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Electrorotation of Deformable Fluid Droplets
JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.
206, 10 –18 (1998)
CS985716
Electrorotation of Deformable Fluid Droplets Sonja Krause1 and Preeti Chandratreya2 Rensselaer Polytechnic Institute, Department of Chemistry, Troy, New York 12180 Received February 5, 1997; accepted June 19, 1997
consider what happens when such a sphere, suspended in a liquid in a uniform electric field, suffers a slight rotational perturbation. Jones (2, 3) has discussed the conditions under which this perturbation results in a stable rotation in a DC field. Assuming that the sphere is isotropic, one considers the charge relaxation time in the solid sphere t2 and in the liquid t1 such that
An equilibrium phase separated ternary system of polystyrene (PS)/polydimethylsiloxane (PDMS)/p-xylene was prepared, and the PS-rich phase was dispersed as droplets in a matrix of the PDMS-rich phase. The system was placed between vertical electrodes and the droplets rotated around a vertical axis perpendicular to the electric field direction in 4 – 8-kV cm21 DC fields; in 2– 4-kV cm21, 0.1-Hz AC fields; and in 4-kV cm21, 1-MHz AC fields, in some cases stopping, restarting, and changing the direction of rotation. They rotated less than a quarter of a turn back and forth in 1–10-Hz, 2– 4-kV cm21 fields and did not rotate at all in 1-kHz fields. Rotational velocities measured in the DC field were in agreement with an existing theory; those measured in the 0.1-Hz AC field and estimated in the 1-MHz AC field were in direct disagreement with a different existing theoretical treatment. When the PDMS-rich droplets were dispersed into a matrix of the PS-rich phase, the droplets elongated in the field direction in a 2-kV/cm, 0.1-Hz electric field. Occasionally a portion of the matrix phase broke off into the PDMS-rich droplet, rotated for a while, and then rejoined the matrix phase. © 1998 Academic Press Key Words: polystyrene; polydimethylsiloxane; electrorotation.
t1 5
e1 x1
and
e2 , x2
[1]
where e1 and e2 are the permittivities and x1 and x2 are the conductivities of the liquid and the solid sphere, respectively. When the charge relaxation time of the solid is less than that of the liquid, that is, t2 , t1, a normal dipole is generated in the sphere, that is, a negative charge is generated on the surface of the sphere nearest the positive electrode, whereas a positive charge is generated on the surface of the sphere nearest the negative electrode. These are stable to small rotational perturbations; that is, the sphere will return to its original orientation after the perturbation. On the other hand, when t2 . t1, that is, when the relaxation time of the solid particle is greater than that of the liquid, then a reverse dipole is generated on the sphere; these are unstable to small rotational perturbations, and the sphere will continue to rotate in a DC field if it is above a threshold value (E 0 ) thresh. Equation [2] is different from either of the equations shown by Jones (2, 3); those equations are dimensionally inaccurate. We used the derivation shown in Ref. 3 to develop the equation
INTRODUCTION
Electrorotation of solid spherical objects suspended in liquids has been discovered and rediscovered for over a century. One of the earliest studies was reported by Quincke (1) in 1896. He suspended macroscopic objects, many of them spherical, made of different materials like quartz, topaz, tourmaline, flint glass, and sulfur, in different solvents using a string or wire and applied a DC electric field that could be as high as 2 kV/cm. The spheres rotated back and forth in the applied fields with the axis of rotation vertical and perpendicular to the field direction when the electrodes were vertical; the spheres could not complete their rotations because of the torsion in the string or wire on which they were suspended. Rotation was observed in different solvents and was reproducible at different times, but it was not observed when the spheres were suspended in air. In order to explain this “Quincke rotation” in the case of a particle as symmetrical as a sphere, it has been necessary to
2 5 ~E 0! thresh
4 3
F
S
x2 2e1 11 e2 2x1 t1 e1 1 2 t t 2 MW
h1 1 1
S
D
DG
,
[2]
where tMW is the Maxwell–Wagner interfacial polarization relaxation time
1
To whom correspondence should be addressed. Present address: Raychem Corporation, 300 Constitution Drive, MS 122/ 6403, Menlo Park, CA 94025.
t MW 5
2
0021-9797/98 $25.00 Copyright © 1998 by Academic Press All rights of reproduction in any form reserved.
t2 5
10
e2 1 2e1 . x2 1 2x1
[3]
ELECTROROTATION OF DEFORMABLE FLUID DROPLETS
The angular velocity of rotation of the spheres in a DC field VDC is then V DC 5
1
t MW
Î
E 20 21 2 ~E 0! thresh
[4]
The electrorotation of very heterogeneous spherical objects like biological cells in a uniform AC electric field was observed much later than Quincke rotation in a uniform DC field; it occurs only if two cells are close to each other or if a single cell is near an electrode. Among the first to note biological cell electrorotation were Teixeira-Pinto et al. (4) who were studying Euglena and amoebae. Among later observers were Zimmerman et al. (5), who observed electrorotation of various kinds of biological cells such as mesophyll protoplast cells of Avena sativa (at 20–40 kHz), erythrocytes and ghost cells (at 80–100 kHz), and yeast cells (at 140–180 kHz). Electrorotation of biological cells was attributed by Pohl and Crane (6) to either deposition of charges on the cell surfaces or interaction of the alternating field with the oscillation of dipoles within the cells. On the other hand, Holzapfel et al. (7), who also studied the rotation of protoplasts of A. Sativa, concluded that the rotation was caused by the interactions of dipole fields generated in each cell by the applied electric field. If two such cells are close to each other, each is affected by the neighboring dipole field as well as by the external AC field. In such cases, a torque will be exerted by each cell on the other, resulting in rotation whose velocity depends on the relative positions of the cells. Rotation of individual cells, with no other cells in the vicinity, was not observed. Recently, Hu et al. (8, 9) observed electrorotation of monodisperse polystyrene spheres in a dilute aqueous electrolyte solution in 0.2–3 MHz, electric fields in the kilovolts per centimeter range. The polystyrene spheres rotated when they were close to each other in bands aligned at an angle of 645° relative to the electric field direction. The rotation of the polystyrene spheres was attributed to the existence of a phase lag between the applied field and the induced dipole moments in the spheres; this phase lag causes neighboring particles to exert on each other torques that cause the spinning of the spherical particles. Thus, there appear to be two sorts of rotation mechanisms for spherical objects in electric fields, one for the rotation of isolated spheres (Quincke rotation) in different solvents in DC fields and at least one other for the rotation of biological cells and charged polystyrene spheres that are close to each other in aqueous solution in AC fields. In the course of other work on the effect of electric fields on dispersions of deformable liquid droplets in a nonaqueous medium, we have observed the deformation and rotation of isolated droplets in both DC and AC electric fields. MATERIALS AND METHODS
Polystyrene (PS) was obtained from the Dow Chemical Company, Midland, MI; polydimethysiloxane (PDMS) was
11
obtained from Gelest, Inc., Tullytown, PA; and a block copolymer, PS-b-PDMS was obtained from T.W. Smith, Xerox Corporation, Rochester, NY. Polystyrene standards for GPC calibration were from Scientific Polymer Products, Inc., Ontario, NY, and Arro Laboratories, Inc., Joliet, IL. HPLC grade p-xylene and deuterated chloroform (99.8 atom % D with no internal reference) were from Aldrich Chemical Company, Milwaukee, WI. Coexisting phases in the PS/PDMS/p-xylene system were prepared by first dissolving the two polymers individually in p-xylene in the range 15–20% wt/vol. The two polymer solutions were then mixed in various proportions to give a turbid two-phase mixture. After shaking the turbid mixture vigorously, it was allowed to reach equilibrium over a period of 6 – 8 weeks at 30°C until two clear phases had separated. The molecular weights and molecular weight distributions of the homopolymers were measured using a Viscotek Model 100 gel-permeation chromatograph (GPC) from the Viscotek Corporation, Houston, TX. This GPC used two linear (universal) columns from American Polymer Standards Corporation, Cleveland, OH, and had two detectors connected in series, a Viscotek differential viscometer and a Knauer Model 198 (Knauer, GmbH, Germany) differential refractometer, which were interfaced with a computer to analyze the data using the UNICAL GPC software provided by the Viscotek Corporation. Polystyrene standards were used to calibrate the instrument which was then capable of measuring true molecular weight distributions of the polymers. Densities were measured using a DMA 48 Density Meter, PAAR, Graz, Austria. The interfacial tensions between the coexisting phases were measured at 30 6 0.5°C using a SITE 04 Spinning Drop Interfacial Tensiometer, Kru¨ss USA, Ridge, NY. The dielectric constants of the two coexisting phases were measured at 1 kHz at room temperature using a 1260 Impedance/Gain-Phase Analyzer, Solartron, Inc., Allentown, PA. The conductivities of the two phases were obtained by measuring their resistances using a Keithley Electrometer Model 6517, Keithley Instruments, Inc., Cleveland, OH. The viscosities of the coexisting phases were measured at 30 6 0.5°C using a Brookfield Model DV II1 Digital Cone/Plate Viscometer, Brookfield Engineering Laboratories, Inc., Stoughton, MA. The cone spindles used were CP-40 which can measure the viscosity range of 0.30 –1,028 cP and CP-51 which measures the viscosity range of 4.85–16,180 cP. The compositions of the coexisting phases were obtained by evaporating the solvent from a weighed sample of each phase to constant weight, dissolving the resulting film in deuterated chloroform (CDCl3) and then determining the relative wt % of the polymers in each phase by 1H NMR. The NMR measurements were performed on a 200-MHz Varian V1-200 Fourier Transform Spectrometer. The 6.2–7.4 ppm aromatic peak of PS and the 0–0.2 ppm CH3 peak of PDMS were used for the calculations. The experimental cell for electric field experiments was a microscope glass slide with copper electrodes glued on using
12
KRAUSE AND CHANDRATREYA
FIG. 1. The experimental cell with electrodes and sample on the microscope stage (a) as viewed from above and (b) as viewed from the side.
commercially available “super glue.” The gap between the electrodes was ;0.5 cm in all experiments and was measured using a reticule attached to the eyepiece of an optical microscope, a Leitz Laborlux 12 Pol (Leica, St. Gallen, Switzerland). The cell was placed on a Teflon base and held by two metal clips which also provided electrical connections between the electrodes and the power supply. A small amount of one of the coexisting phases was placed between the electrodes using a syringe, and then small droplets of the other phase were injected into the phase already between the electrodes using a syringe with a very small needle. A glass window was used as a lid to cover the cell to slow down the evaporation rate as well as prevent the solvent vapors from damaging the optics of the microscope. Schematics of the cell and experimental setup are shown in Figs. 1 and 2. The experiments were carried out on the optical microscope stage usually using a 43 magnification objective lens, sometimes a 323 lens. The eyepiece magnification was always 103. A Hamamatsu CCD camera (Hamamatsu Corporation, Bridgewater, NJ) was attached to the microscope, connected to a Sony VCR, and interfaced with a Pentium PC. The experiments were taped, and single frames (30
FIG. 2.
frames to a second) were later analyzed using Optimas Image Analysis Software (Optimas Corporation, Bothell, WA) versions 4.0, 5.2, and 6.0 at various times. A Trek model 610C high-voltage power supply/amplifier from Trek, Inc., Medina, NY, was used to provide both DC and AC voltages. It has a DC output voltage range of 0 –10 kV at a current of 0 –2 mA. An SRS Model DS335 Synthesized Function Generator, Stanford Research Systems, Sunnyvale, CA, was connected to the Trek 610C to provide an AC voltage signal, 0.1 Hz to 1 MHz, 0 –10 kV. RESULTS AND DISCUSSION
Polymers and Coexisting Phases The PS sample used had a weight average molecular weight, M w 5 2.0 3 10 5 , M w/M n 5 2.9 (M n is the number average molecular weight), whereas the PDMS had M w 5 2.0 3 10 3 , M w/M n 5 1.2. The block copolymer had a PS block molecular weight of 5.0 3 104, and a PDMS block molecular weight of
The overall experimental setup for electrorotation studies.
ELECTROROTATION OF DEFORMABLE FLUID DROPLETS
13
FIG. 4. An isolated PS-rich droplet in the PDMS-rich matrix phase in a 5-kV cm21 DC field. The counterclockwise direction of rotation as seen from above is shown by the curved arrow.
FIG. 3. Ternary phase diagram of PS/PDMS/p-xylene. Two experimentally determined tie lines are shown; the dotted curve indicates the approximate position of the binodal. The two small points above the binodal denote mixtures that remained clear, single-phase solutions.
1.0 3 104, as measured by the supplier. Compositions of two sets of coexisting phases were obtained and these, together with the compositions of some single-phase solutions, were used to construct the partial phase diagram shown in Fig. 3. One may note that the PS-rich phase, in each case, contained some PDMS, but the PDMS-rich phase contained no PS within experimental error. This is characteristic of systems that contain a high polymer, in this case the PS, and an oligomer, in this case the PDMS. The measured properties of the coexisting phases used for the electric field experiments are shown in Table 1. The very low value of the interfacial tension between the coexisting phases was to be expected because these coexisting phases had compositions close to that of the critical point in the phase diagram (Fig. 3). It is well known that the interfacial TABLE 1 Measured Properties of the Coexisting Phases Used for Electric Field Experiments Property Dielectric constant Conductivity Viscosity Density Interfacial tension a b
PS-rich phase
PDMS-rich phase
2.3 2.4 2.3 3 10211 2.4 3 10211 92.6a 19.2b 0.8955 0.8744 2.1 3 1025
Units
V21 m21 cP g cm23 N m21
This viscosity was constant up to a shear rate of 192 s21. This viscosity was almost constant up to a shear rate of 384 s21.
tension between coexisting phases decreases as the tie lines move closer to the critical point because the interfacial tension must vanish at the critical point. Furthermore, we have noted similar low interfacial tensions between coexisting phases in polymer/polymer/solvent systems in previous work from this laboratory (10, 11). PS-Rich Droplets in the PDMS-Rich Phase in a DC Field When a DC field up to 8 kV cm21 was applied, the droplets almost always rotated continuously in the counterclockwise direction between vertical electrodes as seen from above. A change in the rotation direction was observed only when a droplet first started rotating in the clockwise direction; then it eventually stopped and began rotating counterclockwise. The droplets slowly migrated toward the positive electrode and then spread over the electrode surface. When the field was turned off, sections of the droplets retracted and regained their spherical shape. If the field was reapplied at this point, the droplets were thrown off the electrode surface, the rotations were restarted, and the droplets again migrated slowly toward the positive electrode. Thus, it is obvious that the droplets had a slight negative charge. Because the droplets deformed slightly in the electric field, it was possible to measure their rotational velocity by following a single droplet over time using a frame-by-frame analysis of the videotape recording of the droplet rotation. Figure 4 shows a single droplet during rotation in a DC field. One may note that the droplet is only very slightly deformed so that it was imperative to use image analysis to follow the inclination of its longest axis with respect to the applied field direction. Actually, it was easier to follow the rotation using the analysis shown in Fig. 5. A pseudo-deformation, called here simply deformation, defined by the length of the largest droplet di-
14
KRAUSE AND CHANDRATREYA
FIG. 5. Cartoon of droplet rotation and the X and Y axes whose lengths were measured to calculate the deformation X/Y as a function of time for calculation of the angular rotational velocity of a droplet.
mension in the field direction X divided by the length of the largest droplet deformation perpendicular to the field direction Y, was measured and plotted versus time frame, as shown in Fig. 6. At 4 kV cm21, as shown in Fig. 6, the droplets rotated at about 0.75 rotations s21 5 4.7 radians s21 in the initial stages of the observations. One may note that the rotational velocity decreased with time; this is because the solvent slowly evaporated during the course of the experiment. This caused an increase in the viscosity of the matrix phase which then slowed the rotational velocity. At 8 kV cm21, the droplets initially rotated at a velocity of approximately 1.5 rotations s21 5 9.4 radians s21. If the direction and magnitude of the elongation of the droplets could have changed very rapidly with respect to the angular velocity of the droplets, we could not have evaluated their angular velocity in the manner shown in Fig. 5. It is possible to estimate the relaxation time for elongation t using the relationship found by Nishiwaki et al. (12):
t5
r 0~2 h 2 1 h 1! g
[5]
where r 0 is the original radius of the spherical droplet and g is the interfacial tension between the two phases. All parameters except r 0 can be found in Table 1. Because our droplets had various dimensions, we shall use a range from 10 –100 mm for r 0 . Using these parameters, we obtain a range of 0.1–1 s for t. Because these values have the same order of magnitude as the speed of rotation of the droplets, it is quite likely that small changes in droplet elongation occurred during the rotation of the droplets, but these changes did not reach steady state. Thus, the droplets rotated faster than they could change their deformation, and we were able to determine the rotational velocity of our droplets. In order to compare the rotational data observed for our deformable PS-rich droplets with the predictions for Quincke rotation of hard spheres using Eqs. [1], [2], and [4], we again used the data shown in Table 1. Equation [1] leads to tPS 5 0.88 s21 and tPDMS 5 0.49 s21, where the PS subscript and the PDMS subscript refer to the PS-rich and the PDMS-rich phase, respectively. Because tPS . tPDMS, Quincke rotation can be expected when the PS-rich droplets are injected into the PDMS-rich phase. Using the tMW 5 0.58 s21 calculated using Eq. [3], Eq. [2] predicts a threshold electric field for rotation, (E 0 ) thresh 5 1.3 kV cm21. This is consistent with our observations, especially since no rotation was observed in this system at 500 V cm21. Table 2 shows that our deformable droplets obeyed the predictions for Quincke rotation from Eq. [4] within our rather large experimental error in spite of the fact that the droplets had a slightly ellipsoidal shape while rotating. Figure 6 shows that the droplets appeared to be slightly elongated in the direction of the electric field at most times during the rotation. It is possible to calculate the direction in
FIG. 6. Deformation, as determined in the manner of Fig. 5, as a function of time of a PS-rich droplet in the PDMS-rich matrix phase in a 4-kV cm21 DC electric field.
15
ELECTROROTATION OF DEFORMABLE FLUID DROPLETS
TABLE 2 Experimental and Calculated Rotational Velocities of the PSRich Droplets in the PDMS-Rich Matrix in DC Electric Fields Rotational velocity (rad s21) Electric field (kV cm21)
Experimental
Calculated from Eq. [4]
4 8
4.7 8.7
5.0 10.5
which nonrotating droplets are expected to elongate in a DC field using the leaky dielectric theory of Torza et al. (13), in which a discrimination function F is used to determine whether a droplet suspended in a matrix will elongate in a direction parallel to the applied field direction, forming a prolate ellipsoid, or in a direction perpendicular to the applied field direction, forming an oblate ellipsoid. This discrimination function, written here for AC as well as DC fields, is given by R~11 l 1 14! 1 R 2@15~ l 1 1! 1 q~19 l 1 16!# 1 15a 2v 20~1 1 l !~1 1 2q! , F512 5~1 1 l !@~2 R 1 1! 2 1 a 2v 20~q 1 2! 2# [6] where R5
x1 , x2
q5
e2 , e1
l5
h2 , h1
[7]
and v0 is the frequency of the applied electric field, zero in a DC field. When F . 0, one expects elongation parallel to the field direction (a prolate ellipsoid); F 5 0, one expects an undeformed drop (drop remains spherical); F , 0, one expects elongation perpendicular to the field direction (an oblate ellipsoid). In the present case, F , 0 so that elongation perpendicular to the applied field direction is predicted, even though this was not often observed during rotation. Elongation perpendicular to the applied field direction was, however, observed (1) after all the solvent had evaporated and (2) just before counterclockwise rotation set in after a droplet that started with clockwise rotation had stopped rotating. That is, elongation perpendicular to the applied field direction occurred only when the droplets were not rotating. The virtual absence of perpendicular elongation during rotation is probably connected to the fact that the rotation of the droplet interferes with the electrohydrodynamic flow along the interface between the droplet and the matrix phase. This flow is responsible for the electrohydrodynamic stresses that lead to F , 0 and thus elongation of deformable
droplets perpendicular to the field direction (13). When this flow is absent, only those electrical stresses that are responsible for droplet elongation parallel to the electric field direction are left, and the droplets elongate in this direction. PS-Rich Droplets in the PDMS-Rich Phase in an AC Field This system was examined at frequencies ranging from 0.1 Hz to 1 MHz. When an AC field of 2 kV cm21 and 0.1 Hz was applied, the dispersed droplets of the PS-rich phase moved slowly from one electrode to another. At the same time, the droplets started and stopped rotating periodically. The different droplets all started rotating almost at the same time and stopped at exactly the same time. Looking down on the droplets from above, the rotation direction could be either clockwise or counterclockwise. In some cases, the droplets changed direction alternately with each rotation period. But this pattern was not always observed, and the droplets did not necessarily change the direction of rotation after each period. Some neighboring droplets rotated in the same direction (Fig. 7a), and some rotated in opposite directions (Fig. 7b) at the same time. Figure 7 shows the rotation of clusters of PS-rich droplets, but isolated droplets rotated also. The rotational velocity of these droplets was measured in the same manner as those that rotated in a DC field (using the analysis of Fig. 5). Figures 8 and 9 show the results of such an analysis on a single droplet. In this case, as in a DC field, the velocity of rotation became less as the solvent evaporated. The initial rotational velocity, 3.8 or 3.0 rotations s21, that is, 24 or 19 rad s21, depended on whether one used the first half rotation or the first full rotation shown in Fig. 9. The deformations of the droplets were larger than in the DC field (see also Fig. 6). The rotational velocities, obtained at 0.1 Hz and 2 kV/cm, were much larger than those obtained in DC fields of 4 and 8 kV/cm. We have used the theoretical treatment by Turcu (14), specifically devised for the rotation of single solid spheres in an AC electric field to calculate an expected rotational velocity even though our system violates one of the assumptions made by Turcu, at least in the 0.1-Hz electric field. That is, some time-dependent terms in his electric torque expression are neglected after assuming that the rotational velocity of the sphere, VAC, is much less than the radial frequency of the applied electric field, v0 5 0.63 radians s21 at 0.1 Hz. Neglecting this problem, the criteria for rotation are the same as those in DC fields, as discussed in relation to Eq. [1]. The expressions 2 5 ~E 0! thresh
and
2 h 1~1 1 2 R! 2 3 e 1t 1R|qR 2 1|
F
3 e 1x 1x 2~ t 2 2 t 1! E 20 v 0t MW V AC 5 2 h 1~ x 2 1 2 x 1!~ e 2 1 2 e 1! ~ v 0t MW! 2 1 1
[8]
G
[9]
16
KRAUSE AND CHANDRATREYA
FIG. 7. Clusters of PS-rich droplets in the PDMS-rich matrix phase in a 2-kV cm21, 0.1-Hz electric field. As shown by the curved arrows, some of the droplets rotated (a) in the same direction; others rotated (b) in the opposite direction.
were used to calculate the threshold field for rotation at 0.1 Hz and the rotational velocity. Although Eqs. [2] and [8] look very different, they give almost the same value for the threshold field for rotation in a 0.1 Hz and a DC electric field, 1.3 kV cm21. All our observations at 0.1 Hz were made at fields greater than this. Equation [9] gives a value of 1.21 rad s21 for the expected angular velocity of the droplet; that is, it predicts that the droplet will rotate more slowly in a 0.1-Hz field than in a DC field. In fact, the rotation was much faster in the 0.1-Hz field than in the DC field, as already discussed. Perhaps one should not make too much of this discrepancy between theory and experiment for three reasons: (1) the theory is for rigid spheres, whereas our droplets were not only deformable but noticeably deformed; (2) the rotational velocity of the droplets was fast enough to violate one of the assumptions of the theory; and (3) the rotational velocity predicted by the theory violates the same assumption.
Figure 8 shows that the rotation of the PS-rich droplets was not continuous at 0.1 Hz. The rotation of each droplet started and stopped at intervals; the stop arrows on Fig. 8 are just about 150 frames apart, that is, 5 seconds. A 0.1-Hz electric field reverses and becomes zero at intervals of exactly 5 seconds; it is thus reasonable to say that the droplets stopped rotating every time the field became zero. The start arrows on Fig. 8, which show the times when the droplet started rotating again, are always at a time when the droplet deformation was less than one. That is, when the droplets were not rotating, they always elongated perpendicular to the electric field direction, just as they did in DC fields. This should be expected from Eq. [6] in which the discrimination function F remains negative at all finite frequencies for this system, although it approaches zero at infinite frequency. The implication of the approach to zero of F at high frequencies is that, even though elongation perpendicular to the electric field direction is expected at all obtainable frequencies, this elongation should decrease as the frequency increases. At a higher electric field, 4 kV cm21, than that used to obtain Figs. 7–9, the droplet rotation could not be analyzed because the droplets moved very rapidly back and forth between the electrodes. At AC frequencies 1–10 Hz and 2– 4 kV cm21, the PS-rich droplets rotated from about one sixth to one quarter of a rotation before stopping and reversing the direction of rotation. At 10 Hz, the droplets moved rapidly back and forth between the electrodes, parallel to the electric field direction. When the frequency of the applied electric field was increased to 1 kHz, the droplets stopped migrating back and forth between the electrodes and did not rotate. At 1 MHz, the PS-rich droplets again moved slowly toward one electrode, the electrode opposite to that to which they migrated when the frequency was 0.1 Hz. Switching the electrical terminals of the electrodes resulted in switching the direction of migration at both frequencies. The reason for this migration in AC fields has not been explained. The droplets did not rotate at 1 MHz and 2 kV cm21, but they did rotate when the field was increased to 4 kV cm21. The rotation was usually continuous, but occasionally the rotation stopped and restarted in the opposite direction. The fluid motion outside the droplet, which could be observed by watching the dust particles in the system, simultaneously changed direction. Before the rotation began, the droplets elongated perpendicular to the electric field direction, as was also observed in DC and 0.1-Hz electric fields. At 1 MHz, the droplet deformations were very small and always parallel to the field direction while the droplets were rotating; the rotational velocity was quite rapid but could not be evaluated in the manner shown for the droplets that rotated in a DC or 0.1-Hz AC field. It is interesting to note that Eq. [9] shows clearly that solid spheres should rotate more and more slowly as the frequency of the electric field increased. In the case of a 1-MHz AC field, the calculated radial frequency is 6.3 3 1026 rad s21, much too
ELECTROROTATION OF DEFORMABLE FLUID DROPLETS
17
FIG. 8. Deformation, determined in the manner of Fig. 5, as a function of time of a PS-rich droplet in the PDMS-rich matrix phase in a 2 kV cm21, 0.1-Hz AC electric field. The arrows labeled start and stop denote the times at which the droplet started and stopped rotating.
slow to be observed by our methods. At 1 MHz, the assumption made in the derivation of Eqs. [8] and [9], that the angular velocity of the sphere is much less than the angular frequency of the applied field, is no longer violated as it is at 0.1 Hz. Nevertheless, Eq. [9] gives a result that is many orders of magnitude too slow. Thus, the theoretical treatment of electrorotation of isolated solid spheres in AC fields by Turcu (14) does not work at all for our deformable droplets. When a small amount, up to 0.2 wt%, of the diblock copolymer was added to the PS-rich phase before it was injected into the PDMS-rich phase, the droplets bounced back and forth between the electrodes in a 0.1-Hz, 2-kV cm21 electric field and rotated partially just like the droplets without block copol-
FIG. 9. Deformation, determined in the manner of Fig. 5, as a function of time of a PS-rich droplet in the PDMS-rich matrix phase in a 2-kV cm21, 0.1-Hz AC electric field during a single complete rotation of the droplet.
ymer in a 1-Hz electric field. Droplets that were not rotating had much larger elongations perpendicular to the electric field direction than those without added block copolymer; this was to be expected because the magnitude of the elongation increases as the interfacial tension between the phases decreases. An A–B block copolymer added to a phase-separated mixture containing homopolymer A, homopolymer B and a solvent acts like a surface active agent in an oil–water mixture and always decreases the interfacial tension of the system. In this complex electrohydrodynamic system, the added block copolymer probably has other effects which we have not considered here. PDMS-Rich Droplets in the PS-Rich Phase When PDMS-rich droplets were dispersed in the PS-rich matrix phase and a DC field was applied, the droplets deformed parallel to the electric field direction to form prolate ellipsoids. In this case, Eq. [6] gives the discrimination function F . 0, which indicates that the deformation is in the direction that should be expected. These droplets did not rotate, as expected from the fact that tPDMS , tPS. When a 0.1-Hz AC field was used, one could occasionally observe a peculiar phenomenon. At various times, a small droplet broke off from the PS-rich matrix phase and began to rotate within the larger deformed PDMS-rich droplet in the same way as injected PS-rich droplets rotated in the PDMSrich matrix at the same frequency of applied field. After a while, the rotating PS-rich droplet moved back to the matrix phase and reunited with it. Figure 10 shows a deformed PDMS-rich droplet and another one with a rotating PS-rich droplet inside. Although there is a considerable body of liter-
18
KRAUSE AND CHANDRATREYA
dispersed as droplets in a matrix of the PDMS-rich phase between two vertical electrodes. The droplets rotated around a vertical axis, perpendicular to the electric field direction in 4 – 8 kV cm21 DC fields; in 2– 4 kV cm21, 0.1-Hz AC fields; and in 4 kV cm21, 1-MHz AC fields. In the 0.1-Hz electric fields, the droplets stopped rotating every 5 seconds, the exact interval between the times at which the electric field becomes zero. At all frequencies, when the droplets were not rotating, they always elongated perpendicular to the electric field direction as could be predicted from the Torza et al. (13) leaky dielectric theory. The droplets rotated less than a quarter of a turn back and forth in 2– 4 kV cm21, 1–10 Hz fields, and did not rotate at all in 1-kHz fields. Rotation velocities measured in the DC field were in agreement with a theoretical treatment for electrorotation of solid spheres in DC fields by Jones (2), whereas those measured in the 0.1-Hz AC field and estimated in the 1-MHz AC field were in direct disagreement with a theoretical treatment for electrorotation of solid spheres in AC fields by Turcu (14). When the PDMS-rich droplets were dispersed into a matrix of the PS-rich phase, the droplets elongated in the field direction in a 2 kV/cm, 0.1-Hz electric field. Occasionally, a portion of the matrix phase broke off into the PDMS-rich droplet, rotated for a while, and then rejoined the matrix phase. ACKNOWLEDGMENTS This paper is based upon work partly supported by the Office of Naval Research under Grant N00014-95-0136 and partly by the National Science Foundation under Grant DMR-9521265. Many thanks are due to the Anton Paar Company for a partial subsidy for the purchase of the density meter used in this work.
REFERENCES
FIG. 10. (a) An isolated PDMS-rich droplet in the PS-rich matrix phase in a 2-kV cm21, 0.1-Hz AC field. (b) A PS-rich droplet rotating counterclockwise as seen from above in an isolated PDMS-rich droplet in the PS-rich matrix phase in the same electric field. The curved arrow shows the direction of rotation of the droplet.
ature on the breakup of fluid-dispersed phases in an electric field, as summarized, for example, by Saville (15), we are not aware of any work on the “pinching off” of portions of the continuous phase in order to become dispersed in the original dispersed phase. SUMMARY
An equilibrium phase-separated ternary system of PS/ PDMS/p-xylene was prepared, and the PS-rich phase was
1. Quincke, G., Ann. Phys. Chem. 11, 27 (1896). 2. Jones, T. B., IEEE Trans. IAS 1A-20 (4), 845 (1984). 3. Jones, T. B., “Electromechanics of Particles.” Cambridge Univ. Press, New York, 1995. 4. Teixeira-Pinto, A. A., Nejelski, L. L., Cutler, J. L., and Heller, J. H., Exp. Cell. Res. 20, 548 (1960). 5. Zimmermann, U., Vienken, J., and Pilwat, G., Z. Naturforsch. 36c, 173 (1981). 6. Pohl, H. A., and Crane, J. S., Biophys. J. 11, 711 (1971). 7. Holzapfel, C., Vienken, J., and Zimmermann, U., J. Membr. Bio. 67, 13 (1982). 8. Hu, Y., Glass, J. L., Griffith, A. E., and Fraden, S., J. Chem. Phys. 100, 4674 (1994). 9. Hu, Y., Fraden, S., Glass, J. L., and Wenner, L. E., Mat. Res. Soc. Symp. Proc. 289, 25 (1993). 10. Venugopal, G., and Krause, S., Macromolecules 25, 4626 (1992). 11. Xi, K., and Krause, S., Macromolecules 31, 3974 (1998). 12. Nishiwaki, T., Adachi, K., and Kotaka, T., Langmuir 4, 170 (1988). 13. Torza, S., Cox, R. G., and Mason, S. G., Phil. Trans. R. Soc. London 269, 259 (1971). 14. Turcu, I., J. Phys. A Math. Gen. 20, 3301 (1987). 15. Saville, D. A., Ann. Rev. Fluid Mech. 29, 27 (1997).
206, 10 –18 (1998)
CS985716
Electrorotation of Deformable Fluid Droplets Sonja Krause1 and Preeti Chandratreya2 Rensselaer Polytechnic Institute, Department of Chemistry, Troy, New York 12180 Received February 5, 1997; accepted June 19, 1997
consider what happens when such a sphere, suspended in a liquid in a uniform electric field, suffers a slight rotational perturbation. Jones (2, 3) has discussed the conditions under which this perturbation results in a stable rotation in a DC field. Assuming that the sphere is isotropic, one considers the charge relaxation time in the solid sphere t2 and in the liquid t1 such that
An equilibrium phase separated ternary system of polystyrene (PS)/polydimethylsiloxane (PDMS)/p-xylene was prepared, and the PS-rich phase was dispersed as droplets in a matrix of the PDMS-rich phase. The system was placed between vertical electrodes and the droplets rotated around a vertical axis perpendicular to the electric field direction in 4 – 8-kV cm21 DC fields; in 2– 4-kV cm21, 0.1-Hz AC fields; and in 4-kV cm21, 1-MHz AC fields, in some cases stopping, restarting, and changing the direction of rotation. They rotated less than a quarter of a turn back and forth in 1–10-Hz, 2– 4-kV cm21 fields and did not rotate at all in 1-kHz fields. Rotational velocities measured in the DC field were in agreement with an existing theory; those measured in the 0.1-Hz AC field and estimated in the 1-MHz AC field were in direct disagreement with a different existing theoretical treatment. When the PDMS-rich droplets were dispersed into a matrix of the PS-rich phase, the droplets elongated in the field direction in a 2-kV/cm, 0.1-Hz electric field. Occasionally a portion of the matrix phase broke off into the PDMS-rich droplet, rotated for a while, and then rejoined the matrix phase. © 1998 Academic Press Key Words: polystyrene; polydimethylsiloxane; electrorotation.
t1 5
e1 x1
and
e2 , x2
[1]
where e1 and e2 are the permittivities and x1 and x2 are the conductivities of the liquid and the solid sphere, respectively. When the charge relaxation time of the solid is less than that of the liquid, that is, t2 , t1, a normal dipole is generated in the sphere, that is, a negative charge is generated on the surface of the sphere nearest the positive electrode, whereas a positive charge is generated on the surface of the sphere nearest the negative electrode. These are stable to small rotational perturbations; that is, the sphere will return to its original orientation after the perturbation. On the other hand, when t2 . t1, that is, when the relaxation time of the solid particle is greater than that of the liquid, then a reverse dipole is generated on the sphere; these are unstable to small rotational perturbations, and the sphere will continue to rotate in a DC field if it is above a threshold value (E 0 ) thresh. Equation [2] is different from either of the equations shown by Jones (2, 3); those equations are dimensionally inaccurate. We used the derivation shown in Ref. 3 to develop the equation
INTRODUCTION
Electrorotation of solid spherical objects suspended in liquids has been discovered and rediscovered for over a century. One of the earliest studies was reported by Quincke (1) in 1896. He suspended macroscopic objects, many of them spherical, made of different materials like quartz, topaz, tourmaline, flint glass, and sulfur, in different solvents using a string or wire and applied a DC electric field that could be as high as 2 kV/cm. The spheres rotated back and forth in the applied fields with the axis of rotation vertical and perpendicular to the field direction when the electrodes were vertical; the spheres could not complete their rotations because of the torsion in the string or wire on which they were suspended. Rotation was observed in different solvents and was reproducible at different times, but it was not observed when the spheres were suspended in air. In order to explain this “Quincke rotation” in the case of a particle as symmetrical as a sphere, it has been necessary to
2 5 ~E 0! thresh
4 3
F
S
x2 2e1 11 e2 2x1 t1 e1 1 2 t t 2 MW
h1 1 1
S
D
DG
,
[2]
where tMW is the Maxwell–Wagner interfacial polarization relaxation time
1
To whom correspondence should be addressed. Present address: Raychem Corporation, 300 Constitution Drive, MS 122/ 6403, Menlo Park, CA 94025.
t MW 5
2
0021-9797/98 $25.00 Copyright © 1998 by Academic Press All rights of reproduction in any form reserved.
t2 5
10
e2 1 2e1 . x2 1 2x1
[3]
ELECTROROTATION OF DEFORMABLE FLUID DROPLETS
The angular velocity of rotation of the spheres in a DC field VDC is then V DC 5
1
t MW
Î
E 20 21 2 ~E 0! thresh
[4]
The electrorotation of very heterogeneous spherical objects like biological cells in a uniform AC electric field was observed much later than Quincke rotation in a uniform DC field; it occurs only if two cells are close to each other or if a single cell is near an electrode. Among the first to note biological cell electrorotation were Teixeira-Pinto et al. (4) who were studying Euglena and amoebae. Among later observers were Zimmerman et al. (5), who observed electrorotation of various kinds of biological cells such as mesophyll protoplast cells of Avena sativa (at 20–40 kHz), erythrocytes and ghost cells (at 80–100 kHz), and yeast cells (at 140–180 kHz). Electrorotation of biological cells was attributed by Pohl and Crane (6) to either deposition of charges on the cell surfaces or interaction of the alternating field with the oscillation of dipoles within the cells. On the other hand, Holzapfel et al. (7), who also studied the rotation of protoplasts of A. Sativa, concluded that the rotation was caused by the interactions of dipole fields generated in each cell by the applied electric field. If two such cells are close to each other, each is affected by the neighboring dipole field as well as by the external AC field. In such cases, a torque will be exerted by each cell on the other, resulting in rotation whose velocity depends on the relative positions of the cells. Rotation of individual cells, with no other cells in the vicinity, was not observed. Recently, Hu et al. (8, 9) observed electrorotation of monodisperse polystyrene spheres in a dilute aqueous electrolyte solution in 0.2–3 MHz, electric fields in the kilovolts per centimeter range. The polystyrene spheres rotated when they were close to each other in bands aligned at an angle of 645° relative to the electric field direction. The rotation of the polystyrene spheres was attributed to the existence of a phase lag between the applied field and the induced dipole moments in the spheres; this phase lag causes neighboring particles to exert on each other torques that cause the spinning of the spherical particles. Thus, there appear to be two sorts of rotation mechanisms for spherical objects in electric fields, one for the rotation of isolated spheres (Quincke rotation) in different solvents in DC fields and at least one other for the rotation of biological cells and charged polystyrene spheres that are close to each other in aqueous solution in AC fields. In the course of other work on the effect of electric fields on dispersions of deformable liquid droplets in a nonaqueous medium, we have observed the deformation and rotation of isolated droplets in both DC and AC electric fields. MATERIALS AND METHODS
Polystyrene (PS) was obtained from the Dow Chemical Company, Midland, MI; polydimethysiloxane (PDMS) was
11
obtained from Gelest, Inc., Tullytown, PA; and a block copolymer, PS-b-PDMS was obtained from T.W. Smith, Xerox Corporation, Rochester, NY. Polystyrene standards for GPC calibration were from Scientific Polymer Products, Inc., Ontario, NY, and Arro Laboratories, Inc., Joliet, IL. HPLC grade p-xylene and deuterated chloroform (99.8 atom % D with no internal reference) were from Aldrich Chemical Company, Milwaukee, WI. Coexisting phases in the PS/PDMS/p-xylene system were prepared by first dissolving the two polymers individually in p-xylene in the range 15–20% wt/vol. The two polymer solutions were then mixed in various proportions to give a turbid two-phase mixture. After shaking the turbid mixture vigorously, it was allowed to reach equilibrium over a period of 6 – 8 weeks at 30°C until two clear phases had separated. The molecular weights and molecular weight distributions of the homopolymers were measured using a Viscotek Model 100 gel-permeation chromatograph (GPC) from the Viscotek Corporation, Houston, TX. This GPC used two linear (universal) columns from American Polymer Standards Corporation, Cleveland, OH, and had two detectors connected in series, a Viscotek differential viscometer and a Knauer Model 198 (Knauer, GmbH, Germany) differential refractometer, which were interfaced with a computer to analyze the data using the UNICAL GPC software provided by the Viscotek Corporation. Polystyrene standards were used to calibrate the instrument which was then capable of measuring true molecular weight distributions of the polymers. Densities were measured using a DMA 48 Density Meter, PAAR, Graz, Austria. The interfacial tensions between the coexisting phases were measured at 30 6 0.5°C using a SITE 04 Spinning Drop Interfacial Tensiometer, Kru¨ss USA, Ridge, NY. The dielectric constants of the two coexisting phases were measured at 1 kHz at room temperature using a 1260 Impedance/Gain-Phase Analyzer, Solartron, Inc., Allentown, PA. The conductivities of the two phases were obtained by measuring their resistances using a Keithley Electrometer Model 6517, Keithley Instruments, Inc., Cleveland, OH. The viscosities of the coexisting phases were measured at 30 6 0.5°C using a Brookfield Model DV II1 Digital Cone/Plate Viscometer, Brookfield Engineering Laboratories, Inc., Stoughton, MA. The cone spindles used were CP-40 which can measure the viscosity range of 0.30 –1,028 cP and CP-51 which measures the viscosity range of 4.85–16,180 cP. The compositions of the coexisting phases were obtained by evaporating the solvent from a weighed sample of each phase to constant weight, dissolving the resulting film in deuterated chloroform (CDCl3) and then determining the relative wt % of the polymers in each phase by 1H NMR. The NMR measurements were performed on a 200-MHz Varian V1-200 Fourier Transform Spectrometer. The 6.2–7.4 ppm aromatic peak of PS and the 0–0.2 ppm CH3 peak of PDMS were used for the calculations. The experimental cell for electric field experiments was a microscope glass slide with copper electrodes glued on using
12
KRAUSE AND CHANDRATREYA
FIG. 1. The experimental cell with electrodes and sample on the microscope stage (a) as viewed from above and (b) as viewed from the side.
commercially available “super glue.” The gap between the electrodes was ;0.5 cm in all experiments and was measured using a reticule attached to the eyepiece of an optical microscope, a Leitz Laborlux 12 Pol (Leica, St. Gallen, Switzerland). The cell was placed on a Teflon base and held by two metal clips which also provided electrical connections between the electrodes and the power supply. A small amount of one of the coexisting phases was placed between the electrodes using a syringe, and then small droplets of the other phase were injected into the phase already between the electrodes using a syringe with a very small needle. A glass window was used as a lid to cover the cell to slow down the evaporation rate as well as prevent the solvent vapors from damaging the optics of the microscope. Schematics of the cell and experimental setup are shown in Figs. 1 and 2. The experiments were carried out on the optical microscope stage usually using a 43 magnification objective lens, sometimes a 323 lens. The eyepiece magnification was always 103. A Hamamatsu CCD camera (Hamamatsu Corporation, Bridgewater, NJ) was attached to the microscope, connected to a Sony VCR, and interfaced with a Pentium PC. The experiments were taped, and single frames (30
FIG. 2.
frames to a second) were later analyzed using Optimas Image Analysis Software (Optimas Corporation, Bothell, WA) versions 4.0, 5.2, and 6.0 at various times. A Trek model 610C high-voltage power supply/amplifier from Trek, Inc., Medina, NY, was used to provide both DC and AC voltages. It has a DC output voltage range of 0 –10 kV at a current of 0 –2 mA. An SRS Model DS335 Synthesized Function Generator, Stanford Research Systems, Sunnyvale, CA, was connected to the Trek 610C to provide an AC voltage signal, 0.1 Hz to 1 MHz, 0 –10 kV. RESULTS AND DISCUSSION
Polymers and Coexisting Phases The PS sample used had a weight average molecular weight, M w 5 2.0 3 10 5 , M w/M n 5 2.9 (M n is the number average molecular weight), whereas the PDMS had M w 5 2.0 3 10 3 , M w/M n 5 1.2. The block copolymer had a PS block molecular weight of 5.0 3 104, and a PDMS block molecular weight of
The overall experimental setup for electrorotation studies.
ELECTROROTATION OF DEFORMABLE FLUID DROPLETS
13
FIG. 4. An isolated PS-rich droplet in the PDMS-rich matrix phase in a 5-kV cm21 DC field. The counterclockwise direction of rotation as seen from above is shown by the curved arrow.
FIG. 3. Ternary phase diagram of PS/PDMS/p-xylene. Two experimentally determined tie lines are shown; the dotted curve indicates the approximate position of the binodal. The two small points above the binodal denote mixtures that remained clear, single-phase solutions.
1.0 3 104, as measured by the supplier. Compositions of two sets of coexisting phases were obtained and these, together with the compositions of some single-phase solutions, were used to construct the partial phase diagram shown in Fig. 3. One may note that the PS-rich phase, in each case, contained some PDMS, but the PDMS-rich phase contained no PS within experimental error. This is characteristic of systems that contain a high polymer, in this case the PS, and an oligomer, in this case the PDMS. The measured properties of the coexisting phases used for the electric field experiments are shown in Table 1. The very low value of the interfacial tension between the coexisting phases was to be expected because these coexisting phases had compositions close to that of the critical point in the phase diagram (Fig. 3). It is well known that the interfacial TABLE 1 Measured Properties of the Coexisting Phases Used for Electric Field Experiments Property Dielectric constant Conductivity Viscosity Density Interfacial tension a b
PS-rich phase
PDMS-rich phase
2.3 2.4 2.3 3 10211 2.4 3 10211 92.6a 19.2b 0.8955 0.8744 2.1 3 1025
Units
V21 m21 cP g cm23 N m21
This viscosity was constant up to a shear rate of 192 s21. This viscosity was almost constant up to a shear rate of 384 s21.
tension between coexisting phases decreases as the tie lines move closer to the critical point because the interfacial tension must vanish at the critical point. Furthermore, we have noted similar low interfacial tensions between coexisting phases in polymer/polymer/solvent systems in previous work from this laboratory (10, 11). PS-Rich Droplets in the PDMS-Rich Phase in a DC Field When a DC field up to 8 kV cm21 was applied, the droplets almost always rotated continuously in the counterclockwise direction between vertical electrodes as seen from above. A change in the rotation direction was observed only when a droplet first started rotating in the clockwise direction; then it eventually stopped and began rotating counterclockwise. The droplets slowly migrated toward the positive electrode and then spread over the electrode surface. When the field was turned off, sections of the droplets retracted and regained their spherical shape. If the field was reapplied at this point, the droplets were thrown off the electrode surface, the rotations were restarted, and the droplets again migrated slowly toward the positive electrode. Thus, it is obvious that the droplets had a slight negative charge. Because the droplets deformed slightly in the electric field, it was possible to measure their rotational velocity by following a single droplet over time using a frame-by-frame analysis of the videotape recording of the droplet rotation. Figure 4 shows a single droplet during rotation in a DC field. One may note that the droplet is only very slightly deformed so that it was imperative to use image analysis to follow the inclination of its longest axis with respect to the applied field direction. Actually, it was easier to follow the rotation using the analysis shown in Fig. 5. A pseudo-deformation, called here simply deformation, defined by the length of the largest droplet di-
14
KRAUSE AND CHANDRATREYA
FIG. 5. Cartoon of droplet rotation and the X and Y axes whose lengths were measured to calculate the deformation X/Y as a function of time for calculation of the angular rotational velocity of a droplet.
mension in the field direction X divided by the length of the largest droplet deformation perpendicular to the field direction Y, was measured and plotted versus time frame, as shown in Fig. 6. At 4 kV cm21, as shown in Fig. 6, the droplets rotated at about 0.75 rotations s21 5 4.7 radians s21 in the initial stages of the observations. One may note that the rotational velocity decreased with time; this is because the solvent slowly evaporated during the course of the experiment. This caused an increase in the viscosity of the matrix phase which then slowed the rotational velocity. At 8 kV cm21, the droplets initially rotated at a velocity of approximately 1.5 rotations s21 5 9.4 radians s21. If the direction and magnitude of the elongation of the droplets could have changed very rapidly with respect to the angular velocity of the droplets, we could not have evaluated their angular velocity in the manner shown in Fig. 5. It is possible to estimate the relaxation time for elongation t using the relationship found by Nishiwaki et al. (12):
t5
r 0~2 h 2 1 h 1! g
[5]
where r 0 is the original radius of the spherical droplet and g is the interfacial tension between the two phases. All parameters except r 0 can be found in Table 1. Because our droplets had various dimensions, we shall use a range from 10 –100 mm for r 0 . Using these parameters, we obtain a range of 0.1–1 s for t. Because these values have the same order of magnitude as the speed of rotation of the droplets, it is quite likely that small changes in droplet elongation occurred during the rotation of the droplets, but these changes did not reach steady state. Thus, the droplets rotated faster than they could change their deformation, and we were able to determine the rotational velocity of our droplets. In order to compare the rotational data observed for our deformable PS-rich droplets with the predictions for Quincke rotation of hard spheres using Eqs. [1], [2], and [4], we again used the data shown in Table 1. Equation [1] leads to tPS 5 0.88 s21 and tPDMS 5 0.49 s21, where the PS subscript and the PDMS subscript refer to the PS-rich and the PDMS-rich phase, respectively. Because tPS . tPDMS, Quincke rotation can be expected when the PS-rich droplets are injected into the PDMS-rich phase. Using the tMW 5 0.58 s21 calculated using Eq. [3], Eq. [2] predicts a threshold electric field for rotation, (E 0 ) thresh 5 1.3 kV cm21. This is consistent with our observations, especially since no rotation was observed in this system at 500 V cm21. Table 2 shows that our deformable droplets obeyed the predictions for Quincke rotation from Eq. [4] within our rather large experimental error in spite of the fact that the droplets had a slightly ellipsoidal shape while rotating. Figure 6 shows that the droplets appeared to be slightly elongated in the direction of the electric field at most times during the rotation. It is possible to calculate the direction in
FIG. 6. Deformation, as determined in the manner of Fig. 5, as a function of time of a PS-rich droplet in the PDMS-rich matrix phase in a 4-kV cm21 DC electric field.
15
ELECTROROTATION OF DEFORMABLE FLUID DROPLETS
TABLE 2 Experimental and Calculated Rotational Velocities of the PSRich Droplets in the PDMS-Rich Matrix in DC Electric Fields Rotational velocity (rad s21) Electric field (kV cm21)
Experimental
Calculated from Eq. [4]
4 8
4.7 8.7
5.0 10.5
which nonrotating droplets are expected to elongate in a DC field using the leaky dielectric theory of Torza et al. (13), in which a discrimination function F is used to determine whether a droplet suspended in a matrix will elongate in a direction parallel to the applied field direction, forming a prolate ellipsoid, or in a direction perpendicular to the applied field direction, forming an oblate ellipsoid. This discrimination function, written here for AC as well as DC fields, is given by R~11 l 1 14! 1 R 2@15~ l 1 1! 1 q~19 l 1 16!# 1 15a 2v 20~1 1 l !~1 1 2q! , F512 5~1 1 l !@~2 R 1 1! 2 1 a 2v 20~q 1 2! 2# [6] where R5
x1 , x2
q5
e2 , e1
l5
h2 , h1
[7]
and v0 is the frequency of the applied electric field, zero in a DC field. When F . 0, one expects elongation parallel to the field direction (a prolate ellipsoid); F 5 0, one expects an undeformed drop (drop remains spherical); F , 0, one expects elongation perpendicular to the field direction (an oblate ellipsoid). In the present case, F , 0 so that elongation perpendicular to the applied field direction is predicted, even though this was not often observed during rotation. Elongation perpendicular to the applied field direction was, however, observed (1) after all the solvent had evaporated and (2) just before counterclockwise rotation set in after a droplet that started with clockwise rotation had stopped rotating. That is, elongation perpendicular to the applied field direction occurred only when the droplets were not rotating. The virtual absence of perpendicular elongation during rotation is probably connected to the fact that the rotation of the droplet interferes with the electrohydrodynamic flow along the interface between the droplet and the matrix phase. This flow is responsible for the electrohydrodynamic stresses that lead to F , 0 and thus elongation of deformable
droplets perpendicular to the field direction (13). When this flow is absent, only those electrical stresses that are responsible for droplet elongation parallel to the electric field direction are left, and the droplets elongate in this direction. PS-Rich Droplets in the PDMS-Rich Phase in an AC Field This system was examined at frequencies ranging from 0.1 Hz to 1 MHz. When an AC field of 2 kV cm21 and 0.1 Hz was applied, the dispersed droplets of the PS-rich phase moved slowly from one electrode to another. At the same time, the droplets started and stopped rotating periodically. The different droplets all started rotating almost at the same time and stopped at exactly the same time. Looking down on the droplets from above, the rotation direction could be either clockwise or counterclockwise. In some cases, the droplets changed direction alternately with each rotation period. But this pattern was not always observed, and the droplets did not necessarily change the direction of rotation after each period. Some neighboring droplets rotated in the same direction (Fig. 7a), and some rotated in opposite directions (Fig. 7b) at the same time. Figure 7 shows the rotation of clusters of PS-rich droplets, but isolated droplets rotated also. The rotational velocity of these droplets was measured in the same manner as those that rotated in a DC field (using the analysis of Fig. 5). Figures 8 and 9 show the results of such an analysis on a single droplet. In this case, as in a DC field, the velocity of rotation became less as the solvent evaporated. The initial rotational velocity, 3.8 or 3.0 rotations s21, that is, 24 or 19 rad s21, depended on whether one used the first half rotation or the first full rotation shown in Fig. 9. The deformations of the droplets were larger than in the DC field (see also Fig. 6). The rotational velocities, obtained at 0.1 Hz and 2 kV/cm, were much larger than those obtained in DC fields of 4 and 8 kV/cm. We have used the theoretical treatment by Turcu (14), specifically devised for the rotation of single solid spheres in an AC electric field to calculate an expected rotational velocity even though our system violates one of the assumptions made by Turcu, at least in the 0.1-Hz electric field. That is, some time-dependent terms in his electric torque expression are neglected after assuming that the rotational velocity of the sphere, VAC, is much less than the radial frequency of the applied electric field, v0 5 0.63 radians s21 at 0.1 Hz. Neglecting this problem, the criteria for rotation are the same as those in DC fields, as discussed in relation to Eq. [1]. The expressions 2 5 ~E 0! thresh
and
2 h 1~1 1 2 R! 2 3 e 1t 1R|qR 2 1|
F
3 e 1x 1x 2~ t 2 2 t 1! E 20 v 0t MW V AC 5 2 h 1~ x 2 1 2 x 1!~ e 2 1 2 e 1! ~ v 0t MW! 2 1 1
[8]
G
[9]
16
KRAUSE AND CHANDRATREYA
FIG. 7. Clusters of PS-rich droplets in the PDMS-rich matrix phase in a 2-kV cm21, 0.1-Hz electric field. As shown by the curved arrows, some of the droplets rotated (a) in the same direction; others rotated (b) in the opposite direction.
were used to calculate the threshold field for rotation at 0.1 Hz and the rotational velocity. Although Eqs. [2] and [8] look very different, they give almost the same value for the threshold field for rotation in a 0.1 Hz and a DC electric field, 1.3 kV cm21. All our observations at 0.1 Hz were made at fields greater than this. Equation [9] gives a value of 1.21 rad s21 for the expected angular velocity of the droplet; that is, it predicts that the droplet will rotate more slowly in a 0.1-Hz field than in a DC field. In fact, the rotation was much faster in the 0.1-Hz field than in the DC field, as already discussed. Perhaps one should not make too much of this discrepancy between theory and experiment for three reasons: (1) the theory is for rigid spheres, whereas our droplets were not only deformable but noticeably deformed; (2) the rotational velocity of the droplets was fast enough to violate one of the assumptions of the theory; and (3) the rotational velocity predicted by the theory violates the same assumption.
Figure 8 shows that the rotation of the PS-rich droplets was not continuous at 0.1 Hz. The rotation of each droplet started and stopped at intervals; the stop arrows on Fig. 8 are just about 150 frames apart, that is, 5 seconds. A 0.1-Hz electric field reverses and becomes zero at intervals of exactly 5 seconds; it is thus reasonable to say that the droplets stopped rotating every time the field became zero. The start arrows on Fig. 8, which show the times when the droplet started rotating again, are always at a time when the droplet deformation was less than one. That is, when the droplets were not rotating, they always elongated perpendicular to the electric field direction, just as they did in DC fields. This should be expected from Eq. [6] in which the discrimination function F remains negative at all finite frequencies for this system, although it approaches zero at infinite frequency. The implication of the approach to zero of F at high frequencies is that, even though elongation perpendicular to the electric field direction is expected at all obtainable frequencies, this elongation should decrease as the frequency increases. At a higher electric field, 4 kV cm21, than that used to obtain Figs. 7–9, the droplet rotation could not be analyzed because the droplets moved very rapidly back and forth between the electrodes. At AC frequencies 1–10 Hz and 2– 4 kV cm21, the PS-rich droplets rotated from about one sixth to one quarter of a rotation before stopping and reversing the direction of rotation. At 10 Hz, the droplets moved rapidly back and forth between the electrodes, parallel to the electric field direction. When the frequency of the applied electric field was increased to 1 kHz, the droplets stopped migrating back and forth between the electrodes and did not rotate. At 1 MHz, the PS-rich droplets again moved slowly toward one electrode, the electrode opposite to that to which they migrated when the frequency was 0.1 Hz. Switching the electrical terminals of the electrodes resulted in switching the direction of migration at both frequencies. The reason for this migration in AC fields has not been explained. The droplets did not rotate at 1 MHz and 2 kV cm21, but they did rotate when the field was increased to 4 kV cm21. The rotation was usually continuous, but occasionally the rotation stopped and restarted in the opposite direction. The fluid motion outside the droplet, which could be observed by watching the dust particles in the system, simultaneously changed direction. Before the rotation began, the droplets elongated perpendicular to the electric field direction, as was also observed in DC and 0.1-Hz electric fields. At 1 MHz, the droplet deformations were very small and always parallel to the field direction while the droplets were rotating; the rotational velocity was quite rapid but could not be evaluated in the manner shown for the droplets that rotated in a DC or 0.1-Hz AC field. It is interesting to note that Eq. [9] shows clearly that solid spheres should rotate more and more slowly as the frequency of the electric field increased. In the case of a 1-MHz AC field, the calculated radial frequency is 6.3 3 1026 rad s21, much too
ELECTROROTATION OF DEFORMABLE FLUID DROPLETS
17
FIG. 8. Deformation, determined in the manner of Fig. 5, as a function of time of a PS-rich droplet in the PDMS-rich matrix phase in a 2 kV cm21, 0.1-Hz AC electric field. The arrows labeled start and stop denote the times at which the droplet started and stopped rotating.
slow to be observed by our methods. At 1 MHz, the assumption made in the derivation of Eqs. [8] and [9], that the angular velocity of the sphere is much less than the angular frequency of the applied field, is no longer violated as it is at 0.1 Hz. Nevertheless, Eq. [9] gives a result that is many orders of magnitude too slow. Thus, the theoretical treatment of electrorotation of isolated solid spheres in AC fields by Turcu (14) does not work at all for our deformable droplets. When a small amount, up to 0.2 wt%, of the diblock copolymer was added to the PS-rich phase before it was injected into the PDMS-rich phase, the droplets bounced back and forth between the electrodes in a 0.1-Hz, 2-kV cm21 electric field and rotated partially just like the droplets without block copol-
FIG. 9. Deformation, determined in the manner of Fig. 5, as a function of time of a PS-rich droplet in the PDMS-rich matrix phase in a 2-kV cm21, 0.1-Hz AC electric field during a single complete rotation of the droplet.
ymer in a 1-Hz electric field. Droplets that were not rotating had much larger elongations perpendicular to the electric field direction than those without added block copolymer; this was to be expected because the magnitude of the elongation increases as the interfacial tension between the phases decreases. An A–B block copolymer added to a phase-separated mixture containing homopolymer A, homopolymer B and a solvent acts like a surface active agent in an oil–water mixture and always decreases the interfacial tension of the system. In this complex electrohydrodynamic system, the added block copolymer probably has other effects which we have not considered here. PDMS-Rich Droplets in the PS-Rich Phase When PDMS-rich droplets were dispersed in the PS-rich matrix phase and a DC field was applied, the droplets deformed parallel to the electric field direction to form prolate ellipsoids. In this case, Eq. [6] gives the discrimination function F . 0, which indicates that the deformation is in the direction that should be expected. These droplets did not rotate, as expected from the fact that tPDMS , tPS. When a 0.1-Hz AC field was used, one could occasionally observe a peculiar phenomenon. At various times, a small droplet broke off from the PS-rich matrix phase and began to rotate within the larger deformed PDMS-rich droplet in the same way as injected PS-rich droplets rotated in the PDMSrich matrix at the same frequency of applied field. After a while, the rotating PS-rich droplet moved back to the matrix phase and reunited with it. Figure 10 shows a deformed PDMS-rich droplet and another one with a rotating PS-rich droplet inside. Although there is a considerable body of liter-
18
KRAUSE AND CHANDRATREYA
dispersed as droplets in a matrix of the PDMS-rich phase between two vertical electrodes. The droplets rotated around a vertical axis, perpendicular to the electric field direction in 4 – 8 kV cm21 DC fields; in 2– 4 kV cm21, 0.1-Hz AC fields; and in 4 kV cm21, 1-MHz AC fields. In the 0.1-Hz electric fields, the droplets stopped rotating every 5 seconds, the exact interval between the times at which the electric field becomes zero. At all frequencies, when the droplets were not rotating, they always elongated perpendicular to the electric field direction as could be predicted from the Torza et al. (13) leaky dielectric theory. The droplets rotated less than a quarter of a turn back and forth in 2– 4 kV cm21, 1–10 Hz fields, and did not rotate at all in 1-kHz fields. Rotation velocities measured in the DC field were in agreement with a theoretical treatment for electrorotation of solid spheres in DC fields by Jones (2), whereas those measured in the 0.1-Hz AC field and estimated in the 1-MHz AC field were in direct disagreement with a theoretical treatment for electrorotation of solid spheres in AC fields by Turcu (14). When the PDMS-rich droplets were dispersed into a matrix of the PS-rich phase, the droplets elongated in the field direction in a 2 kV/cm, 0.1-Hz electric field. Occasionally, a portion of the matrix phase broke off into the PDMS-rich droplet, rotated for a while, and then rejoined the matrix phase. ACKNOWLEDGMENTS This paper is based upon work partly supported by the Office of Naval Research under Grant N00014-95-0136 and partly by the National Science Foundation under Grant DMR-9521265. Many thanks are due to the Anton Paar Company for a partial subsidy for the purchase of the density meter used in this work.
REFERENCES
FIG. 10. (a) An isolated PDMS-rich droplet in the PS-rich matrix phase in a 2-kV cm21, 0.1-Hz AC field. (b) A PS-rich droplet rotating counterclockwise as seen from above in an isolated PDMS-rich droplet in the PS-rich matrix phase in the same electric field. The curved arrow shows the direction of rotation of the droplet.
ature on the breakup of fluid-dispersed phases in an electric field, as summarized, for example, by Saville (15), we are not aware of any work on the “pinching off” of portions of the continuous phase in order to become dispersed in the original dispersed phase. SUMMARY
An equilibrium phase-separated ternary system of PS/ PDMS/p-xylene was prepared, and the PS-rich phase was
1. Quincke, G., Ann. Phys. Chem. 11, 27 (1896). 2. Jones, T. B., IEEE Trans. IAS 1A-20 (4), 845 (1984). 3. Jones, T. B., “Electromechanics of Particles.” Cambridge Univ. Press, New York, 1995. 4. Teixeira-Pinto, A. A., Nejelski, L. L., Cutler, J. L., and Heller, J. H., Exp. Cell. Res. 20, 548 (1960). 5. Zimmermann, U., Vienken, J., and Pilwat, G., Z. Naturforsch. 36c, 173 (1981). 6. Pohl, H. A., and Crane, J. S., Biophys. J. 11, 711 (1971). 7. Holzapfel, C., Vienken, J., and Zimmermann, U., J. Membr. Bio. 67, 13 (1982). 8. Hu, Y., Glass, J. L., Griffith, A. E., and Fraden, S., J. Chem. Phys. 100, 4674 (1994). 9. Hu, Y., Fraden, S., Glass, J. L., and Wenner, L. E., Mat. Res. Soc. Symp. Proc. 289, 25 (1993). 10. Venugopal, G., and Krause, S., Macromolecules 25, 4626 (1992). 11. Xi, K., and Krause, S., Macromolecules 31, 3974 (1998). 12. Nishiwaki, T., Adachi, K., and Kotaka, T., Langmuir 4, 170 (1988). 13. Torza, S., Cox, R. G., and Mason, S. G., Phil. Trans. R. Soc. London 269, 259 (1971). 14. Turcu, I., J. Phys. A Math. Gen. 20, 3301 (1987). 15. Saville, D. A., Ann. Rev. Fluid Mech. 29, 27 (1997).