figure 2c. The electric charge carriers are generated owing to the locally intense electric field, E. The charge cloud drifts towards the liquid layer and grounded electrode. The surface charge density, ρs, accumulates over the liquid interface until it asymptotically saturates. After reaching the surface charge saturation limit, the excess charges generated by the corona discharge are repelled from the droplet surface owing to the electric field developed by the deposited charges. However, as the dielectric liquid is not ideal, some electric charges migrate to the grounded electrode across the liquid film through the conduction mechanism and the corona discharge needs to supply new charge carriers to compensate for the lost charges. Under the quasi-steady-state conditions, the supplied and migrated charges reach equilibrium. The normal component of the electric field produces an interfacial electrophoretic pressure, ρsE, acting towards the substrate. This electric pressure, resulting from the surface charge–electric field interaction, creates a uniform pressure gradient in the radial direction. The net visible effect of this phenomenon is the axisymmetric radial spreading of the droplet. This hypothetical explanation of the spreading mechanism will be discussed through various experiments in this paper.
In order to investigate the effect of ion injection strength on the spreading process, two different hypodermic needles were used to create different corona currents at the same gap spacing. Figure 3 shows a comparison between the V –I characteristics obtained for the modified 15 and 22 g hypodermic needles with S = 25 mm electrode gap spacing. As the sum of the two principal radii of the corona electrode curvature affects the strength of ion injection and corona current, the V –I characteristics of both needles at a given electrode separation are different. It should be noted that the V –I characteristics do not show measurable changes in the presence and absence of a thin dielectric film. Therefore, figure 3 represents universal V –I characteristics for all experimental data presented in this paper.
Figure 4 shows the dynamic evolution of a large and medium silicone oil droplet in the presence of corona discharge for both the 15 and 22 g hypodermic needles. According to figure 4, a significant expansion can be achieved at 15 kV applied voltage using both the needles. Comparing the corona characteristics for these needles at the same applied voltage and electrode gap spacing, one can notice that the corona current becomes almost double when the 22 g electrode is used. Therefore, using a fine needle at a fixed applied voltage leads to a higher electric field, an increased ion injection strength and thus a larger corona current, which corresponds to a larger rate of interfacial charge accumulation and droplet spreading ratio. As it was shown before, the spreading process for both the needles has two distinct regions: (i) rapid expansion at the beginning of discharge exposure and (ii) long-term evolution, which is remarkably slower. The experimental observations revealed that the fine emitter needle, 22 g, gives a faster expansion rate in both the regions, while the coarser needle, 15 g, results in a lower rate of expansion.
4. Effect of electric field strength
The effect of electric field strength on droplet spreading dynamics is shown in figure 5. The electrode voltage was kept constant (V =15 kV) during the experiment and the electrode spacing was varied between 25 and 62 mm. As it can be seen in figure 5, decreasing the electrode separation results in a faster spreading process. Here the enhanced electric field increases the current, elevates the rate of interfacial charging, produces higher electrical pressure and increases the spreading rate. However, as for the reduced gap spacing, the effective region of discharge (corona cone) becomes narrower, the electric field and surface charge deposition is more concentrated around the centre region (Warburg's law; Warburg (1899, 1927)), and is weaker at the rim of the film (Jones 1997). Therefore, it may be expected that the corona discharge at reduced gap spacing leads to a less uniform expanded films. Qualitative observation of Fizeau interferograms confirmed that for the larger gap spacing, the fringe patterns become more regular and a more uniform film spreading was obtained. Furthermore, for the smaller electrode separations (S<25 mm), the random irregularities of the high-voltage emitter electrode and its associated non-uniformities in the electric field presumably become of increasing importance and the resulting film is more likely to be non-uniform.
5. Effect of droplet size and electrical conductivity on the spreading process
Figures 6 and 7 depict the dynamic evolution of droplets with different sizes at different electric field strengths. As the capillary length, , is approximately 1.5 mm, droplets with initial size Do>3 mm can be classified as large droplets. In order to investigate the effect of droplet dimension, small, medium and large droplets were generated and expanded. The spreading ratio of different size droplets falls approximately on a universal curve for both electric field strengths. This means that the spreading rate in the presence of corona discharge is not sensitive to the droplet volume. Later, we will revisit this experimental observation analytically.
In order to investigate the importance of the physical properties of the dielectric liquid, such as electrical conductivity and viscosity some additional demonstrative experiments were performed. To increase the electrical conductivity of silicone oil from 5.5×10−9 S m−1 to approximately 5×10−5 S m−1, a solution of silicone oil and 10−1 mol AOT was prepared. When using the 22 g electrode at V =15 kV, I=23 μA and S=40 mm, only a minor depression over the droplet interface was observed, but the droplet contact line remained unchanged and no spreading was recorded. The minor depression of interface at the centre region can be an indication of a slight electrical pressure, which weakly affects the semi-conductive liquid interface. Repeating the above experiment using silicone oil solution with 10−2 mol AOT (σ∼1×10−6 S m−1), one can obtain very slow film spreading. As shown in figure 8, the spreading rate in both short-term and long-term regimes is significantly slower than that of pure oil with the same corona discharge parameters. These observations confirm the effect of elevated electrical conductivity on scaling down the electrical pressure generation and reduced rate of spreading explained earlier. The same experiment was repeated with a glycerine droplet, which has a viscosity one order of magnitude higher than that of silicone oil and its electrical conductivity is near the conductivity of 10−2 mol AOT solution. The corona discharge at 15 kV showed no change of glycerine droplet interface. However, after increasing the applied voltage to 18 kV, a similar minor interfacial depression was detected and contact line position was slightly changed. For an extreme case, where the liquid is electrically conductive (water droplet), even the strongest corona discharge showed no measurable interfacial distortion. As observed with the glycerine droplet, both electrical conductivity and viscosity of the liquid play an important role in determining the spreading rate of a film exposed to the corona discharge. All of these evidences confirm the effect of elevated electrical conductivity on scaling down the electrical pressure generation and the reducing rate of spreading as postulated earlier.
6. Mathematical model
In order to understand the spreading mechanism, the hydrodynamics of the droplet distortion in the presence of corona discharge was simplified to the classical Stefan's squeezing flow of a fluid between two parallel discs through an external force (Stefan 1874). An analogy between the current problem and classical squeeze flow can be made by replacing the external mechanical pressure with the electric pressure (figure 9). One minor difference is the boundary condition imposed at the droplet interface. For the classical Stefan problem, no-slip conditions are applied over both the plates, whereas in the current problem, the boundary conditions at the interface can be primarily assumed to be free boundary conditions (stress-free boundary). Because of the symmetry of the squeezing flow between parallel discs, all other aspects of both problems can be considered identical. The simplifying assumptions are as follows:
— the film thickness h(t) is uniform in the radial direction,
— according to the approximation normally assumed in lubrication theory, the spreading process is quasi-steady (Cameron 1966),
— the electric pressure, ρsE, is uniformly distributed over the liquid surface and is treated as an external pressure exerted on the liquid film interface,
— all the edge effects, including the effect of surface charge migration from the contact line to the ground, are neglected,
— convection terms are assumed to be negligible,
— the inertia effect is important for large droplets only in short time evolution,
— in the first approximation, the effect of surface tension and droplet interfacial curvature are assumed to be negligible,
— the electrohydrodynamic instabilities owing to the surface charge are neglected,
— the film thickness was assumed to be very small with respect to the gap length during the evolution process. However, the corona discharge over the interface was assumed to be strong enough, so that the current density change owing to the film thickness variation becomes negligible. Applying both conditions, the implicit current density expression presented in the study of Mott & Gurney (1964) is reduced to the well-known Mott's steady-state space–charge-limited conduction (SCLC; Vega & Pérez (2002)),
— the electric pressure has two components: (i) electric pressure owing to the surface charge (electrophoretic component) and (ii) electric pressure owing to the interfacial jump in 1/2εE2 (dielectrophoretic component). According to the observations, as the electrostatic field in the absence of ion injection does not influence the interface, the dielectrophoretic component was neglected, and
— the main voltage drop is assumed to occur in air gap. As the liquid layer is thin, the voltage drop across the dielectric film is assumed to be small with respect to that of the electrode spacing.
The Stephan's equation for squeezing flow assuming a constant force F, Newtonian liquid of viscosity μ and a constant volume V 0, and partially filled between two parallel discs with separation h is as follows (Dienes & Klemm 1946): 6.1 The differential equation (6.1) has been obtained from the simplified lubrication theory for quasi-steady state neglecting the inertia effects (Leider & Bird 1974). Assuming that the net electric pressure, ρsEn, resulting from the surface charge density, ρs, and the normal electric field, En, are uniform, and considering the hydrodynamic analogy between the mechanical and electrical pressures: 6.2 As the droplet volume, V 0, is constant during the evolution, the dynamic spreading diameter, D(t), can be obtained as: 6.3 Knowing the electric pressure exerted on the interface and maximum spontaneous spreading of the droplet deposited over the solid substrate, Ds, the dynamic behaviour of the droplet having an initial diameter of Do can be predicted from equation (6.3). However, estimation of the electric pressure is difficult as a portion of the deposited surface charge density migrates towards the grounded substrate through the conduction mechanism as the liquid was assumed to be an ohmic (‘leaky’) dielectric. Moreover, the current density, surface charge and electric field have a non-uniform distribution over the interface. Additional difficulty is owing to the fact that the thickness of the dielectric layer may affect this distribution. As a first approximation, the surface charge density distribution can be assumed to be uniform over the interface (see the electronic supplementary material). For moderate or strong ion injections, the interfacial charge and current density rapidly reach the saturation level and become constant. This regime is the so-called SLCC regime (Mott & Gurney 1964). As the voltage drop mainly occurs across the air gap, the voltage drop in the thin film is assumed to be much smaller than that across the air gap. Therefore, the effect of film thickness on the current density and electric pressure can be neglected. This assumption is particularly valid for thin dielectric liquid films (h≪S), which are exposed to stronger corona discharge at larger electrode separations (Vega & Pérez 2002). Current density, surface charge density and electric field in the presence of uniform electric field are saturated and they can be expressed as (Mott & Gurney 1964; Vega & Pérez 2002): 6.4 6.5 and 6.6 where V is the voltage applied to the corona electrode, S is the electrode gap spacing, Ka and εa are the ion mobility of the positive ions and electrical permittivity of the air, respectively and σf is the electrical conductivity of the dielectric liquid. The electric field enhancement coefficient, α equals 1 for strong uniform ion injection in an infinite film layer (Mott's steady-state SCLC; Mott & Gurney (1964), Vega & Pérez (2002)). The details of the field enhancement coefficient calculation can be found in the electronic supplementary material. Substituting the surface charge density and the electric field expressions into equation (6.4), the dynamic behaviour of the film diameter can be expressed as: 6.7 The spreading ratio can be calculated as: 6.8 According to equation (6.8), the dynamic spreading ratio is not a function of the droplet initial size. This conclusion is consistent with the universal spreading ratio for different droplet sizes presented earlier in figures 6 and 7. The interested reader can find a detailed description of the analytical approach in the electronic supplementary material.
7. Analytical results versus experimental measurements
A typical comparison between the analytical and experimental behaviour of the film diameter, D(t), is presented in figure 10. The predicted trend of the dynamic evolution of the film diameter is very similar to the experimental results. The comparison between the analytical and experimental diameters for the long-term regime shows a maximum 4 per cent difference, while for the short-term the deviation is less than 16 per cent. The analytical overestimation of the film diameter in the short-term range is believed to be owing to the fact that equation (6.7) is sensitive to the model simplifications at the very beginning of the evolution process. These assumptions are for instance, neglection of the surface tension force and dynamic contact angle effects, temporal/convective inertia, spatial variation of surface charge density and electric field and the effect of dielectric film thickness (which is thicker in the early stage of spreading). It should also be mentioned that the electric pressure is slightly overestimated owing to neglecting the voltage drop across the film. This leads to an overestimation of current density, and consequently electric pressure and dynamic spreading, particularly when the dielectric layer is comparatively thicker. Better agreement could be found by applying the Vega's and Pérez's approaches (Vega & Pérez 2002) for current density and interfacial charge estimation at the very beginning of evolution (t<5 s). Considering so many idealizations of the model, the analytical expression shows relatively good agreement with the experimental results and supports the above spreading mechanism hypothesis.
A corona discharge-assisted technique for uniform spreading of a dielectric droplet is reported. The mechanism of spreading process was investigated through several qualitative and quantitative experiments. It was found that the spreading process is governed by the interfacial charge accumulation resulting from the imposed corona discharge. We also demonstrated that the current problem is analogous to the classical Stefan's lubrication theory for squeezing liquid flow between parallel discs. Using this analogy, the dynamic spreading ratio was correlated to the electrical pressure and the electrical/thermophysical properties of the fluids. The analytical model showed good agreement with the experimental results especially for long-term film expansion and supports the postulated spreading mechanism.
- Received April 5, 2011.
- Accepted June 1, 2011.
- This journal is © 2011 The Royal Society