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Rose-window instability in low conducting liquids
Journal ot
ELECTROSTATICS ELSEVIER
Journal of Electrostatics 40&41 (1997) 141-146
Rose-window instability in low conducting liquids A. T. P6rez Departamento de Electr6niea y Electromagnetismo. Facultad de Ffsica. Avenida Reina Mercedes sin. ~I012 Sevilla, 8pain
Rose-window instability appears when corona discharge from a needle is applied over a thin layer of low conducting liquid. When the tip is at high voltage a corona current passes through the liquid and surface deformations appears due to the destabilization of the liquid surface. This instability gives place to a pattern, referred to as rose-window, whose cells are six to ten times greater than the layer thickness. 1
INTRODUCTION
Rose-window instability appears when a layer of low conducting liquid is subjected to an electric current. This current produces an accumulation of electric charge in the vicinity of the liquid surface and the electric field acts onto the accumulated charge. Above a certain threshold the surface deforms and a pattern appears with a typical length scale five to six times larger than the layer depth. If the current source is corona discharge from a needle, only a circular region of the surface receives enough current to be unstable, the outer part of the liquid remains undeformed. The liquid surface has the appearance of a rose-window in a Gothic Cathedral (see Figure 1). The rose-window instability may or may not be coexistent with the Electrohydrodynamic (EHD) instability due to unipolar injection, which is a well known phenomenon [1,2] induced in this case by corona discharge [31. The ions injected at the point accumulate near the liquid surface, because their mobility is several orders of magnitude smaller in liquid than in air. This is equivalent to the injection of ions achieved by other processes and leads to a space charge distribution in the liquid volume. The situation This work was carried out with financial support from DGICYT (Spanish Government Agency) under contract PB93-1182. 0304-3886/97/$17.00 © Elsevier Science B.V. All fights reserved.
PII S0304-3886(97)00028-4
142
A.T. P$rez/Journal of Electrostatics 40&41 (1997) 141-146
liql
electrode
Figure 1. Rose-window instability in silicone oil. The circular frame is 5 cm in diameter, and the liquid layer depth about 1 mm.
Figure 2. Experimental Set up.
is that of the so called Space Charge Limited Current (SCLC), i. e., the charge density is so large that the electric field becomes much smaller near the injecting surface than in the liquid volume. The relevant parameter in the stability of a liquid layer subjected to unipolar injection, for SCLC, is T = EVt/IQy, where e is the dielectric constant, ~ the voltage drop across the liquid layer, Kt the ion mobility in the liquid and y the viscosity. Below a critical value Tc the liquid is at rest and the current density is given by j -- 9KiEVt~/8d 3. When using corona current ~ is controlled through the current density and this is related to the electric field at the tip rather than at the surface. The instability appears for a certain value of jd 3, and therefore is more sensitive to the value of V, the voltage at the point, than to the mean electric field V/L. The aim of this paper is to present some experimental results concerning the onset of rose-window instability, even though the instability of the volume space charge due to unipolar injection may be present. 2
EXPERIMENTAL
SET U P
Figure 2 shows the experimental set up. A point electrode, whose tip has a radius of about 10 #m, is placed above a flat circular electrode of 8 cm diameter. A ring in Plexiglas of 5 cm diameter, thickness and width of 1 mm, is glued to the electrode. The tip is connected to a high voltage source and the plate to an electrometer, so as to measure the current through the point-plate gap for each voltage. We dispose a fixed amount of liquid onto the flat electrode with the help of a pipette. The volume of disposed liquid serves to control the thickness of the layer, denoted by d in what follows.
A. T PErez/Journal of Electrostatics 40&41 (1997) 141-146
143
CoO lens light I source I
lens _-'~/_ Jmem.
l I
"1I photodiode
'ioI~O
~_ 20 4'.0
81o
i (10"7A) Figure 3. Optical system for the detection oi the surface deformation.
Figure 4. Photodiode current versus corona current. The point to plane distance is L = 25.8 mm and the liquid thickness is 1 mm.
This thickness is typically between 0.2 and 1 mm. The distance from the point to the plate is another variable in the set up. We will denote it by L and ranges from 2 to 50 cm. An optical system serves to detect the surface deformation. The light from a mercury vapor lamp is focused onto the surface (see Figure 3). The reflected beam is collected by a photodiode. When the surface is deformed, the beam is deflected and the current through the photodiode changes. Figure 4 plots the photodiode current versus the corona current for a typical situation. The change of slope indicates the instability threshold. The liquid we have used more extensively is silicone oil of viscosity 50 centistokes and conductivity less than 10-13S/m. 3
RESULTS
Figures 5 and 6 show the critical voltages and currents for the onset of the rose-window instability in silicone oil as functions of the point-to-plane distance (L). Both values, I and V, are increasing functions of L, which is in contrast with the case of the instability due to unipolar injection. The latter appears for more or less the same value of the current, quite independently of L. The rose-window instability appears, at a first glance, to occur for a given value of the electric field at the surface (E ,,, V/L). The trend of both figures suggest that the cause of instability is the electric pressure acting on the charge surface. The higher the current, more charged is the surface, and less electric field is needed to exert the same pressure. This effect is clearly visible if we compare the instability for both polarities.
144
A.T. P~rez/Journal of Electrostatics 40&41 (1997) 141-146
Negative polarity produces higher corona current and the voltage required for desestabilize the surface is smaller. Figures 7 and 8 show critical voltages and currents as a function of the the liquid layer thickness. Both of t h e m are decreasing functions of d. This behavior can be qualitatively understood analyzing the electric pressure exerted onto the undeformed surface. From Equation (14) in [4] the following relation holds for SCLC and d/L < < 1:
j 1+
(L)
= -~IQeo---~
(1)
where Ka is the ion mobility in air. The electric pressure exerted on the liquid is d
j
Pe = f q(z)E(z) dz = --~td
(2)
0
and using (1):
V2 __ 8KzPeL2 L ( ~ Kaeo _d 3 9Kaeo d 1+ Kte (L)
(3)
From Equation (3) V is a decreasing function of d, for a constant pressure, if 2 > (Kaeo/KtE)l/2(d/L) 3/2, which is the case in our experiment. Also from Equation (2) is j ,~ 1/d, for a constant pressure. In fact, the slope in Figure 8 is close to - 1 , as expected. We have observed the rose-window instability in other liquids like castor oil and corn oil, with conductivities two orders of magnitude higher, but it is not observable for more conducting liquids like water or glycerin. 4
DISCUSSION
Two first attempts to explain the rose-window instability can be found in [5] and [4]. In [5] we used the Melcher's relation [6] for the instability of liquid surface subjected to an electric field. From it, the instability threshold is Pe --- 2 Pv/-~-q "" 20N/m 2, where Pe is the electrical pressure onto the surface and 7 the surface tension. The electrical pressure is estimated as asV/L, where as is the charge per unit area in the liquid layer as = fod q(z)dz, being q(z) the space charge density. In the simple case of plane to plane geometry the calculations give as = (2jed/Kl) 1/2 and from
A.T. P$rez l Journal of Electrostatics 40&41 (1997) 141-146
8
145
PointnegaUve] • I-
7
Point i ~ a t i v e
10
+ /
<8 ~-~6j
++."
4
o
•
• 04"4"*
~'~'~'~o
•
L(mm)
L(mm) Figure 5. Critical voltage versus point-to-plane distance for silicone oil I m m depth.
Figure 6. Critical current versus point-to-plane distance for silicone oil I m m depth. 100
•
•
~nt~
÷
~ n t ~
Point negative
Pointpositive
,4-
<1o 0
4.
0
4. •
'4"
4"
4.
........ l '0:5'
1:0'
1:5 ' 2'.0 ' 2:5
d(mm) Figure 7. Critical voltage versus liquid layer thickness. (+) L = 28.2 mm, (o) L = 40 mm. it the instability is expected for
d(mm) Figure 8. Critical current versus liquid layer thickness. (+) L = 28.2 mm, (o) L = 40 mm.
v~-d V -- 4.7 (SI units)
(4)
Our measurements (Figures 5 to 8) give Vr-~V/L = 4 4- 1 and the expression describe qualitatively the trend of the four curves. Atten and Koulova [4] have obtained another criterion on a more rigorous base. They analyzed the instability of the liquid layer calculating the electric pressure for a deformation of the surface in the long wave approximation also in plane to plane geometry. Their conclusion is that the instability threshold is j ,,~ Klpg. This dependence is compatible with the observed behavior in V and I, since V is related to I through d and L. But the dependence on the geometrical factors d and L obtained by these
146
A.T. Pjrez/Journal of Electrostatics 40&41 (1997) 141-146
authors is more intricate and our set of measurements is not fine enough to be conclusive. In conclusion it seems clear that rose-window instability is related to the fact that the electric field pushes the charged liquid surface. The fingerprint of this instability is that, just above the onset, the size of the cells is a few times the liquid depth. The electric field at the liquid surface plays an important role, and the instability is easy to observe if the distance pointto-plane is small. In addition, the liquid has to be low conducting, in order to support a charge of the same sign than the corona current. References
[1] P.K. Watson, J. M. Schneider and H. R. Till, The Physics of Fluids 13, 1955 (1970). [2] P. Atten and R. Moreau, Journal de M~canique 11,471 (1972). [3] B. Malraison and P. Atten, J. Phys. III (France) 1, 1243 (1991). [4] P. Atten and D. Koulova-Nenova, in 12th ICDL (96CH35981, Roma, Italy, 1996), pp. 476-479. [5] A. T. P~rez, in 12th ICDL (96CH35981, Roma, Italy, 1996), pp. 126129. [6] J. R. Melcher, Field-coupled surface waves (MIT Press, Cambridge, Massachussetts, 1963).
ELECTROSTATICS ELSEVIER
Journal of Electrostatics 40&41 (1997) 141-146
Rose-window instability in low conducting liquids A. T. P6rez Departamento de Electr6niea y Electromagnetismo. Facultad de Ffsica. Avenida Reina Mercedes sin. ~I012 Sevilla, 8pain
Rose-window instability appears when corona discharge from a needle is applied over a thin layer of low conducting liquid. When the tip is at high voltage a corona current passes through the liquid and surface deformations appears due to the destabilization of the liquid surface. This instability gives place to a pattern, referred to as rose-window, whose cells are six to ten times greater than the layer thickness. 1
INTRODUCTION
Rose-window instability appears when a layer of low conducting liquid is subjected to an electric current. This current produces an accumulation of electric charge in the vicinity of the liquid surface and the electric field acts onto the accumulated charge. Above a certain threshold the surface deforms and a pattern appears with a typical length scale five to six times larger than the layer depth. If the current source is corona discharge from a needle, only a circular region of the surface receives enough current to be unstable, the outer part of the liquid remains undeformed. The liquid surface has the appearance of a rose-window in a Gothic Cathedral (see Figure 1). The rose-window instability may or may not be coexistent with the Electrohydrodynamic (EHD) instability due to unipolar injection, which is a well known phenomenon [1,2] induced in this case by corona discharge [31. The ions injected at the point accumulate near the liquid surface, because their mobility is several orders of magnitude smaller in liquid than in air. This is equivalent to the injection of ions achieved by other processes and leads to a space charge distribution in the liquid volume. The situation This work was carried out with financial support from DGICYT (Spanish Government Agency) under contract PB93-1182. 0304-3886/97/$17.00 © Elsevier Science B.V. All fights reserved.
PII S0304-3886(97)00028-4
142
A.T. P$rez/Journal of Electrostatics 40&41 (1997) 141-146
liql
electrode
Figure 1. Rose-window instability in silicone oil. The circular frame is 5 cm in diameter, and the liquid layer depth about 1 mm.
Figure 2. Experimental Set up.
is that of the so called Space Charge Limited Current (SCLC), i. e., the charge density is so large that the electric field becomes much smaller near the injecting surface than in the liquid volume. The relevant parameter in the stability of a liquid layer subjected to unipolar injection, for SCLC, is T = EVt/IQy, where e is the dielectric constant, ~ the voltage drop across the liquid layer, Kt the ion mobility in the liquid and y the viscosity. Below a critical value Tc the liquid is at rest and the current density is given by j -- 9KiEVt~/8d 3. When using corona current ~ is controlled through the current density and this is related to the electric field at the tip rather than at the surface. The instability appears for a certain value of jd 3, and therefore is more sensitive to the value of V, the voltage at the point, than to the mean electric field V/L. The aim of this paper is to present some experimental results concerning the onset of rose-window instability, even though the instability of the volume space charge due to unipolar injection may be present. 2
EXPERIMENTAL
SET U P
Figure 2 shows the experimental set up. A point electrode, whose tip has a radius of about 10 #m, is placed above a flat circular electrode of 8 cm diameter. A ring in Plexiglas of 5 cm diameter, thickness and width of 1 mm, is glued to the electrode. The tip is connected to a high voltage source and the plate to an electrometer, so as to measure the current through the point-plate gap for each voltage. We dispose a fixed amount of liquid onto the flat electrode with the help of a pipette. The volume of disposed liquid serves to control the thickness of the layer, denoted by d in what follows.
A. T PErez/Journal of Electrostatics 40&41 (1997) 141-146
143
CoO lens light I source I
lens _-'~/_ Jmem.
l I
"1I photodiode
'ioI~O
~_ 20 4'.0
81o
i (10"7A) Figure 3. Optical system for the detection oi the surface deformation.
Figure 4. Photodiode current versus corona current. The point to plane distance is L = 25.8 mm and the liquid thickness is 1 mm.
This thickness is typically between 0.2 and 1 mm. The distance from the point to the plate is another variable in the set up. We will denote it by L and ranges from 2 to 50 cm. An optical system serves to detect the surface deformation. The light from a mercury vapor lamp is focused onto the surface (see Figure 3). The reflected beam is collected by a photodiode. When the surface is deformed, the beam is deflected and the current through the photodiode changes. Figure 4 plots the photodiode current versus the corona current for a typical situation. The change of slope indicates the instability threshold. The liquid we have used more extensively is silicone oil of viscosity 50 centistokes and conductivity less than 10-13S/m. 3
RESULTS
Figures 5 and 6 show the critical voltages and currents for the onset of the rose-window instability in silicone oil as functions of the point-to-plane distance (L). Both values, I and V, are increasing functions of L, which is in contrast with the case of the instability due to unipolar injection. The latter appears for more or less the same value of the current, quite independently of L. The rose-window instability appears, at a first glance, to occur for a given value of the electric field at the surface (E ,,, V/L). The trend of both figures suggest that the cause of instability is the electric pressure acting on the charge surface. The higher the current, more charged is the surface, and less electric field is needed to exert the same pressure. This effect is clearly visible if we compare the instability for both polarities.
144
A.T. P~rez/Journal of Electrostatics 40&41 (1997) 141-146
Negative polarity produces higher corona current and the voltage required for desestabilize the surface is smaller. Figures 7 and 8 show critical voltages and currents as a function of the the liquid layer thickness. Both of t h e m are decreasing functions of d. This behavior can be qualitatively understood analyzing the electric pressure exerted onto the undeformed surface. From Equation (14) in [4] the following relation holds for SCLC and d/L < < 1:
j 1+
(L)
= -~IQeo---~
(1)
where Ka is the ion mobility in air. The electric pressure exerted on the liquid is d
j
Pe = f q(z)E(z) dz = --~td
(2)
0
and using (1):
V2 __ 8KzPeL2 L ( ~ Kaeo _d 3 9Kaeo d 1+ Kte (L)
(3)
From Equation (3) V is a decreasing function of d, for a constant pressure, if 2 > (Kaeo/KtE)l/2(d/L) 3/2, which is the case in our experiment. Also from Equation (2) is j ,~ 1/d, for a constant pressure. In fact, the slope in Figure 8 is close to - 1 , as expected. We have observed the rose-window instability in other liquids like castor oil and corn oil, with conductivities two orders of magnitude higher, but it is not observable for more conducting liquids like water or glycerin. 4
DISCUSSION
Two first attempts to explain the rose-window instability can be found in [5] and [4]. In [5] we used the Melcher's relation [6] for the instability of liquid surface subjected to an electric field. From it, the instability threshold is Pe --- 2 Pv/-~-q "" 20N/m 2, where Pe is the electrical pressure onto the surface and 7 the surface tension. The electrical pressure is estimated as asV/L, where as is the charge per unit area in the liquid layer as = fod q(z)dz, being q(z) the space charge density. In the simple case of plane to plane geometry the calculations give as = (2jed/Kl) 1/2 and from
A.T. P$rez l Journal of Electrostatics 40&41 (1997) 141-146
8
145
PointnegaUve] • I-
7
Point i ~ a t i v e
10
+ /
<8 ~-~6j
++."
4
o
•
• 04"4"*
~'~'~'~o
•
L(mm)
L(mm) Figure 5. Critical voltage versus point-to-plane distance for silicone oil I m m depth.
Figure 6. Critical current versus point-to-plane distance for silicone oil I m m depth. 100
•
•
~nt~
÷
~ n t ~
Point negative
Pointpositive
,4-
<1o 0
4.
0
4. •
'4"
4"
4.
........ l '0:5'
1:0'
1:5 ' 2'.0 ' 2:5
d(mm) Figure 7. Critical voltage versus liquid layer thickness. (+) L = 28.2 mm, (o) L = 40 mm. it the instability is expected for
d(mm) Figure 8. Critical current versus liquid layer thickness. (+) L = 28.2 mm, (o) L = 40 mm.
v~-d V -- 4.7 (SI units)
(4)
Our measurements (Figures 5 to 8) give Vr-~V/L = 4 4- 1 and the expression describe qualitatively the trend of the four curves. Atten and Koulova [4] have obtained another criterion on a more rigorous base. They analyzed the instability of the liquid layer calculating the electric pressure for a deformation of the surface in the long wave approximation also in plane to plane geometry. Their conclusion is that the instability threshold is j ,,~ Klpg. This dependence is compatible with the observed behavior in V and I, since V is related to I through d and L. But the dependence on the geometrical factors d and L obtained by these
146
A.T. Pjrez/Journal of Electrostatics 40&41 (1997) 141-146
authors is more intricate and our set of measurements is not fine enough to be conclusive. In conclusion it seems clear that rose-window instability is related to the fact that the electric field pushes the charged liquid surface. The fingerprint of this instability is that, just above the onset, the size of the cells is a few times the liquid depth. The electric field at the liquid surface plays an important role, and the instability is easy to observe if the distance pointto-plane is small. In addition, the liquid has to be low conducting, in order to support a charge of the same sign than the corona current. References
[1] P.K. Watson, J. M. Schneider and H. R. Till, The Physics of Fluids 13, 1955 (1970). [2] P. Atten and R. Moreau, Journal de M~canique 11,471 (1972). [3] B. Malraison and P. Atten, J. Phys. III (France) 1, 1243 (1991). [4] P. Atten and D. Koulova-Nenova, in 12th ICDL (96CH35981, Roma, Italy, 1996), pp. 476-479. [5] A. T. P~rez, in 12th ICDL (96CH35981, Roma, Italy, 1996), pp. 126129. [6] J. R. Melcher, Field-coupled surface waves (MIT Press, Cambridge, Massachussetts, 1963).