Geometric properties of the patterns observed in low conducting liquids under corona discharge

Journal of Electrostatics 66 (2008) 178–183 Geometric properties of the patterns observed in low conducting liquids under corona discharge Francisco Vega Reyes a,b, , Francisco J. Garcı´a a,c a Departamento de Electro ´nica y Electromagnetismo, Facultad de Fı ´sica, Avda. Reina Mercedes s/n, 41012 Sevilla, Spain b Fı ´sica Teo ´rica, Universidad de Extremadura, Avda. de Elvas s/n, 06071 Badajoz, Spain c Departamento de Fı ´sica Aplicada I, E.U.I.T.A., Universidad de Sevilla, Carretera de Utrera, km. 1, 41013 Sevilla, Spain Received 30 June 2007; received in revised form 3 December 2007; accepted 9 December 2007 Available online 17 January 2008 Abstract We describe an experimental investigation of the different patterns that appear in thin liquid layers subjected to strong electric fields and a corona discharge. The study is motivated by recent observations [F. Vega Reyes, F.J. Garcı´a, Pattern imaging of primary and secondary instabilities, J. Fluid Mech. 549 (2006) 61–69] of important disparities that appear in the behaviour of these patterns in liquids with different properties. Two main groups of long-wave hydrodynamic instabilities were detected in this system: primary and secondary instabilities. We present the first statistics of the geometric forms of the patterns cells, as a function of the properties of liquids. We also show that the primary instability may be divided into two subgroups, having different pattern characteristic geometries. r2007 Elsevier B.V. All rights reserved. Keywords: Electrohydrodynamic; Instability; Primary; Secondary; Geometry; Patterns 1. Introduction The study of electrohydrodynamic (EHD) instabilities has generated many research works, in part because EHDs has many industrial applications, such as spraying, ink jet printing, pumping, electrostatic lithography, etc. [1–3]. More specifically, over the past few decades, several studies have dealt with the patterns of EHD instabilities observed in experiments with low conducting liquid layers under corona discharge [1,4–6]. Two characteristic length scales were observed in the patterns found in experiments with silicone and castor oils. The first pattern was observed only in silicone oil, having its cells a size of 101times the liquid layer thickness (i.e., a short wavelength instability). This type of instability (usually called ‘‘classical EHD instability’’ in unipolar injection problems) has been extensively studied in former experimental and theoretical works [7]. The other pattern length scale is larger (on the order of the liquid layer thickness), and it does not correspond to the so-called classical EHD instability. The long-wave patterns were observed in both silicone and castor oils. These were all called generically ‘‘rose-window’’. In a recent work [1], the use of a variety of additional liquids allowed us to show that long-wave patterns should be classified as two different types of EHD instabilities. In the first group, the long-wave instability is the only instability that eventually appears in the system (for this reason is called primary instability [1]), and in the second group the long-wave instability occurs only after the short wavelength instability, and destroying its pattern (secondary instability [1]). Furthermore, the diversity of the patterns among the different liquids was more complex than expected according to the initial observations. For example, we detected significant differences in the geometry of the primary instability cells depending on the liquid viscosity and the intensity of the electric field. In this work, we present a more detailed study of the geometric properties of the patterns observed in our previous work [1]. We expect to confirm and refine the classification of the instabilities. We also seek to determine a common behaviour in the primary instability thresholds, ARTICLE IN PRESS www.elsevier.com/locate/elstat 0304-3886/$ - see front matter r2007 Elsevier B.V. All rights reserved. doi:10.1016/j.elstat.2007.12.001  Corresponding author at: Fı´sica Teo ´rica, Universidad de Extremadura, Avda. de Elvas s/n, 06071 Badajoz, Spain. Tel.: +34 924 289540; fax: +34 924 289651. E-mail address: [email protected] (F. Vega Reyes). and to look for a first experimental evidence of an electricpressure driven instability mechanism, as predicted in our former theoretical work [8]. The ‘‘rose-window’’ patterns have remained theoretically unexplained for a long time. Although we deal with a standard academic problem, that of a hydrodynamic instability in a liquid layer, caused by an electrostatic action, no theoretical input of the problem had been attempted for many years. The difficulty is in the solutions of electric fields in the presence of space charge sources [7], which are much less known and less generally used in theoretical research in the field of hydrodynamics, in contrast with the case of electric fields generated without space charge sources. Therefore, our experimental study provides input in a basic problem, which also has technical applications [1], and on which little is known at a theoretical level. 2. Experimental set-up and pattern imaging The typical experimental set-up consists of a thin metallic point above a circular, plane, transparent electrode supporting a thin liquid film (1 mm thick). The distance h between both electrodes is typically of 4 cm. The metallic point is made of tungsten and has a tip radius of about 10 mm and the transparent circular electrode (4.5 cm diameter) is made of glass coated with a thin conducting film (ITO). A corona discharge is produced by a large electric potential difference between both electrodes, typically around 3.5 kV in our set-up (Fig. 1). If the electric field is further increased, an EHD instability pattern appears in the liquid surface. This pattern occurs at extremely low electric currents (102nA). Images of the instability patterns are taken from beneath the circular transparent electrode. For instability threshold detection, we use an imaging technique that is able to quantify the liquid interface deformation [1]. This is achieved with the aid of SUSAN image filters (from Smallest Univalued Segment Assimilating Nucleus [9]). The SUSAN filters work by counting the number of pixels Npat the optical edges (maxima of light intensity gradient), that in our set-up coincide with the cells borders [1] (in our images the cells borders are much darker than the centres of the cells). Therefore, Npis a measure of instability development (because it measures the total perimeter of all cells in the pattern). A typical experiment results in a series of images, being the images taken for each value of the applied voltage Vp(the voltage is increased 0.1 kV, at constant time intervals of 1.0 min). For low voltage Vp, the liquid surface is plane (no instability) and Npis constant because optical edges are only produced by the border of the circular electrode. When increasing voltage, the first variation of the number of pixels in the image edges is due to instability development and thus indicates the instability threshold voltage Vc. More details on the experimental set-up, the liquid properties, the image processing and instability pictures can be found in [1]. Technical details on the functionalities of SUSAN image filters may be found in [9]. 3. Results Following our former theoretical analysis [10],we consider the parameter C0, which is the ratio of the nonohmic characteristic conduction time over the ohmic charge relaxation time. We find that a primary instability appears for C041 (ohmic regime) and a secondary instability appears for C0o1 (non-ohmic regime). In our preceding experimental work we found this parameter to describe correctly the experimental behaviour of the instabilities [1]. Thus, the long-wave instabilities may be classified into two main groups depending on the whether or not a first and different short wavelength instability appears previously [1]. In previous works [5,6] only hexagonal-like cells were observed in rose-window patterns. In our last work, however, [1], with the use of liquids having property values in different ranges of magnitude, we observed a much more complex scenario. Specifically, we detected significant numbers of non-hexagonal cells (with other regular or irregular forms). We also observed a great variety of behaviours inside each of these two main groups, notably in the primary instability. In order to obtain a more detailed and verified classification of the instabilities, we compared the properties of the patterns in the primary and secondary instabilities. Thus, we have tracked the evolution of the number of cells Nas a function of the applied voltage Vp, from values where the instability is incipient up to values where the instability becomes well developed. These measurements provide information about the evolution of the characteristic wavelength of the instabilities, wherein the wavelength decreases as the number of cells increases. We are particulary interested in comparing the behaviour of the number of cells in the primary and secondary instabilities. We present measurements of Nin Fig. 2 for two liquids ARTICLE IN PRESS metallic point (injecting electrode) liquid ions black panel plexiglas ammeter camera air V A transparent electrode p Fig. 1. Experimental set-up. F. Vega Reyes, F.J. Garcı ´a / Journal of Electrostatics 66 (2008) 178–183 179 having similar viscosity (50 cSt), specifically, silicone oil (secondary instability, s¼1:31013 O1m1) and a sunflower oil (primary instability, s¼6:41011 O1m1). The difference between the primary and secondary instabilities is very apparent. In the secondary instability the number of cells does not vary significatively when we vary the liquid volume (i.e., the layer thickness). Conversely, we observed a dramatic change in the behaviour of N, as the liquid volume is changed, in the sunflower oil primary instability. As shown in Fig. 2(b), for sunflower oil layers of 1.5 and 2.0 ml, the number of cells is very small compared to the silicone oil secondary instability but for 1.0 ml there is a steep increase in N, up to values similar to those in the silicon oil secondary instability. It is thus reasonable to think there is a transition from the behaviour of the primary instability to that of the secondary instability when the liquid layer becomes thin enough, because C0decreases if layer thickness is decreased [10]. Let us now analyse the number of cells classified by their polygonal forms. Fig. 3 presents the results for silicone oil at 1.5 and 2.0 ml of liquid volume and Fig. 4 presents the results for sunflower oil, 1.0 and 1.5 ml of liquid volume. For silicone oil, we do not find significative differences between the graphs for both volumes. The hexagonally shaped cells prevail very strongly over the rest of possibilities, even for the highest values of Vp. (Note: for silicone oil, we could not decrease the liquid volume to 1.0 ml because the long-wave instability in this liquid (non-ohmic regime) disappears below a critical liquid layer thickness [10].) Fig. 4 (sunflower oil, primary instability) shows that the steep transition to secondary-instability-like behaviour for the thinner liquid layer involves also the number of cells for each type of polygonal form. In effect, as shown in Fig. 4, the polygonal forms switch from predominantly irregular in the 1.5 ml volume liquid layer (and thicker) to predominantly hexagonal in the thinner liquid layer (a volume of 1.0 ml). We also shown in Fig. 5 (viscous silicone oils with volume of 2.0 ml) that in primary instability, there is a ARTICLE IN PRESS 0 20 40 60 80 100 120 5678910 Vp (kV) 1.0 ml 1.5 ml 2.0 ml 0 20 N (cells)N (cells) 40 60 80 100 5678910 Vp (kV) 1.5 ml 2.0 ml Fig. 2. Total number of cells Nas a function of applied voltage Vp. Tip- to-plane distance h¼4:0 cm for different liquid volumes. In all figures 1.0 ml of liquid volume corresponds to a liquid layer 0.63 mm thick (and, for increasing liquid volume, thickness increases proportionally). (a) Silicone oil (n¼50 cSt). (b) Sunflower oil (n¼52:4 cSt). 0 10 20 30 N (cells)N (cells) 40 50 60 70 80 5678910 Vp (kV) Pentagons Hexagons 0 10 20 30 40 50 60 5678910 Vp (kV) Squared Pentagons Hexagons Fig. 3. Number of cells in a secondary instability (silicone oil, n¼50 cSt) classified by type as a function of applied voltage. Tip-to-plane distance h¼4:0 cm. (a) Liquid volume of 1.5 ml. (b) Liquid volume 2.0 ml. F. Vega Reyes, F.J. Garcı ´a / Journal of Electrostatics 66 (2008) 178–183180 significative number of hexagonal cells in the pattern, although coexisting now with other shapes, also if C041 decreases approaching to 1 due to a decrease in viscosity instead of layer thickness (C0decreases also if viscosity decreases [1,10]). In Fig. 6 we show images of this transition in the pattern geometry for a primary instability, from a regular hexagonal-like (C0\1) pattern to an irregular pattern (C0b1). These observations suggest the possibility that the instability in sunflower oil has already become secondary for C0\1. This would cast doubt on the C0-based classification of the instabilities, and it seems plausible that we might have not noticed the first short wavelength instability. (The interface deformation due to the classical EHD instability is very weak and it would have a very small wavelength, around 0.05 mm for this liquid volume [7]). To clarify this point, we analyse another characteristic behaviour of the primary instability. Following a theoretical analysis of linear instability, the voltage threshold Vc for primary instability should increase if the distance between both electrodes increases [8].Fig. 7 shows the instability threshold values in experiments with several liquids, including sunflower oil with a volume of 1.0 ml. Here we see that for primary instability the instability threshold voltage Vcis always increasing, showing the primary instability typical behaviour with tip-to-plane distance h. (The secondary instability may behave in an opposite way, showing ranges of decreasing Vcwith increasing h[10]). It is important to remark that secondary-instability-like behaviour with C041 occurs in all cases only for values of Vpconsiderably higher than instability threshold Vc (see Figs. 4(a) and 5(b)). Furthermore, in the threshold the interface deformation is always irregular for C041. In the primary instability threshold, we found no closed cells for any of the volume values; i.e., the primary instability critical values have still an infinite wavelength, as predicted by previous theoretical work on linear EHD instabilities [8]. Thus, we may confirm that the instability for all liquids with C041 appears in the threshold as a primary instability, as expected [1]. We have also observed the transitions in the pattern geometry for primary instability ARTICLE IN PRESS 0 20 40 N (cells)N (cells) 60 80 100 5678910 Irregular Squared Pentagons Hexagons 0 5 10 15 20 25 30 35 5678910 Vp (kV) Vp (kV) Irregular Squared Pentagons Hexagons Fig. 4. Number of cells in a primary instability (sunflower oil) classified by type as a function of applied voltage. Tip-to-plane distance h¼4:0 cm. (a) Liquid volume 1.0 ml. (b) Liquid volume 1.5 ml. Fig. 5. Number of cells in the (viscous) silicone oils primary instability classified by type as a function of applied voltage. Tip-to-plane distance h¼4:0 cm, liquid volume vol:¼2:0 ml. (a) Silicone oil with viscosity n¼104cSt. (b) Silicone oil with n¼103cSt. F. Vega Reyes, F.J. Garcı ´a / Journal of Electrostatics 66 (2008) 178–183 181 behave analogously in other liquids where the primary instability is present. Examples include olive, corn oils and ohmic isopars (the same liquids used in our previous work [1]). 4. Conclusion We have presented a series of measurements of the number of cells in the long-wave instability patterns that appear in low conducting liquids under corona discharge, and we have grouped them by geometric shape. In the case of secondary instability, the cells are always predominantly hexagonal for all liquids and degree of instability development (i.e., near or far from the instability threshold). In primary instability, we have detected two clearly distinct subgroups (see Fig. 6). One subgroup applies to liquids near the transition region to secondary instability (1oC0o100). In these liquids the non-linear instability has secondary-like behaviour, i.e., the total number of cells (i.e., instability characteristic wavelength) remains of the same order for varying voltage, and the cells are always predominantly hexagonal. The second subgroup of primary instability occurs in liquids far from the transition region to secondary instability (C04100), where the behaviour of the non-linear instability is very different to that of the secondary instability. Specifically, there are a great variety of polygonal forms in the patterns which eventually turn predominantly irregular and chaotic. We have also confirmed that, in effect, all instabilities with C041, regardless of subgroup, are primary, by analysing the behaviour of the instability threshold voltages as a function of the interelectrode distance. We complete and improve the classification depicted in our former work [1]. These new results provide useful insight for future studies on EHD instabilities in systems under corona discharge, for which in most cases there is still an important lack of experimental data and theoretical background. Acknowledgements The authors acknowledge financial support from the Spanish Ministerio de Ciencia y Tecnologı´a through contracts BFM-2003-01739 and also (only F.V.R.) FIS2004-01399 and through ‘‘Juan de la Cierva’’ research program. References [1] F. Vega Reyes, F.J. Garcı ´a, Pattern imaging of primary and secondary instabilities, J. Fluid Mech. 549 (2006) 61–69. [2] J.R. Melcher, G.I. Taylor, Electrohydrodynamics: a review of the role of interfacial shear stresses, Annu. Rev. Fluid Mech. 1 (1969) 111–146. [3] A. Castellanos (Ed.), Electrohydrodynamics, Springer, Vien, New York, 1998. [4] C.S. Herrick, Electroconvection cells in dielectric liquids interfaced with conducting liquids, Proc. R. Soc. London A 336 (1974) 487–494. [5] A.T. Ahmed El-Haddad, J. Fornaze ´ro, G. Me ´snard, Me ´canismes de base de la de ´formation cellulaire d’une couche mince isolante ARTICLE IN PRESS Fig. 6. Images of the primary instability subgroups transition (liquid: sunflower oil , Vp¼9:0 kV, h¼4:0 cm), from the hexagonal pattern with smaller cells in a thin layer ðvol:¼1:0mlÞ, in (a), to the irregular pattern with larger cells in a thicker layer ðvol:¼2:0mlÞ, in (b). 2.5 3 3.5 4 4.5 5 5.5 6 h (cm) 4 5 6 7 8 Vc (kV) Fig. 7. Long-wave instability thresholds versus interelectrode distance h for sunflower oil and the series of silicone oils. Sunflower oil (x): vol:¼1:0 ml. In the silicone oil series ðvol:¼2:0mlÞ. Viscosities: 50 cSt (m), 103cSt (n), 104cSt (&). In all liquids, the instability shows the predicted behaviour for a primary instability except for silicone oil 50 cSt (m), where the instability is secondary, as expected. F. Vega Reyes, F.J. Garcı ´a / Journal of Electrostatics 66 (2008) 178–183182 liquide soumise a `un flux de charges, Phys. Chem. 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