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Langmuir 2005, 21, 790-792
Notes Droplet Deformation of a Low-Molecular-Weight System in an Alternating Current Electric Field Nikolaos Bentenitis, Sonja Krause,* and Khaoula Benghanem Department of Chemistry and Chemical Biology, Rensselaer Polytechnic Institute, Troy, New York 12180 Received June 17, 2004. In Final Form: October 15, 2004
1. Introduction When an uncharged liquid droplet is suspended in another liquid under a uniform external electric field, there is a discontinuity of the field at the droplet interface. In 1962, several liquid-liquid systems studied by Allan and Mason1 produced droplets with deformations parallel to the applied electric field (prolate deformation); however, other systems produced droplets which deformed perpendicular to the electric field direction (oblate deformation). To explain this phenomenon, Taylor2 proposed the leaky dielectric model, LDM, which takes into account the small conductivities exhibited by all dielectric fluids. When an electric field is applied to such liquids, free charge appears at the droplet interface; the electric field that acts on this charge sets the fluids in motion, and toroidal circulation patterns are formed inside and outside the droplet. Such circulations were observed experimentally by Taylor2 who used particles of bismuth oxychloride dry powder. To model the physical problem mathematically, Taylor assumed that the shape of a distorted droplet was a slightly deformed sphere. His model predicted both oblate and prolate distortions, depending on the properties of the fluids, in agreement with the experiments by Allan and Mason. The properties of the fluids involved were their conductivities, permittivities, viscosities, and interfacial tension. Taylor focused on predicting the direction of deformation (either prolate or oblate) and not on describing the magnitude of the deformation quantitatively. The first quantitative comparisons between experiments and the LDM were reported by Torza et al.3 who also extended the LDM to alternating electric fields. A prediction of the Torza et al. extension was that a drop that is subjected to a sinusoidal electric field of angular frequency ω will deform in a sinusoidal manner with a frequency 2ω. To the best of our knowledge, this prediction has not been tested experimentally. In this note we discuss a system for which the prediction of sinusoidal deformation was tested. 1.1. Basic Equations. Torza et al.3 considered a droplet of radius r0 which carried no net electric charge and was suspended in a neutrally buoyant condition in another immiscible fluid. The system was subjected to a uniform (1) Allan, R. S.; Mason, S. G. Proc. R. Soc. London, Ser. A 1962, 267, 45-61. (2) Taylor, G. I. Proc. R. Soc. London, Ser. A 1966, CCXCI, 159-166. (3) Torza, S.; Cox, R. O.; Mason, S. G. Philos. Trans. R. Soc. London, Ser. A 1971, 269, 295-319.
electric field whose strength E, far from the droplet, varied with time according to
E ) E0 cos ωt
(1)
where ω ) 2πν is the angular frequency of the field, ν is the frequency of the field, and E0 is the peak value of the field. Torza et al.3 assumed that the permittivities d and m and the conductivities σd and σm of the droplet and the medium, respectively, were independent of ν. Under the influence of the electric field, the droplet was distorted into an ellipsoid. The authors used the droplet deformation, D,
a-b D) a+b
(2)
where a and b are the axes of the ellipsoidal drop parallel and perpendicular to the applied field, respectively. When the droplet is deformed in the direction of the electric field, 0 < D < 1, and when the droplet is deformed in the direction perpendicular to the electric field, -1 < D < 0. Because the droplet deformation is a result of the competition between the applied electric field that tends to deform the drop and the interfacial tension that tends to keep the droplet spherical, a useful dimensionless parameter that expresses the magnitude of this interaction is the electrical capillary number Ce, which is defined as
dr0E02 Ce ) γ
(3)
The LDM predicts the relation between the electrical capillary number, Ce, and the droplet distortion D using the following ratios:
S)
m d
(4a)
R)
σd σm
(4b)
M)
µm µd
(4c)
The deformation D(t) of a distorted droplet in an alternating current (ac) field can be written as the sum of a timeindependent deformation, Ds, and a time-dependent deformation, Dt(t)
D(t) ) Ds + Dt(t)
(5)
The LDM predicts that
Ds )
9 Φ SC 16 s e
I cos(2ωt + β) Dt(t) ) Ds Φsx1 + k2 λ22
10.1021/la048495j CCC: $30.25 © 2005 American Chemical Society Published on Web 12/17/2004
(6a) (6b)
Notes
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When eqs 5 and 6 are combined,
D(t) )
[
9 Φ 1+ 16 s
I Φsx1 + k2 λ22
]
cos(2ωt + β) SCe (7)
where
ωµmr0 γ
(8a)
(16M + 19)(3M + 2) 20(M + 1)
(8b)
k) λ2 ) and
Φs ) 1 - {S2R(11 + 14M) + 15S2(1 + M) + S(19 + 16M) + 15R2Sτ2ω2(1 + M)(S + 2)}/{5(1 + M)[S2(2 + R)2 + R2τ2ω2(1 + 2S)2]} (9a)
Figure 1. Top and front view of the cell used for observation of droplet deformation.
I2 ) Φs2 + {R2Sτ2ω2(SR - 1)2(19 + 16M)[20(M + 1) - S(4M + 1)]}/{25(M + 1)2[S2(2 + R)2 + R2τ2ω2(1 + 2S)2]2} (9b) τ)
m σd
(9c)
As can be seen from eqs 8 and 9, Φs, I, k, and λ2 are functions of the conductivities, permittivities, viscosities of the fluids, the interfacial tension between the fluids, the radius of the undeformed drop, and the applied frequency. For a particular combination of fluids, their values depend on the radius of the undeformed drop and the applied frequency. That is also the case for the phase angle, β; however, the equations for its calculation are not given here because they will not be used. 2. Experimental Section 2.1. Materials. An aqueous solution of 52 168 ppm KCl, sold as a conductivity standard for the calibration of conductivity cells, was obtained from Cole-Palmer Instrument Co., Vernon Hills, IL, and epoxidized linseed oil (EpoxLinsOil, CAS-RN: 801611-3) was obtained from Union Carbide, Danbury, CT. 2.2. Material Characterization. The viscosities of both fluids were measured at different shear rates using a Brookfield LVDV-II+ cone and plate viscometer (Brookfield Engineering Laboratories, Stoughton, MA). For EpoxLinsOil, the viscosity changed as the shear rate changed (shear thinning behavior), so the measured viscosities were extrapolated to zero shear rate. The density of the fluids was measured with a Paar DMA 48 densitometer (Ashland, VA) at 30 °C. The dielectric constant of EpoxLinsOil was measured at a frequency of 1 MHz and a voltage of 30 mV (peak-to-peak), using a Hewlett-Packard 4192ALF impedance analyzer (HewlettPackard, Palo Alto, CA) and a calibrated dielectric sensor designed by Kranbuehl et al.4 The sensor consisted of two gold electrodes deposited in an interdigitated configuration on a nonconductive substrate. For the KCl solution, direct measurement of the dielectric constant was not possible because of the high conductivity of the solution. It was estimated using (see ref 5)
soln ) solv + 2δc
(10)
where soln is the dielectric constant of the KCl solution, solv is (4) Kranbuehl, D. E.; Delos, S. E.; Jue, P. K. Polymer 1986, 27, 1118. (5) Hasted, J. B.; Ritson, D. M.; Collie, C. H. J. Chem. Phys. 1948, 16, 1-21.
Figure 2. Schematic of the image acquisition system used for observations of droplet deformation: (1) microscope, (2) CCD camera, (3) time code generator, (4) videocassette recorder, (5) video printer, (6) monitor, and (7) computer with frame grabber and image analysis software. the dielectric constant of the solvent (76.55 for water at 30 °C),6 δ is a constant for the electrolyte (-5.5 for KCl), and c its molar concentration. Ordinary conductivity meters were not appropriate for EpoxLinsOil because the values of its conductivity were extremely low. An Emcee 1154 Cell&Precision conductivity meter (Emcee Electronics, Inc., Venice, FL) was used. This instrument satisfied the requirements set by the ASTM D-4308 standard for conductivity measurements. 2.3. Apparatus Used in Experiments under an Electric Field. Figure 1 shows a schematic of the cell used. It consisted of a cylindrical piece of Teflon with a 0.6 × 2.5 cm2 rectangular slit in the middle. Two stainless steel electrodes were pushed into each side of the slit and held within two cuts in the Teflon. The whole cell fit onto a glass Petri dish. A wire was soldered onto each of the cell electrodes, and the wires were connected to a Trek 610C high voltage supply/amplifier (Trek, Inc., Medina, NY). The Trek 610C amplified an ac voltage generated by a Stanford Research Systems DS335 synthesized function generator (Stanford Research Systems, Sunnyvale, CA). EpoxLinsOil was poured into the cell, and a drop of the aqueous KCl solution was then injected in the EpoxLinsOil matrix via a micropipet. The cell was placed on the stage of a Leitz Laborlux 12 Pol S microscope (Wild Leitz, Inc., Rockleigh, NJ) as shown in Figure 2. The pictures from the microscope were captured by a Hamamatsu Digital XC-77 charge-coupled device (CCD) camera (Hamamatsu Corp., Bridgewater, NJ), which magnified the images by a factor of 20. The output was sent to a TC-3 SMTPE (6) Lide, D. R. CRC Handbook of Chemistry and Physics, 81st ed.; CRC Press: Boca Raton, 2000.
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Notes
Table 1. Electrohydrodynamic Properties of Liquids and Solutionsa fluid
conductivity (σ/pS‚m-1)
permittivity /0
KCl 8.00 × 1012 68.9 EpoxLinsOil (1.46 ( 0.1) × 103 6.18 ( 0.03 a
viscosity (µ/cP)
density (F/g‚cm-3)
0.81 ( 0.01 339 ( 2
0.996 0.965
Error limits correspond to 95% confidence intervals.
time code generator/reader (Burst Electronics, Corrales, NM), that added a time code to the signal. The resulting image was sent to a Sony SVO-9500MD videocassette recorder (Sony Corp., Tokyo, Japan) and from there to a Sony PVM-1343MD Trinitron color video monitor. Each image was then sent to a computer through the VP-1300-60 Imaging Technologies color frame grabber (Imaging Technologies, Inc.). The frame grabber encoded the input as 8-bit gray scale images with dimensions of 640 × 512 pixels. The images were analyzed to obtain the length of the major and minor axes of the distorted ellipsoidal droplets with the Optimas Image Analysis 6.1 software (Optimas Corp., Bothell, WA).
3. Results and Discussion 3.1. Material Properties. The measured electrical and hydrodynamic properties are given in Table 1; these give S ) 8.97 × 10-2, R ) 5.47 × 109, and M ) 4.17 × 102 for the system. Although the interfacial tension of several polymeric fluid pairs has been measured with a spinning drop tensiometer in our laboratory (see ref 7), the high viscosity of the EpoxLinsOil precluded the use of this technique. Table 1 shows a difference in the density of the KCl solution and that of EpoxLinsOil, but the relatively high viscosity of EpoxLinsOil prevented the KCl drop from sedimenting. Furthermore, although EpoxLinsOil exhibited shear thinning behavior, the LDM, which was developed for Newtonian fluids, was still used for the system, because the maximum estimated Reynolds number for the system was of the order of 10-4. 3.2. Oscillatory Deformation. Figure 3 suggests that the response of the KCl droplet to an applied electric field was close to a sinusoidal change of its deformation, D, as predicted from the Torza et al.3 theory. When the experimental value of D was fitted to a sinusoidal equation, the resulting period of oscillation was 5.0 s within experimental error. Because the period of the applied electric field was 10 s (frequency 0.10 Hz), the droplet deformed with twice the frequency of the electric field as expected, because it is only the magnitude and not the direction of the electric field that determines the extent of the deformation. For this system, the radius of the undeformed drop was 315 µm and, therefore, SCe ) 0.109. The interfacial tension between the conductivity standard and a different batch of EpoxLinsOil was γ ) 0.0043 N‚m-1. Using the same value for the batch of EpoxLinsOil used in this experiment, the Torza et al. model predicts Φ ) 1.000 and I/[Φs(1 +
Figure 3. Deformation, D, of a droplet of the aqueous KCl solution in the matrix of EpoxLinsOil versus time in a 0.10 Hz, 1.65 kV‚cm-1 ac field. The line shows predictions of the LDM for γ ) 0.0043 N‚m- 1.
k2 λ22)1/2] ) 0.999 and there is a good fit with the data, as shown in Figure 3. (The phase lag, β, was adjusted to 3.985 rad so that the peaks of the experimental data match the peaks of the predictions of the Torza et al. model.) Figure 3 also shows that the deformation was higher than the prediction of the Torza et al. model when the deformation had its greatest values. This shows that the LDM, which was developed for small droplet deformations only, is not applicable to deformations greater than 0.12 in this system. Several researchers7-9 who performed experiments in direct current (dc) electric fields observed even larger disagreements between experiments and the LDM. Such studies and observations in this work motivated us to develop a model that predicts higher droplet deformations that occur at large dc electric fields. That model will be presented in another publication. Acknowledgment. This paper is based on work partly supported by the National Science Foundation under Grant DMR-9521265. The authors thank Keith J. Nelson for the use of the conductivity apparatus in his laboratory. K.B., who was a high school student when she obtained these data, thanks the Camille and Henry Dreyfous Foundation for financial support. LA048495J (7) Xi, K.; Krause, S. Macromolecules 1998, 31, 3974-3984. (8) Tsukada, T.; Yamamoto, Y.; Katayama, T.; Hozawa, M. J. Chem. Eng. Jpn. 1993, 26, 698-703. (9) Ha, J.-W.; Yang, S.-M. J. Fluid Mech. 2000, 405, 131-156.
