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Electrical charging of a conducting water droplet in a dielectric fluid on the electrode surface
Journal of Colloid and Interface Science 322 (2008) 617–623
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Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Electrical charging of a conducting water droplet in a dielectric fluid on the electrode surface Yong-Mi Jung, Hyun-Chang Oh, In Seok Kang ∗ Department of Chemical Engineering, Pohang University of Science and Technology San31, Hyoja-dong, Pohang 790-784, South Korea
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 15 November 2007 Accepted 7 April 2008 Available online 11 April 2008
It has been conceived that a charged droplet driven by Coulombic force can be used as a droplet-based microreactor. As a basic research for such applications, electrical charging of a conducting water droplet is studied experimentally. The effects of electric field, medium viscosity, and droplet size are investigated. It is found that the amount of electrical charging increases with the droplet size and the electric field. However, the medium viscosity does not have a significant effect in the range of the present study. A scaling law is derived from the experimental results. Unlike the case of a perfect conductor, the estimated amount of electrical charge (Q est ) of a water droplet is proportional to the 1.59 power of the droplet radius (R) and the 1.33 power of the electric field strength (E). (For a spherical perfect conductor, Q is proportional to R 2 and E.) In order to understand these differences, numerical simulations are performed for the idealized droplets of perfect conductor. Comparison of the numerical and experimental results suggests that the differences are mainly due to incomplete charging of a water droplet resulted from the combined effect of electrochemical reaction at electrode and the relatively low conductivity of water. © 2008 Elsevier Inc. All rights reserved.
Keywords: Charging Coulombic force Dielectric fluid Charged conducting droplet
1. Introduction When a charged conducting droplet is suspended in an immiscible fluid under an external electric field, it can move quickly toward the electrode with opposite charges due to Coulombic force. This behavior of a charged droplet attracts much attention owing to its potential applications in various fields such as the bio-MEMS [1,2]. Recently, a droplet has also been considered as a candidate for a “micro-reactor” in microfabricated devices. Electrical manipulation of a charged droplet may be used in the design of such systems. Related to the behaviors of a conducting droplet under electric field, we have essentially two issues. The first is how much charge a droplet can acquire at the electrode. The other is how the droplet deforms and moves inside the fluid. The second issue has been studied extensively both experimentally and theoretically [3–5]. However, the first issue of charging process of a fluid particle has hardly been considered. This fact provides the major motivation of the present work. Regarding the problem of charging of a particle at electrode, studies have mostly been limited to the solid particle cases. Charging of a micron-sized metal particle at the electrode was studied experimentally by Cho [6]. In the paper, two mechanisms of charging, the contact potential charging and the induction charging, are
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Corresponding author. Fax: +82 54 279 5528. E-mail address: [email protected] (I.S. Kang).
0021-9797/$ – see front matter doi:10.1016/j.jcis.2008.04.019
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considered. The experimental results show that the contact potential charging may be neglected when the particle size is larger than one micron. Also shown is the theoretical result for the amount of induced charge Q = 1.65 × 4πε R 2 E 0 ,
(1)
where R is the radius of the particle and E 0 is the electric field between the charged plates. The theoretical result can be derived for a perfectly conducting sphere contacting the electrode and the exact value of the coefficient 1.65 is π 2 /6 (see also [7]). Since Cho’s work, there have been several related works. Khayari and Perez [7] used Eq. (1) to estimate how much charge is leaked out during the motion of a charged particle inside the surrounding fluid. They observed the vertical motion of a spherical metallic ball bouncing between two horizontal plates. When the applied voltage is high enough for the electric force to overcome gravity, the ball rises and moves up through the liquid. Then the ball falls down as its charge leaks away. They extended their experiment to the drop problem in which a drop is attached to the bottom electrode with a certain contact area before the liftoff [8]. They considered the effects of adhesion force and deformation of a drop on the liftoff electric field strength. But they did not pay attention to the charging process of a drop at the electrode. Very recently, Hase et al. performed the experiments by using two electrodes in the horizontal configuration [9]. They observed rhythmic motion of a small water droplet under a DC electric field. They found that there are three distinct modes of oscillation: stationary state, small oscillation over some part of the electrode gap,
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Fig. 1. The schematic view of the experimental setup. The cell sketch is not to scale.
and large oscillation over the whole gap. Based on the experimental results, they prepared a state diagram that shows the three modes. To understand the phenomena, they solved an equation of motion for a droplet by using the charge leakage model. As the initial charge condition, they used the value estimated by Eq. (1). But again, they have not considered the characteristics of the electrical charging process of a droplet at the electrode. A bit earlier, Eow et al. [10] performed experiments on the oscillatory motion of a conducting drop between electrodes. They considered the cases of quite large drops. Their experimental results show that the measured velocity of a drop is smaller than that predicted by Eq. (1) for a spherical metal particle of the same equivalent radius. Although they have noticed the difference in the moving velocities, they have not studied the difference in the characteristics of the charging process. As seen above, the charging process itself of a conducting droplet at the electrode has not been studied systematically. Therefore, in this paper, we study experimentally the electrical charging process of a micro-sized conducting water droplet immersed in very insulating silicon oil. From the experimental results, we attempt to obtain a quantitative relationship in the form of a scaling law between the amount of charging and the key parameters. For better understanding, we also observe very carefully the behaviors of a droplet near the moment of contact to the electrode surface. A high speed camera is used with 5000 fps. Compared to a spherical perfect conductor, a water droplet manifests a few distinct features such as ionization reaction at electrode, finite (relatively low) conductivity, and the shape deformation. In order to clarify these effects by comparison, numerical simulation is also performed for a perfectly conducting droplet with various predefined shapes. 2. Materials and methods 2.1. Experimental setup and conditions In the experiment, we measure the velocity of a droplet moving between the electrodes. The velocity is converted to the amount of a charge as will be shown later. The schematic diagram of the experimental setup is shown in Fig. 1. A transparent rectangular acryl test cell is placed between a light source and a microscope. To avoid thermal effect, a cold light source is used. The rectangular glass test cell, which can grab electrodes in both walls, is 10 ( L ) × 30 ( W ) × 45 ( H ) mm3 in the internal dimension. Two copper electrodes are connected to a DC voltage power supply (Trek. Inc.) to apply an external electric field. To observe deformation and translation of a droplet, a high speed CCD camera (Photron Fastcam 1024 PCI Model 100 K) at 5000 fps is mounted on the microscope. The velocity is extracted using the image analysis techniques. Single DI (de-ionized) water droplet and silicon oil (KF-96 series produced from Shin-Etsu Silicone) are used as the conducting droplet and the dielectric fluid medium respectively. Because the silicon oil has almost the same density as water, it is assumed that the effect of the gravity is negligible. To inject a micro-sized water droplet in the cell, a micro-pipette (Eppendorf) is used.
The experimental procedure is as follows. We prepare silicon oil in the test cell. Then a DC electric field is applied at both electrodes in the test cell with silicon oil. We put a single water droplet in the test cell using a micropipette. At that time, the experiment should be carried out as quickly as possible after silicon oil is prepared in the test cell. This caution is taken to minimize the absorbed moisture, which changes the electrical conductivity of silicon oil. The caution is taken also to exclude any unexpected property change of silicon oil by the high electric field. To observe the effects of various parameters, the viscosity of medium, the size of droplet, and the applied voltage are changed systematically. We use three kinds of silicon oils (ε = 2.75ε0 , σ 10−13 S/m, ρ = 957.24 kg/m3 , where ε0 = 8.854 × 10−12 F/m) with viscosities of 50, 260, and 1000 cS. The volumes of the deionized water droplets (ε = 77.75ε0 , σw = 1.2 × 10−4 S/m, μw = 8.90 × 10−4 kg/m s, ρw = 997.75 kg/m3 ) are varied with 0.2, 0.4, 0.8, and 1.6 μl. The corresponding drop radii are 363, 457, 576, and 726 μm, respectively. Also, we apply the external voltages of 2, 2.5, 3, and 4 kV across the fixed distance between two electrodes of 10 mm. 2.2. Estimation of amount of charge on the droplet An initially uncharged conducting water droplet suspended in silicon oil starts to move slowly when a DC electric field is applied. At the moment a droplet touches the electrode surface, electrical charging occurs between the electrode and the droplet. Then the droplet, which has the charges of the same sign as the electrode, is repulsed and moves toward the opposite electrode. A droplet, which arrives at the opposite electrode, discharges its own charges and acquires new charges from the electrode. Then the droplet is also repulsed and translated. This process repeats itself. This is the so-called contact charging process. For the amount of electrical charging, direct measurement is most desirable. However, in the present problem, accurate direct measurement is extremely difficult. The contact time is very short (the order of micro seconds) and transferred charge amount is too little (the order of 10−11 C). So, we take an alternative way in which the drop translation velocity is measured to estimate the charge amount. When the droplet size is small enough, the shape is spherical and the inertial effect may be neglected. In the middle region of the two electrodes, it can be assumed that the electric field is uniform and the droplet moves with a constant velocity. We consider now the force balance for the droplet
F = F D + FQ = 0 .
(2)
In the above, the drag force is assumed to be given by the Hadamard–Rybczynski solution [11] FD = 4πμUc
with c =
3λ + 2 2(λ + 1)
, λ=
μw μ
(3)
and the electrical force term is given by FD = Q E∞ .
(4)
From the force balance, we have the following formula for estimation of the amount of charge in terms of droplet velocity Q est =
4πμaU c E∞
.
(5)
In the above, Q est is the estimated electrical charge on the droplet, E ∞ is the magnitude of the applied external field, μw is the viscosity of water inside the droplet, μ is the viscosity of the medium, a is the droplet radius, U is the droplet velocity.
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2.3. Assumptions To use the force balance equation (2) (or Eq. (5)), several assumptions should be satisfied. First, the Reynolds number should be much less than unity. The Reynolds number based on the experimental parameters in our system ranges from 0.0002 to 0.1120. Second, we need the assumption that charges do not leak away during the droplet travels between the electrodes [7,9]. If the charge relaxation time of silicon oil is much longer than the droplet-traveling-time during its half cycle, the assumption is reasonable. The charge relaxation time of the silicon oil used in our experiment is estimated to be
τ=
ε 2.75 × 8.854 × 10−12 = ≈ 243.5 s. σ 10−13
(6)
The electrical permittivity and the conductivity of silicon oil are chosen as ε = 2.75ε0 with ε0 = 8.854 × 10−12 F/m and σ 10−13 S/m, respectively. They are referenced from the data of the specification of Shin-Etsu Silicone. In experiments, the traveling time of the water droplet between the electrodes is from 0.144 to 4.664 s. This indicates that the charge leakage is negligible in our experiments. In the present study, we use the fixed values of the physicochemical parameters for the analysis of experimental results. However, it is well known that these parameters, especially the conductivity, may vary easily and the results may be influenced. Thus we have paid special attention to these effects. But no significant change has been observed for the droplet moving velocity during many repeated cycles of motion between the electrodes. Third, deformation of the droplet from spherical shape must be small. As will be shown below, this condition is also satisfied reasonably well. The interfacial tension between the water droplet and the silicon oil (in the case of no electric field) can be estimated by using the formula
√
γAB = γA + γB − 2 γA γB =
√
√
2
γA − γB .
(7)
The surface tension of silicon oil is about 20.5 mN/m from the specification data sheet and that of DI water is generally known as 72 mN/m. From Eq. (7), the interfacial tension between silicon oil and a water droplet is estimated to be about 15.7 mN/m. Now, the electrical Weber number can be estimated. The Weber number is the ratio of the deforming electric force to the restoring interfacial tension force and is defined as We(e) =
ε E 2a , γ
(8)
where a is the characteristic radius of the droplet and γ is the interfacial tension. For the experimental parameters of this study, the Weber number ranges from 0.0225 to 0.1801. This indicates that the drop deformation is small. The interfacial tension may also be estimated by using the shape of a drop deformed in an electric field. Kim et al. [5] showed that, in a uniform electric field with a small Weber number, a conducting drop (charged or uncharged) is deformed into the shape (in dimensionless form) of R (cos θ) = 1 +
3 4
We(e) P 2 (cos θ),
Fig. 2. The sequence of photographs of a droplet moving from the left (positive electrode) to the right (negative electrode).
(9)
where R is the radial coordinate of the drop surface, P 2 is the Legendre polynomial, and the angle is measured from the electric field direction. The pictures of deformed droplets indicate that the value of the interfacial tension estimated by Eq. (7), 15.7 mN/m, is rather small. The value of the interfacial tension estimated by Eq. (9) is about twice as large as the value estimated by Eq. (7). (The pictures of deformed droplets will be given in the next section.) This fact suggests that the drop deformation is quite small because even the largest Weber number is below 0.1.
Fig. 3. The position of droplet from the positive electrode as a function of time.
The fourth assumption is that the wall effect must be negligible, because Eq. (5) is derived for the case of infinite medium. In view of large dimension of the test cell (10 mm (L) × 30 mm (W ) × 45 mm (H )) compared to the droplet size (maximum radius 726 μm), this effect will not be significant. 3. Results 3.1. Translational motion of a droplet between electrodes After contacting an electrode, the droplet is repulsed and translated toward the opposite electrode. Similarly, after touching the opposite electrode, the droplet is also repulsed and moved back to the original electrode. The droplet has this repeated motion under the electric field formed between two electrodes. Fig. 2 is the sequence of pictures of a half-cycle of the time-periodic motion. The droplet near the electrode becomes elongated and makes a tip facing the electrode. This is because a droplet is deformable and the electric field is very asymmetric near the electrode. When the droplet approaches the electrode, the electric field between the droplet and the electrode surface becomes much stronger than that between the droplet and the opposite electrode. The drop tip becomes sharper and sharper and eventually touches the electrode. At that time, the oppositely-signed charges on the droplet are discharged at the electrode and the same-signed charges move to the droplet. This is the process of electrical charging. Fig. 3 shows the droplet position as a function of time for a typical case. The droplet shows a very regular periodic motion between the electrodes and reaches a constant velocity near the center. The constant velocity is used for estimation of the amount of charge on the droplet by using Eq. (5). In Fig. 4, the average velocities of the droplet are shown according to the moving directions. The cases of starting from the positive electrode are compared with the cases of starting from the negative electrode. The velocity of a droplet starting from the
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Fig. 4. The dependence of the average velocity on the moving directions (+: from the positive electrode, −: from the negative electrode).
Fig. 6. The dependence of the estimated charge ( Q est / Q 0 ) on the electric field (E / E 0 ).
Fig. 5. The dependence of the estimated charge ( Q est / Q 0 ) on the droplet radius (R / R 0 ). The subscript “0” stands for the reference state of E 0 = 2 kV/cm, R 0 = 363 μm, medium viscosity 50 cS with the estimated charge Q 0 = 1.03 × 10−11 C. For more convincing representation, the abscissa variable is modified by multiplying the factors in terms of the kinematic viscosity (ν /ν0 ) and the electric field (E / E 0 ). Fig. 7. The graph of the scaling law summarizing the experimental results.
positive electrode is slightly higher than that from the negative electrode. Unfortunately, a detailed molecular level explanation on this phenomenon is not available now. Further studies are definitely needed. 3.2. The estimation of electrical charge acquired at an electrode As mentioned in Section 2, the measured velocity of a droplet is used to estimate the amount of electrical charge acquired at the starting electrode by Eq. (5). In order to use the equation, as discussed earlier, we need to assume that the charge leakage is negligible, the Reynolds number is small enough, the droplet deformation is negligible, and the wall effect is also negligible. 3.3. The scaling law for the amount of electrical charging
convincing way, the x-axis (abscissa) variable is modified by multiplying the factors in terms of other parameters (the kinematic viscosity ratio and the electric field ratio). In a similar way, the effect of electric field is shown in Fig. 6. From Figs. 5 and 6, we can see that the amount of electrical charging increases with the size of the droplet and the applied electric field strength. On the other hand, the viscosity of the dielectric fluid medium is not a key factor in the electrical charging process for the parameters in the present study. (See the very small exponent for the viscosity ratio in the adjustment of the abscissa variables of Figs. 5 and 6.) Summarizing the experimental results, a scaling law for the amount of the electrical charging has been derived as (see Fig. 7) Q
The experimental results for the amount of electrical charging have been summarized in the form of a scaling law. In order to treat the problem in a dimensionless form, a reference state is defined. As the reference state, we choose the case of volume 0.2 μl (radius 363 μm), applied field 2 kV/m, and medium viscosity 50 cS. For this case, the droplet moving velocity is U = 0.95 cm/s and the estimated charge is Q est = 1.03 × 10−11 C. In Fig. 5, the dependency of charging amount on the droplet radius is shown. In order to show the dependency in a more
Q0
=C
R R0
1.59
E E0
1.33 ,
(10)
where the subscript 0 stands for the reference state (R 0 = 363 μm,
ν0 = 50 cS, E 0 = 2 kV/cm, Q 0 = 1.03 × 10−11 C) and C is a constant that depends on the viscosity ratio to the reference medium viscosity. The value of C changes slightly as, for μ/μ0 = 1, C = 1, for μ/μ0 = 5.2, C = 1.0147, and for μ/μ0 = 20, C = 1.045. Since the viscosity effect is very weak, the viscosity effect is treated in the way shown above.
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Fig. 8. The sequence of photographs that shows the motion and deformation of a droplet near the electrode due to attraction and repulsion by the electrode (the experimental condition: droplet volume 1.6 μl (radius R = 726 μm), applied field 3 kV/cm, viscosity of medium 50 cS).
According to Cho [6], the amount of the induction charging of a spherical perfect conductor is proportional to the square of radius and the electric field (see Eq. (1)). The scaling law for the present water droplet is a bit different from the perfect conductor case in the exponents. In the water droplet case, the exponent for the effect of the droplet radius is decreased (2.0 → 1.59), while the exponent for the effect of the electric field is increased (1.0 → 1.33). The explanations on these differences will be given later after looking at the numerical results for the (idealized) perfectly conducting droplets with predefined shapes. 3.4. The deformation of a droplet near the electrode As a droplet approaches the electrode, the droplet is deformed into an elongated shape as shown in Fig. 8. Because the droplet is a good conductor compared to the dielectric medium, the electric field becomes stronger in the gap between the electrode and the droplet surface. The charges are now concentrated near the tip of the droplet facing the electrode, and the droplet is deformed into the shape with a sharper tip [12]. Therefore the maximum curvature tip is formed at the moment of touching the electrode. Fig. 9 shows the ratio of the maximum diameter to the minimum diameter as a function of time when the droplet is attracted toward the electrode and repulsed from the electrode. The first peak occurs at the moment of touching the electrode. A second small peak is also observed. This is because of an overshoot in deformation. At the moment of repulsion, the droplet is pushed outward very strongly. Especially the part of droplet surface facing the electrode experiences stronger push and is deformed inward accordingly. After a certain time, the inwardly deformed surface comes out again due to the action of surface tension. As the drop travels further, the electric field becomes weaker, and the droplet is less deformed. 4. Discussion 4.1. Comparison of experimental and numerical results As mentioned earlier, numerical simulation has been performed for better understanding of the electrical charging process of a
Fig. 9. The degree of deformation represented by the ratio of the maximum diameter to the minimum diameter (for droplet deformation in Fig. 8).
droplet. In computation, we consider the (idealized) perfect conductor droplets with predefined shapes. By comparing the numerical results with the experimental results, we can find some characteristics of charging process of real (nonideal) water droplets at the electrode. For each experimental parameter set, numerical computations are carried out for the shapes predefined in the following three distinct ways: the experimentally observed droplet shape at the moment of touching the electrode (case (i)), the prolate shape that has the same volume and the same major axis length as the experimentally observed droplet shape (case (ii)), and the spherical shapes that has the same volume as the experimentally observed droplet shape (case (iii)). By comparing the experimental results for a nonideal droplet with the numerical results for the counterpart of ideal perfect conductor of the same shape (case (i)), we can see the combined effect of all other reasons than the droplet shape, such as the ionization reaction at the electrode surface and the finite (relatively low) electric conductivity of water. We can also learn several things by comparing the numerical results for the three cases mentioned
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Fig. 10. The details of the computational domain for numerical simulations of the (idealized) perfectly conducting droplets.
Fig. 11. Comparison of the numerical and experimental results for the electric field effect in the cases of droplet volumes 0.2 and 1.6 μl. (2) The numerical results for the shapes obtained from the experimental images, (") the numerical results for the prolate shapes with the same volume and the length as the experimental images, (Q) the numerical results for the spherical shapes with the same volume as the experimental images, (a) the experimental results for the water droplets in the case of medium viscosity 50 cS.
above. Comparison of the results for spherical shape (case (iii)) and the extended prolate shape (case (ii)) gives us the effect of extended droplet shape on the amount of electrical charging. Comparison of the results for the extended prolate shape (case (ii)) and the experimentally observed shape (case (i)) gives us the effect of the sharp tip of droplet at the moment of touching the electrode. For numerical computation, we use the commercial package of COMSOL Multiphysics. We adopt the (r , z)-axisymmetric coordinate system with its origin at the tip of the droplet as shown in Fig. 10. The governing equation is the Laplace equation for the electric potential outside the droplet. At the boundary, typical wellknown conditions are used. After obtaining the solution of the electric potential, the surface charge density σ is computed by εE · nˆ = σ , where E is the electric field outside the droplet and ε is the electrical permittivity of the outside dielectric fluid with ε = 2.75ε0 . Then the amount of electrical charging is computed by integrating the surface charge density over the droplet surface. In Fig. 11, the effects of electric field on the amount of electrical charging are shown. The experimental results are shown with the inverted solid triangles. The numerical results are represented with closed squares (case (i)), closed circles (case (ii)), and closed triangles (case (iii)). If we first pay attention to the numerical results only, we can notice that there is a considerable effect of extended droplet shape. On the other hand, the sharp tip effect is not so
Fig. 12. Comparison of the numerical and experimental results for the effect of droplet radius in the cases of electric fields 2 and 4 kV/cm. (The same symbols are used as in Fig. 11.)
large as long as the major axis lengths are the same. The effect of extended droplet shape is easy to understand because the electric field near the droplet surface facing the opposite electrode gets stronger as the droplet is more extended. A very important observation from the numerical results is that a perfect conductor droplet (case (i) with closed squares) exhibits the scaling law with the exponent “unity” to the electric field strength just like the well-known theoretical result for the perfect conductor sphere (see Eq. (1)). However, the experimental results for the real water droplet (inverted closed triangles) show that the exponent is in the range of 1.32 to 1.35. (The results for the droplet volumes of 0.4 and 0.8 μl are not given here.) This means that the stronger electric field is more favorable to electrical charging compared to the perfect conductor case. Differently from the case of perfect conductor, several distinct mechanisms (such as the electrolysis reaction at the electrode surface, transport of charges in the forms of ions) may be involved for the charging and electrical conduction processes [13]. Therefore, it is not possible to pinpoint one specific reason behind the difference in the exponent of the scaling. Detailed explanation on the exponent difference is certainly beyond the scope of the present paper and further theoretical and experimental studies of deeper level are definitely needed. Nevertheless, some phenomenological and qualitative speculation may be given. Since the ionization reaction at electrode is involved and the electrical conductivity of the real water droplet is finite and relatively low, the rate of surface reaction and the contact time (or effective charge transport time) are important. For both factors, stronger electric field seems to be favorable. The increased charge transport time may be expected as follows. The electric field between the charged droplet and the counter electrode increases indefinitely as the droplet approaches the electrode. Therefore at a certain critical distance the dielectric fluid breaks down and the charges may flow between the electrode and the droplet. Since the amount of electrical charging increases and the droplet tip becomes sharper as the field strength increases, the critical distance of dielectric breakdown increases as the electric field increases. This fact seems to make a partial contribution to the higher exponent of about 1.33 for the actual water droplet case to the exponent unity in the perfect conductor case. The effects of droplet radius are shown in Fig. 12. As in the case of the electric field effects, the numerical results for the (idealized) perfect conductor droplets show a good agreement with the theoretical prediction in the exponent of scaling (Eq. (1)). The exponent found from the numerical results is very close to 2, which is the theoretical prediction. On the other hand, the nonideal water droplet shows a smaller exponent that ranges from 1.56 to 1.59.
Y.-M. Jung et al. / Journal of Colloid and Interface Science 322 (2008) 617–623
(The results for the electric fields of 2.5 and 3.0 kV/cm are not given here.) This may be explained as follows. Less sufficient charging occurs for a larger water droplet, while complete charging occurs for a perfect conductor droplet regardless of the droplet size. As for the electric field effects, this part also needs further theoretical and experimental studies of deeper level. 4.2. Discussion of the effect of the Weber number on the scaling law In our study, the amount of charge carried by a droplet is estimated by using the Hadamard–Rybczynski solution under the assumption that the droplet shape is spherical in the middle region of the electrode cell. However, a droplet is elongated under electric field and the moving velocity is also changed from the value predicted under the sphere assumption. For this matter, Kim et al. [5] derived a formula that can be reduced to the following form when the viscosity of the droplet phase is negligibly small compared to that of the medium phase.
U = U H−R 1 + 0.6W + O ( W 2 ) ,
(11)
where U H−R is the velocity predicted by the Hadamard–Rybczynski solution and W is the electrical Weber number. Therefore, the value of amount of charge estimated by Eq. (5) is a bit overestimated by a factor (1 + 0.6W ) when the Weber number is small. Since the most deformed is the case of the largest drop, let us apply the correction to the case of Fig. 11. If we assume that the interfacial tension is about 30 mN/m, the corresponding Weber number ranges from 0.0225 to 0.09. If we adjust the experimental values according to the formula (11), the slope must be adjusted from 1.35 to about 1.30. But it must be noted that this is the largest adjustment of the slope. Therefore, when we take into account of the effect of deformation of a droplet in the middle region of the electrode cell, the exponent in the scaling law for the electric field effect may be lowered. However, the adjustment is not so large and the exponent is still larger than about 1.3. 5. Summary In this study, electrical charging of a water droplet at the electrode has been studied. To estimate the amount of charge, the moving velocity of a droplet in the middle region of the electrode cell is measured. Under the assumption that the droplet has a spherical shape and follows the Stokes law, the measured velocity is used to estimate the charge amount by using the force balance. The experimental results show that the electric field strength and the droplet size are the important key factors to the electrical charging process. On the other hand, the viscosity of the medium fluid does not have a significant effect in the parameter range of the present study. The experimental results are summarized in the form of a scaling law. In the scaling law, the amount of electrical charging of a water droplet is proportional to the power 1.59 of
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the droplet radius and the power 1.33 of the electric field strength. In the case of a perfect conductor sphere, the charge is proportional to the square of radius (the power 2) and proportional to the electric field (the power 1). In order to understand these fundamental differences, numerical computations have been performed for the (idealized) perfect conductor droplets with predefined shapes (experimentally obtained shapes, prolate shapes, and spherical shapes). The numerical results reveal an important fact. Perfectly conducting droplets (even with the deformed shapes) follow the scaling law in which the charge amount is proportional to the square of the radius and proportional to the electric field just like the well-known theoretical result for a perfect conductor sphere. So, the fundamental differences are due to other reasons such as the ionization reaction at the electrode and the relatively low electrical conductivity of the water droplet. Since several mechanisms are simultaneously involved, further studies of deeper level are definitely needed for thorough understanding. Finally, a comment should be given to the applicability of the results of the present experimental work. It is thought that the electrical charging phenomenon can be used as a tool of transporting a single micro- or nano-liter droplet in a micro-channel without moving the medium fluid. Especially, when a conducting droplet is used as a tool of transporting bio-particles like DNA molecules, they are safe inside the droplet against the strong external electric field. Inside the conducting droplet, there is virtually no electric field. This fact may also find many applications. Acknowledgments This work was supported by the grant R01-2004-000-10838-0 from KOSEF. This work was also supported by the grant from Center for Ultramicrochemical Process Systems (CUPS) sponsored by KOSEF and the grant by BK21 program of Ministry of Education of Korea. The authors greatly acknowledge the financial supports. References [1] M. Washizu, IEEE Trans. Ind. Appl. 34 (1998) 732. [2] S. Katsura, A. Yamaguchi, H. Inami, S. Matsuura, K. Hirano, A. Mizuno, Electrophoresis 22 (2001) 289. [3] G.I. Taylor, Proc. R. Soc. London A 280 (1964) 383–397. [4] J.-W. Ha, S.-M. Yang, J. Fluid Mech. 405 (2000) 131. [5] J.G. Kim, D.J. Im, Y.M. Jung, I.S. Kang, J. Colloid Interface Sci. 310 (2007) 599. [6] A.Y.H. Cho, J. Appl. Phys. 35 (1964) 2561. [7] A. Khayari, A.T. Perez, IEEE Trans. Dielectr. Electr. Inst. 9 (2002) 589. [8] A. Khayari, A.T. Perez, F.J. Garcia, A. Castellanos, in: Proc. Ann. Report Conf. on Electr. Ins. Dielectr. Phenomena, 2003, pp. 682–685. [9] M. Hase, S.N. Watanabe, K. Yoshiokawa, Phys. Rev. E 74 (2006) 046301. [10] J.S. Eow, M. Ghadiri, A. Sharif, Colloids Surf. A Physicochem. Eng. Aspects 225 (2003) 193. [11] L.G. Leal, Advanced Transport Phenomena: Laminar Flow and Convective Transport Processes, Cambridge Univ. Press, Cambridge, 2007. [12] G.I. Taylor, Proc. R. Soc. London A 291 (1966) 159. [13] J.O’M. Bockris, A.K.N. Reddy, Modern Electrochemistry 1: Ionics, second ed., Plenum, New York, 1998.
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Electrical charging of a conducting water droplet in a dielectric fluid on the electrode surface Yong-Mi Jung, Hyun-Chang Oh, In Seok Kang ∗ Department of Chemical Engineering, Pohang University of Science and Technology San31, Hyoja-dong, Pohang 790-784, South Korea
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 15 November 2007 Accepted 7 April 2008 Available online 11 April 2008
It has been conceived that a charged droplet driven by Coulombic force can be used as a droplet-based microreactor. As a basic research for such applications, electrical charging of a conducting water droplet is studied experimentally. The effects of electric field, medium viscosity, and droplet size are investigated. It is found that the amount of electrical charging increases with the droplet size and the electric field. However, the medium viscosity does not have a significant effect in the range of the present study. A scaling law is derived from the experimental results. Unlike the case of a perfect conductor, the estimated amount of electrical charge (Q est ) of a water droplet is proportional to the 1.59 power of the droplet radius (R) and the 1.33 power of the electric field strength (E). (For a spherical perfect conductor, Q is proportional to R 2 and E.) In order to understand these differences, numerical simulations are performed for the idealized droplets of perfect conductor. Comparison of the numerical and experimental results suggests that the differences are mainly due to incomplete charging of a water droplet resulted from the combined effect of electrochemical reaction at electrode and the relatively low conductivity of water. © 2008 Elsevier Inc. All rights reserved.
Keywords: Charging Coulombic force Dielectric fluid Charged conducting droplet
1. Introduction When a charged conducting droplet is suspended in an immiscible fluid under an external electric field, it can move quickly toward the electrode with opposite charges due to Coulombic force. This behavior of a charged droplet attracts much attention owing to its potential applications in various fields such as the bio-MEMS [1,2]. Recently, a droplet has also been considered as a candidate for a “micro-reactor” in microfabricated devices. Electrical manipulation of a charged droplet may be used in the design of such systems. Related to the behaviors of a conducting droplet under electric field, we have essentially two issues. The first is how much charge a droplet can acquire at the electrode. The other is how the droplet deforms and moves inside the fluid. The second issue has been studied extensively both experimentally and theoretically [3–5]. However, the first issue of charging process of a fluid particle has hardly been considered. This fact provides the major motivation of the present work. Regarding the problem of charging of a particle at electrode, studies have mostly been limited to the solid particle cases. Charging of a micron-sized metal particle at the electrode was studied experimentally by Cho [6]. In the paper, two mechanisms of charging, the contact potential charging and the induction charging, are
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Corresponding author. Fax: +82 54 279 5528. E-mail address: [email protected] (I.S. Kang).
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considered. The experimental results show that the contact potential charging may be neglected when the particle size is larger than one micron. Also shown is the theoretical result for the amount of induced charge Q = 1.65 × 4πε R 2 E 0 ,
(1)
where R is the radius of the particle and E 0 is the electric field between the charged plates. The theoretical result can be derived for a perfectly conducting sphere contacting the electrode and the exact value of the coefficient 1.65 is π 2 /6 (see also [7]). Since Cho’s work, there have been several related works. Khayari and Perez [7] used Eq. (1) to estimate how much charge is leaked out during the motion of a charged particle inside the surrounding fluid. They observed the vertical motion of a spherical metallic ball bouncing between two horizontal plates. When the applied voltage is high enough for the electric force to overcome gravity, the ball rises and moves up through the liquid. Then the ball falls down as its charge leaks away. They extended their experiment to the drop problem in which a drop is attached to the bottom electrode with a certain contact area before the liftoff [8]. They considered the effects of adhesion force and deformation of a drop on the liftoff electric field strength. But they did not pay attention to the charging process of a drop at the electrode. Very recently, Hase et al. performed the experiments by using two electrodes in the horizontal configuration [9]. They observed rhythmic motion of a small water droplet under a DC electric field. They found that there are three distinct modes of oscillation: stationary state, small oscillation over some part of the electrode gap,
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Fig. 1. The schematic view of the experimental setup. The cell sketch is not to scale.
and large oscillation over the whole gap. Based on the experimental results, they prepared a state diagram that shows the three modes. To understand the phenomena, they solved an equation of motion for a droplet by using the charge leakage model. As the initial charge condition, they used the value estimated by Eq. (1). But again, they have not considered the characteristics of the electrical charging process of a droplet at the electrode. A bit earlier, Eow et al. [10] performed experiments on the oscillatory motion of a conducting drop between electrodes. They considered the cases of quite large drops. Their experimental results show that the measured velocity of a drop is smaller than that predicted by Eq. (1) for a spherical metal particle of the same equivalent radius. Although they have noticed the difference in the moving velocities, they have not studied the difference in the characteristics of the charging process. As seen above, the charging process itself of a conducting droplet at the electrode has not been studied systematically. Therefore, in this paper, we study experimentally the electrical charging process of a micro-sized conducting water droplet immersed in very insulating silicon oil. From the experimental results, we attempt to obtain a quantitative relationship in the form of a scaling law between the amount of charging and the key parameters. For better understanding, we also observe very carefully the behaviors of a droplet near the moment of contact to the electrode surface. A high speed camera is used with 5000 fps. Compared to a spherical perfect conductor, a water droplet manifests a few distinct features such as ionization reaction at electrode, finite (relatively low) conductivity, and the shape deformation. In order to clarify these effects by comparison, numerical simulation is also performed for a perfectly conducting droplet with various predefined shapes. 2. Materials and methods 2.1. Experimental setup and conditions In the experiment, we measure the velocity of a droplet moving between the electrodes. The velocity is converted to the amount of a charge as will be shown later. The schematic diagram of the experimental setup is shown in Fig. 1. A transparent rectangular acryl test cell is placed between a light source and a microscope. To avoid thermal effect, a cold light source is used. The rectangular glass test cell, which can grab electrodes in both walls, is 10 ( L ) × 30 ( W ) × 45 ( H ) mm3 in the internal dimension. Two copper electrodes are connected to a DC voltage power supply (Trek. Inc.) to apply an external electric field. To observe deformation and translation of a droplet, a high speed CCD camera (Photron Fastcam 1024 PCI Model 100 K) at 5000 fps is mounted on the microscope. The velocity is extracted using the image analysis techniques. Single DI (de-ionized) water droplet and silicon oil (KF-96 series produced from Shin-Etsu Silicone) are used as the conducting droplet and the dielectric fluid medium respectively. Because the silicon oil has almost the same density as water, it is assumed that the effect of the gravity is negligible. To inject a micro-sized water droplet in the cell, a micro-pipette (Eppendorf) is used.
The experimental procedure is as follows. We prepare silicon oil in the test cell. Then a DC electric field is applied at both electrodes in the test cell with silicon oil. We put a single water droplet in the test cell using a micropipette. At that time, the experiment should be carried out as quickly as possible after silicon oil is prepared in the test cell. This caution is taken to minimize the absorbed moisture, which changes the electrical conductivity of silicon oil. The caution is taken also to exclude any unexpected property change of silicon oil by the high electric field. To observe the effects of various parameters, the viscosity of medium, the size of droplet, and the applied voltage are changed systematically. We use three kinds of silicon oils (ε = 2.75ε0 , σ 10−13 S/m, ρ = 957.24 kg/m3 , where ε0 = 8.854 × 10−12 F/m) with viscosities of 50, 260, and 1000 cS. The volumes of the deionized water droplets (ε = 77.75ε0 , σw = 1.2 × 10−4 S/m, μw = 8.90 × 10−4 kg/m s, ρw = 997.75 kg/m3 ) are varied with 0.2, 0.4, 0.8, and 1.6 μl. The corresponding drop radii are 363, 457, 576, and 726 μm, respectively. Also, we apply the external voltages of 2, 2.5, 3, and 4 kV across the fixed distance between two electrodes of 10 mm. 2.2. Estimation of amount of charge on the droplet An initially uncharged conducting water droplet suspended in silicon oil starts to move slowly when a DC electric field is applied. At the moment a droplet touches the electrode surface, electrical charging occurs between the electrode and the droplet. Then the droplet, which has the charges of the same sign as the electrode, is repulsed and moves toward the opposite electrode. A droplet, which arrives at the opposite electrode, discharges its own charges and acquires new charges from the electrode. Then the droplet is also repulsed and translated. This process repeats itself. This is the so-called contact charging process. For the amount of electrical charging, direct measurement is most desirable. However, in the present problem, accurate direct measurement is extremely difficult. The contact time is very short (the order of micro seconds) and transferred charge amount is too little (the order of 10−11 C). So, we take an alternative way in which the drop translation velocity is measured to estimate the charge amount. When the droplet size is small enough, the shape is spherical and the inertial effect may be neglected. In the middle region of the two electrodes, it can be assumed that the electric field is uniform and the droplet moves with a constant velocity. We consider now the force balance for the droplet
F = F D + FQ = 0 .
(2)
In the above, the drag force is assumed to be given by the Hadamard–Rybczynski solution [11] FD = 4πμUc
with c =
3λ + 2 2(λ + 1)
, λ=
μw μ
(3)
and the electrical force term is given by FD = Q E∞ .
(4)
From the force balance, we have the following formula for estimation of the amount of charge in terms of droplet velocity Q est =
4πμaU c E∞
.
(5)
In the above, Q est is the estimated electrical charge on the droplet, E ∞ is the magnitude of the applied external field, μw is the viscosity of water inside the droplet, μ is the viscosity of the medium, a is the droplet radius, U is the droplet velocity.
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2.3. Assumptions To use the force balance equation (2) (or Eq. (5)), several assumptions should be satisfied. First, the Reynolds number should be much less than unity. The Reynolds number based on the experimental parameters in our system ranges from 0.0002 to 0.1120. Second, we need the assumption that charges do not leak away during the droplet travels between the electrodes [7,9]. If the charge relaxation time of silicon oil is much longer than the droplet-traveling-time during its half cycle, the assumption is reasonable. The charge relaxation time of the silicon oil used in our experiment is estimated to be
τ=
ε 2.75 × 8.854 × 10−12 = ≈ 243.5 s. σ 10−13
(6)
The electrical permittivity and the conductivity of silicon oil are chosen as ε = 2.75ε0 with ε0 = 8.854 × 10−12 F/m and σ 10−13 S/m, respectively. They are referenced from the data of the specification of Shin-Etsu Silicone. In experiments, the traveling time of the water droplet between the electrodes is from 0.144 to 4.664 s. This indicates that the charge leakage is negligible in our experiments. In the present study, we use the fixed values of the physicochemical parameters for the analysis of experimental results. However, it is well known that these parameters, especially the conductivity, may vary easily and the results may be influenced. Thus we have paid special attention to these effects. But no significant change has been observed for the droplet moving velocity during many repeated cycles of motion between the electrodes. Third, deformation of the droplet from spherical shape must be small. As will be shown below, this condition is also satisfied reasonably well. The interfacial tension between the water droplet and the silicon oil (in the case of no electric field) can be estimated by using the formula
√
γAB = γA + γB − 2 γA γB =
√
√
2
γA − γB .
(7)
The surface tension of silicon oil is about 20.5 mN/m from the specification data sheet and that of DI water is generally known as 72 mN/m. From Eq. (7), the interfacial tension between silicon oil and a water droplet is estimated to be about 15.7 mN/m. Now, the electrical Weber number can be estimated. The Weber number is the ratio of the deforming electric force to the restoring interfacial tension force and is defined as We(e) =
ε E 2a , γ
(8)
where a is the characteristic radius of the droplet and γ is the interfacial tension. For the experimental parameters of this study, the Weber number ranges from 0.0225 to 0.1801. This indicates that the drop deformation is small. The interfacial tension may also be estimated by using the shape of a drop deformed in an electric field. Kim et al. [5] showed that, in a uniform electric field with a small Weber number, a conducting drop (charged or uncharged) is deformed into the shape (in dimensionless form) of R (cos θ) = 1 +
3 4
We(e) P 2 (cos θ),
Fig. 2. The sequence of photographs of a droplet moving from the left (positive electrode) to the right (negative electrode).
(9)
where R is the radial coordinate of the drop surface, P 2 is the Legendre polynomial, and the angle is measured from the electric field direction. The pictures of deformed droplets indicate that the value of the interfacial tension estimated by Eq. (7), 15.7 mN/m, is rather small. The value of the interfacial tension estimated by Eq. (9) is about twice as large as the value estimated by Eq. (7). (The pictures of deformed droplets will be given in the next section.) This fact suggests that the drop deformation is quite small because even the largest Weber number is below 0.1.
Fig. 3. The position of droplet from the positive electrode as a function of time.
The fourth assumption is that the wall effect must be negligible, because Eq. (5) is derived for the case of infinite medium. In view of large dimension of the test cell (10 mm (L) × 30 mm (W ) × 45 mm (H )) compared to the droplet size (maximum radius 726 μm), this effect will not be significant. 3. Results 3.1. Translational motion of a droplet between electrodes After contacting an electrode, the droplet is repulsed and translated toward the opposite electrode. Similarly, after touching the opposite electrode, the droplet is also repulsed and moved back to the original electrode. The droplet has this repeated motion under the electric field formed between two electrodes. Fig. 2 is the sequence of pictures of a half-cycle of the time-periodic motion. The droplet near the electrode becomes elongated and makes a tip facing the electrode. This is because a droplet is deformable and the electric field is very asymmetric near the electrode. When the droplet approaches the electrode, the electric field between the droplet and the electrode surface becomes much stronger than that between the droplet and the opposite electrode. The drop tip becomes sharper and sharper and eventually touches the electrode. At that time, the oppositely-signed charges on the droplet are discharged at the electrode and the same-signed charges move to the droplet. This is the process of electrical charging. Fig. 3 shows the droplet position as a function of time for a typical case. The droplet shows a very regular periodic motion between the electrodes and reaches a constant velocity near the center. The constant velocity is used for estimation of the amount of charge on the droplet by using Eq. (5). In Fig. 4, the average velocities of the droplet are shown according to the moving directions. The cases of starting from the positive electrode are compared with the cases of starting from the negative electrode. The velocity of a droplet starting from the
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Fig. 4. The dependence of the average velocity on the moving directions (+: from the positive electrode, −: from the negative electrode).
Fig. 6. The dependence of the estimated charge ( Q est / Q 0 ) on the electric field (E / E 0 ).
Fig. 5. The dependence of the estimated charge ( Q est / Q 0 ) on the droplet radius (R / R 0 ). The subscript “0” stands for the reference state of E 0 = 2 kV/cm, R 0 = 363 μm, medium viscosity 50 cS with the estimated charge Q 0 = 1.03 × 10−11 C. For more convincing representation, the abscissa variable is modified by multiplying the factors in terms of the kinematic viscosity (ν /ν0 ) and the electric field (E / E 0 ). Fig. 7. The graph of the scaling law summarizing the experimental results.
positive electrode is slightly higher than that from the negative electrode. Unfortunately, a detailed molecular level explanation on this phenomenon is not available now. Further studies are definitely needed. 3.2. The estimation of electrical charge acquired at an electrode As mentioned in Section 2, the measured velocity of a droplet is used to estimate the amount of electrical charge acquired at the starting electrode by Eq. (5). In order to use the equation, as discussed earlier, we need to assume that the charge leakage is negligible, the Reynolds number is small enough, the droplet deformation is negligible, and the wall effect is also negligible. 3.3. The scaling law for the amount of electrical charging
convincing way, the x-axis (abscissa) variable is modified by multiplying the factors in terms of other parameters (the kinematic viscosity ratio and the electric field ratio). In a similar way, the effect of electric field is shown in Fig. 6. From Figs. 5 and 6, we can see that the amount of electrical charging increases with the size of the droplet and the applied electric field strength. On the other hand, the viscosity of the dielectric fluid medium is not a key factor in the electrical charging process for the parameters in the present study. (See the very small exponent for the viscosity ratio in the adjustment of the abscissa variables of Figs. 5 and 6.) Summarizing the experimental results, a scaling law for the amount of the electrical charging has been derived as (see Fig. 7) Q
The experimental results for the amount of electrical charging have been summarized in the form of a scaling law. In order to treat the problem in a dimensionless form, a reference state is defined. As the reference state, we choose the case of volume 0.2 μl (radius 363 μm), applied field 2 kV/m, and medium viscosity 50 cS. For this case, the droplet moving velocity is U = 0.95 cm/s and the estimated charge is Q est = 1.03 × 10−11 C. In Fig. 5, the dependency of charging amount on the droplet radius is shown. In order to show the dependency in a more
Q0
=C
R R0
1.59
E E0
1.33 ,
(10)
where the subscript 0 stands for the reference state (R 0 = 363 μm,
ν0 = 50 cS, E 0 = 2 kV/cm, Q 0 = 1.03 × 10−11 C) and C is a constant that depends on the viscosity ratio to the reference medium viscosity. The value of C changes slightly as, for μ/μ0 = 1, C = 1, for μ/μ0 = 5.2, C = 1.0147, and for μ/μ0 = 20, C = 1.045. Since the viscosity effect is very weak, the viscosity effect is treated in the way shown above.
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Fig. 8. The sequence of photographs that shows the motion and deformation of a droplet near the electrode due to attraction and repulsion by the electrode (the experimental condition: droplet volume 1.6 μl (radius R = 726 μm), applied field 3 kV/cm, viscosity of medium 50 cS).
According to Cho [6], the amount of the induction charging of a spherical perfect conductor is proportional to the square of radius and the electric field (see Eq. (1)). The scaling law for the present water droplet is a bit different from the perfect conductor case in the exponents. In the water droplet case, the exponent for the effect of the droplet radius is decreased (2.0 → 1.59), while the exponent for the effect of the electric field is increased (1.0 → 1.33). The explanations on these differences will be given later after looking at the numerical results for the (idealized) perfectly conducting droplets with predefined shapes. 3.4. The deformation of a droplet near the electrode As a droplet approaches the electrode, the droplet is deformed into an elongated shape as shown in Fig. 8. Because the droplet is a good conductor compared to the dielectric medium, the electric field becomes stronger in the gap between the electrode and the droplet surface. The charges are now concentrated near the tip of the droplet facing the electrode, and the droplet is deformed into the shape with a sharper tip [12]. Therefore the maximum curvature tip is formed at the moment of touching the electrode. Fig. 9 shows the ratio of the maximum diameter to the minimum diameter as a function of time when the droplet is attracted toward the electrode and repulsed from the electrode. The first peak occurs at the moment of touching the electrode. A second small peak is also observed. This is because of an overshoot in deformation. At the moment of repulsion, the droplet is pushed outward very strongly. Especially the part of droplet surface facing the electrode experiences stronger push and is deformed inward accordingly. After a certain time, the inwardly deformed surface comes out again due to the action of surface tension. As the drop travels further, the electric field becomes weaker, and the droplet is less deformed. 4. Discussion 4.1. Comparison of experimental and numerical results As mentioned earlier, numerical simulation has been performed for better understanding of the electrical charging process of a
Fig. 9. The degree of deformation represented by the ratio of the maximum diameter to the minimum diameter (for droplet deformation in Fig. 8).
droplet. In computation, we consider the (idealized) perfect conductor droplets with predefined shapes. By comparing the numerical results with the experimental results, we can find some characteristics of charging process of real (nonideal) water droplets at the electrode. For each experimental parameter set, numerical computations are carried out for the shapes predefined in the following three distinct ways: the experimentally observed droplet shape at the moment of touching the electrode (case (i)), the prolate shape that has the same volume and the same major axis length as the experimentally observed droplet shape (case (ii)), and the spherical shapes that has the same volume as the experimentally observed droplet shape (case (iii)). By comparing the experimental results for a nonideal droplet with the numerical results for the counterpart of ideal perfect conductor of the same shape (case (i)), we can see the combined effect of all other reasons than the droplet shape, such as the ionization reaction at the electrode surface and the finite (relatively low) electric conductivity of water. We can also learn several things by comparing the numerical results for the three cases mentioned
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Fig. 10. The details of the computational domain for numerical simulations of the (idealized) perfectly conducting droplets.
Fig. 11. Comparison of the numerical and experimental results for the electric field effect in the cases of droplet volumes 0.2 and 1.6 μl. (2) The numerical results for the shapes obtained from the experimental images, (") the numerical results for the prolate shapes with the same volume and the length as the experimental images, (Q) the numerical results for the spherical shapes with the same volume as the experimental images, (a) the experimental results for the water droplets in the case of medium viscosity 50 cS.
above. Comparison of the results for spherical shape (case (iii)) and the extended prolate shape (case (ii)) gives us the effect of extended droplet shape on the amount of electrical charging. Comparison of the results for the extended prolate shape (case (ii)) and the experimentally observed shape (case (i)) gives us the effect of the sharp tip of droplet at the moment of touching the electrode. For numerical computation, we use the commercial package of COMSOL Multiphysics. We adopt the (r , z)-axisymmetric coordinate system with its origin at the tip of the droplet as shown in Fig. 10. The governing equation is the Laplace equation for the electric potential outside the droplet. At the boundary, typical wellknown conditions are used. After obtaining the solution of the electric potential, the surface charge density σ is computed by εE · nˆ = σ , where E is the electric field outside the droplet and ε is the electrical permittivity of the outside dielectric fluid with ε = 2.75ε0 . Then the amount of electrical charging is computed by integrating the surface charge density over the droplet surface. In Fig. 11, the effects of electric field on the amount of electrical charging are shown. The experimental results are shown with the inverted solid triangles. The numerical results are represented with closed squares (case (i)), closed circles (case (ii)), and closed triangles (case (iii)). If we first pay attention to the numerical results only, we can notice that there is a considerable effect of extended droplet shape. On the other hand, the sharp tip effect is not so
Fig. 12. Comparison of the numerical and experimental results for the effect of droplet radius in the cases of electric fields 2 and 4 kV/cm. (The same symbols are used as in Fig. 11.)
large as long as the major axis lengths are the same. The effect of extended droplet shape is easy to understand because the electric field near the droplet surface facing the opposite electrode gets stronger as the droplet is more extended. A very important observation from the numerical results is that a perfect conductor droplet (case (i) with closed squares) exhibits the scaling law with the exponent “unity” to the electric field strength just like the well-known theoretical result for the perfect conductor sphere (see Eq. (1)). However, the experimental results for the real water droplet (inverted closed triangles) show that the exponent is in the range of 1.32 to 1.35. (The results for the droplet volumes of 0.4 and 0.8 μl are not given here.) This means that the stronger electric field is more favorable to electrical charging compared to the perfect conductor case. Differently from the case of perfect conductor, several distinct mechanisms (such as the electrolysis reaction at the electrode surface, transport of charges in the forms of ions) may be involved for the charging and electrical conduction processes [13]. Therefore, it is not possible to pinpoint one specific reason behind the difference in the exponent of the scaling. Detailed explanation on the exponent difference is certainly beyond the scope of the present paper and further theoretical and experimental studies of deeper level are definitely needed. Nevertheless, some phenomenological and qualitative speculation may be given. Since the ionization reaction at electrode is involved and the electrical conductivity of the real water droplet is finite and relatively low, the rate of surface reaction and the contact time (or effective charge transport time) are important. For both factors, stronger electric field seems to be favorable. The increased charge transport time may be expected as follows. The electric field between the charged droplet and the counter electrode increases indefinitely as the droplet approaches the electrode. Therefore at a certain critical distance the dielectric fluid breaks down and the charges may flow between the electrode and the droplet. Since the amount of electrical charging increases and the droplet tip becomes sharper as the field strength increases, the critical distance of dielectric breakdown increases as the electric field increases. This fact seems to make a partial contribution to the higher exponent of about 1.33 for the actual water droplet case to the exponent unity in the perfect conductor case. The effects of droplet radius are shown in Fig. 12. As in the case of the electric field effects, the numerical results for the (idealized) perfect conductor droplets show a good agreement with the theoretical prediction in the exponent of scaling (Eq. (1)). The exponent found from the numerical results is very close to 2, which is the theoretical prediction. On the other hand, the nonideal water droplet shows a smaller exponent that ranges from 1.56 to 1.59.
Y.-M. Jung et al. / Journal of Colloid and Interface Science 322 (2008) 617–623
(The results for the electric fields of 2.5 and 3.0 kV/cm are not given here.) This may be explained as follows. Less sufficient charging occurs for a larger water droplet, while complete charging occurs for a perfect conductor droplet regardless of the droplet size. As for the electric field effects, this part also needs further theoretical and experimental studies of deeper level. 4.2. Discussion of the effect of the Weber number on the scaling law In our study, the amount of charge carried by a droplet is estimated by using the Hadamard–Rybczynski solution under the assumption that the droplet shape is spherical in the middle region of the electrode cell. However, a droplet is elongated under electric field and the moving velocity is also changed from the value predicted under the sphere assumption. For this matter, Kim et al. [5] derived a formula that can be reduced to the following form when the viscosity of the droplet phase is negligibly small compared to that of the medium phase.
U = U H−R 1 + 0.6W + O ( W 2 ) ,
(11)
where U H−R is the velocity predicted by the Hadamard–Rybczynski solution and W is the electrical Weber number. Therefore, the value of amount of charge estimated by Eq. (5) is a bit overestimated by a factor (1 + 0.6W ) when the Weber number is small. Since the most deformed is the case of the largest drop, let us apply the correction to the case of Fig. 11. If we assume that the interfacial tension is about 30 mN/m, the corresponding Weber number ranges from 0.0225 to 0.09. If we adjust the experimental values according to the formula (11), the slope must be adjusted from 1.35 to about 1.30. But it must be noted that this is the largest adjustment of the slope. Therefore, when we take into account of the effect of deformation of a droplet in the middle region of the electrode cell, the exponent in the scaling law for the electric field effect may be lowered. However, the adjustment is not so large and the exponent is still larger than about 1.3. 5. Summary In this study, electrical charging of a water droplet at the electrode has been studied. To estimate the amount of charge, the moving velocity of a droplet in the middle region of the electrode cell is measured. Under the assumption that the droplet has a spherical shape and follows the Stokes law, the measured velocity is used to estimate the charge amount by using the force balance. The experimental results show that the electric field strength and the droplet size are the important key factors to the electrical charging process. On the other hand, the viscosity of the medium fluid does not have a significant effect in the parameter range of the present study. The experimental results are summarized in the form of a scaling law. In the scaling law, the amount of electrical charging of a water droplet is proportional to the power 1.59 of
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the droplet radius and the power 1.33 of the electric field strength. In the case of a perfect conductor sphere, the charge is proportional to the square of radius (the power 2) and proportional to the electric field (the power 1). In order to understand these fundamental differences, numerical computations have been performed for the (idealized) perfect conductor droplets with predefined shapes (experimentally obtained shapes, prolate shapes, and spherical shapes). The numerical results reveal an important fact. Perfectly conducting droplets (even with the deformed shapes) follow the scaling law in which the charge amount is proportional to the square of the radius and proportional to the electric field just like the well-known theoretical result for a perfect conductor sphere. So, the fundamental differences are due to other reasons such as the ionization reaction at the electrode and the relatively low electrical conductivity of the water droplet. Since several mechanisms are simultaneously involved, further studies of deeper level are definitely needed for thorough understanding. Finally, a comment should be given to the applicability of the results of the present experimental work. It is thought that the electrical charging phenomenon can be used as a tool of transporting a single micro- or nano-liter droplet in a micro-channel without moving the medium fluid. Especially, when a conducting droplet is used as a tool of transporting bio-particles like DNA molecules, they are safe inside the droplet against the strong external electric field. Inside the conducting droplet, there is virtually no electric field. This fact may also find many applications. Acknowledgments This work was supported by the grant R01-2004-000-10838-0 from KOSEF. This work was also supported by the grant from Center for Ultramicrochemical Process Systems (CUPS) sponsored by KOSEF and the grant by BK21 program of Ministry of Education of Korea. The authors greatly acknowledge the financial supports. References [1] M. Washizu, IEEE Trans. Ind. Appl. 34 (1998) 732. [2] S. Katsura, A. Yamaguchi, H. Inami, S. Matsuura, K. Hirano, A. Mizuno, Electrophoresis 22 (2001) 289. [3] G.I. Taylor, Proc. R. Soc. London A 280 (1964) 383–397. [4] J.-W. Ha, S.-M. Yang, J. Fluid Mech. 405 (2000) 131. [5] J.G. Kim, D.J. Im, Y.M. Jung, I.S. Kang, J. Colloid Interface Sci. 310 (2007) 599. [6] A.Y.H. Cho, J. Appl. Phys. 35 (1964) 2561. [7] A. Khayari, A.T. Perez, IEEE Trans. Dielectr. Electr. Inst. 9 (2002) 589. [8] A. Khayari, A.T. Perez, F.J. Garcia, A. Castellanos, in: Proc. Ann. Report Conf. on Electr. Ins. Dielectr. Phenomena, 2003, pp. 682–685. [9] M. Hase, S.N. Watanabe, K. Yoshiokawa, Phys. Rev. E 74 (2006) 046301. [10] J.S. Eow, M. Ghadiri, A. Sharif, Colloids Surf. A Physicochem. Eng. Aspects 225 (2003) 193. [11] L.G. Leal, Advanced Transport Phenomena: Laminar Flow and Convective Transport Processes, Cambridge Univ. Press, Cambridge, 2007. [12] G.I. Taylor, Proc. R. Soc. London A 291 (1966) 159. [13] J.O’M. Bockris, A.K.N. Reddy, Modern Electrochemistry 1: Ionics, second ed., Plenum, New York, 1998.