Langmuir 1989, 5 , 376-384
376
3.5 for sodium oleate a t concentrations ranging from m~l-dm-~. to 5 x The measured isoelectric points of the bubbles were consistently very close to the isoelectric points of the free amine or free acid precipitate, which appear in alkaline solutions for amine and in acid solutions for fatty acids.
This indicates the electric charge of bubbles in such systems results from the armoring of bubbles by the partially hydrophobic surfactant precipitate. Registry No. Dodecylamine hydrochloride, 929-73-7;lauric acid, 143-07-7;sodium oleate, 143-19-1;sodium dodecyl sulfate, 151-21-3;sodium dodecanesulfonate, 2386-53-0.
Electrified Droplet Fission and the Rayleigh Limit Daniel C. Taflin, Timothy L. Ward, and E. James Davis* Department of Chemical Engineering, BF-IO, University of Washington, Seattle, Washington 98195 Received May 20, 1988. I n Final Form: October 17, 1988 The stability of charged evaporating droplets and the characteristics of their rupture have been investigated by trapping a single microdroplet in the superposed ac and dc electrical fields of an electrodynamic balance. By use of the method of optical resonance spectroscopy to measure the size precisely (to better than 1 part in lo4) before and after an explosion and using suspension voltage measurements to determine the charge, droplet fission was examined in ionized and unionized gaseous media. Droplet explosions were observed to occur a t lower charge densities than predicted by the classical Rayleigh theory. For several organic liquids studied in unionized air, the droplet exploded approximately 20% below the predicted limit of surface charge. The observed mass loss varied from about 1.0% to 2.3%, while the charge loss ranged from 10% to 18%. This is the first reported precise measurement of such small mass loss. In a C14contaminated balance chamber (partially ionized gas atmosphere), the explosion was also premature, but the mass loss and charge loss varied greatly. An analysis of the surface charge variations associated with thermal fluctuations was performed to determine if such fluctuations might cause the droplet to explode before the Rayleigh limit was reached, but it appears that the external electrical fields complicate the fission process and have a greater effect on the instability than either possible surface contamination or thermal fluctuations in charge.
Introduction Electrified droplets are encountered in the atmosphere, are involved in metallic powder production, ink-jet printing, microspray lubrication, emulsification, spray painting, and crop spraying, and occur in a variety of industrial processes. Bailey’ recently reviewed the theory and practice of electrostatic spraying. The characteristics of the fission of electrified droplets are of interest in these applications and in fuel droplet breakup. An evaporating, electrically charged drop becomes unstable when the repulsive electrostatic force overcomes the attractive surface tension force. The drop then ruptures, sending out one or more daughter droplets and leaving behind a stable residual drop. Lord Rayleigh2 first derived the criterion of instability for a conducting droplet of radius a and charge q , obtaining q2 = 64n2toya3
(1)
where eo is the permittivity of free space and y is the surface tension of the droplet. Several experimental studies have attempted to verify this prediction. Hendricks3 in 1962 produced charged oil drops and accelerated them through a large potential to measure size and charge. In nearly all cases the charge on a droplet was near or below the Rayleigh limit. Observations of exploding single droplets were made by Doyle, Moffett, and Vonnegut4 using a Millikan-type apparatus. (1) Bailey, A. G. Atomisation Spray Technol. 1986, 2, 95. (2) Rayleigh, Lord Philos. Mag. 1882, 14, 184. (3) Hendricks, C. D., Jr. J . Colloid Sci. 1962, 17, 249. (4) Doyle, A,; Moffett, D. R.; Vonnegut, B. J . Colloid Sci. 1964,19, 136.
0743-7463/S9/2405-0376$01,50/0
They reported that the Rayleigh limit was confirmed for droplets of radius 30-100 pm; however, their method of size determination was crude and resulted in deviations as high as 2C-25%. They further reported that at breakup the parent drop ejected a cloud of very small droplets which carried away about 30% of its charge but relatively little mass. Abbas and Latham5 also used a Millikan condenser together with aerodynamic drag measurements to obtain the size before and after breakup. They reported confirmation of the Rayleigh limit with about scatter. The charge loss was found to be about 25% and the mass loss 20-30%. This relatively large mass loss conflicts with the results of Doyle et al. Ataman and Hanson6 and Schweizer and Hanson7 experimented with single charged droplets in a quadrupole trap of Straubel’s8 design. This quadrupole trap or electrodynamic balance used superposed ac and dc fields to maintain the droplet at the center of the ac ring electrode. The ac field is used to center the droplet, and the dc field is used to balance the gravitational field. A vertical force balance on the droplet gives 4 cOvdc
/ 20 = m&
(2)
Co is a geometrical constant of the balance (C, = 1 for a Millikan cell with flat electrodes of infinite extent), Vd, is the dc voltage required to suspend the droplet, zo is the (5) Abbas, M. A.; Latham, J. J . Fluid Mech. 1967, 30, 663. (6) Ataman, S.; Hanson, D. N. Ind. Eng. Chem. Fundam. 1969,8,833. (7) Schweizer, J. W.; Hanson, D. N. J. Colloid Interface 1971,35,417. (8) Straubel, H. Dechema Monograph 1959, 32, 153.
0 1989 American Chemical Society
Langmuir, Vol. 5, No. 2, 1989 377
Electrified Droplet Fission distance between the end-cap electrodes, mD is the droplet mass, and g is the gravitational acceleration constant. Determination of the balance constant is discussed below. Schweizer and Hanson reported confirmation of the Rayleigh limit (with a spread of f 4 % ) for droplets with a = 7.5-20 pm. The droplets lost 23 f 5% of their charge and 5 f 5% of their mass in the explosions. However, the earlier paper used eq 1 to calibrate the charge measurement, yielding an instrument constant Co = 1.18; the latter paper used a calibration of Co = 1.29, which would result in a 9% lower estimate of charge. This introduces a large uncertainty in the results. Berg, Trainor, and Vaughang reported explosions occurring below the Rayleigh limit. Their results are suspect, though, because they also found that the charge on the droplet increased between explosions, whereas the constancy of charge on a small evaporating droplet has been well established. It is believed that the neglect of small convection currents led to this result. The effect of convection in the balance chamber can be taken into account by including the Stokesian drag force in eq 2 to yield qcovdc/&)= m,g - GapaU,
(3)
where p is the viscosity of the gas surrounding the droplet and Uc is the convective velocity at the center of the balance chamber, considered positive when upward in the formulation of eq 3. We note that for a spherical droplet with density pL the mass can be written mD = 4npLa3/3, and then eq 3 can be rearranged to yield (4) From eq 4 we can expect that for an evaporating droplet with constant surface charge in a steady flow a plot of a2 versus VdJa should yield a straight line with slope 3 q C o / ( 4 m , p ~ )and intercept 9pUc/ ( 2 p ~ g ) . Roulleau and Desbois’O took into account a small, constant convection in their study of evaporating water drops and found that explosions occurred roughly at the Rayleigh limit. This cannot be considered confirmation of eq 1, however, because they, like Ataman and Hanson, used the explosions to calibrate their charge measurement. They reported that the mother drop lost 16-40% of its charge, with an average of 2670, but they were unable to detect the small change in mass. Dawson’l also reported approximate confirmation of eq 1for droplets of radius 40-80 pm, with accompanying mass loss of “at most a few percent”. The charge loss accompanying an explosion appears to vary between 16% and 4070, but the mass loss has generally been too small to measure very precisely. Recently Rhim and c o - ~ o r k e r s at ~ ~the J ~Jet Propulsion Laboratory (JPL) reported the use of a hybrid electrostatic-acoustic technique to levitate charged droplets with diameters of order 1mm. They found that water droplets burst prior to reaching the Rayleigh limit, but they reported that no mass loss occurred. As we shall show, a mass loss of about 2% is to be expected. This corresponds to a size change of only 0.770, so it is not likely that the video camera system used by the JPL group would detect (9) Berg, T. G.; Owe; Trainor, R. J., Jr.; Vaughan, U. J.Atmos. Sci.
such a small size change. It is possible to have charge loss with no mass loss, for in the high electric field surrounding a charged drop gas breakdown can occur. Elghazaly and Castle14 recently analyzed the maximum positive charge a liquid drop can retain without charge loss due to electron avalanches or Rayleigh instability. Electron avalanching is more likely to occur a t subatmospheric pressures, a t higher temperatures, and at higher humidities. Rhim et al. also examined the evaporation of an aqueous solution of a surfactant, finding that a readily observed instability occurred as the drop approached the Rayleigh limit. Measurements of the frequency spectrum for the oscillations detected by light scattering indicated that the instability was composed of 60 Hz and its higher harmonics. The primary source of this 60 Hz was the quadrupole. Rhim et al.13 were not able to quantify their results, but they speculated that the drop became resonant with the quadrupole field without avalanching or reaching the Rayleigh limit. Earlier, Robertson15 had attributed measured charge loss for droplets with diameters between 20 and 100 pm to electron avalanche in the surrounding gas. The literature to date does not provide definitive verification of the Rayleigh limit nor does it provide reliable data on the mass loss associated with the explosion of an electrified droplet. There is better agreement among investigators about the percentage of charge lost, but there is not general agreement about the mechanism of charge loss in view of the fact that there is good evidence that droplets burst prior to reaching the Rayleigh limit. At least three phenomena can lead to charge loss: (i) electron emission, which occurs when the surface field strength, E,, is sufficiently high, (ii) charge and mass loss associated with droplet bursting when it reaches the Rayleigh limit, and (iii) breakdown of the surrounding gas, which occurs when the field strength is sufficiently high. Other mechanisms could also cause charge loss, but they are not likely to have been significant in our experiments. The mechanisms include secondary electron emission, photo electron emission, thermionic emission, evaporation of charged ions, and ambient ion neutralization. The electrical field strength at the surface of a sphere of radius a is given by
E, = q/(4moa2)
(5)
Reist16 pointed out that electron emission can be expected to occur for E, I9.9 X lo7 V/m, which, for a droplet with a radius of 10 pm, corresponds to q I1.10 X lo-’’ C or 6.87 X lo7 elementary charges. For a liquid droplet with y = 30 mN/m and a = 10 pm, the Rayleigh limit predicted using eq 1 occurs for q = 4.10 X C or 2.56 X lo6 elementary charges, which is well below the charge level required for electron emission. Using an available semiempirical equation for the onset of a positive corona, Elghazaly and Castle showed that for a water drop in dry air at ambient conditions the Rayleigh limit will be reached before the corona limit is reached for all droplet radii. In saturated air a water droplet should lose charge by avalanching before the Rayleigh limit is reached for radii from 40 to 120 pm. For analine (y = 42.9 mN/m), n-octane (y = 27.5 mN/m), and ethanol (y = 22.7 mN/m) in dry air a t atmospheric pressure the Rayleigh limits are well below the limit of corona discharge for all radii.
1970,27, 1173.
(10) Roulleau, M.; Desbois, M. J. Atmos. Sci. 1972, 29, 565. (11) Dawson, G. A. J. Geophys. Res. 1973, 78, 6364. (12) Rhim, W. K.; Chung, S. K.; Trinh, E. H.; Elleman, D. D. Muter. Res. SOC.Symp. Proc. 1987,329, 87. (13) Rhim, W. K.; Chung, S. K.; Hyson, M. T.; Elleman, D. D. Muter. Res. SOC.Symp. Proc. 1987, 103, 87.
(14) Elghazaly, H. A.; Castle, G. S. P. Inst. of Phys. Conf. Series No. 85; Sproston, J. L. S., Ed.; IOP Publishing: Bristol, 1987; pp 121-128. (15) Roberston, J. A. Ph.D. Dissertation, University of Illinois, 1969. (16) Reist, P. C. Introduction to Aerosol Science; MacMillan: New York, 1984.
378 Langmuir, Vol. 5 , No. 2, 1989
Taflin et al. LASER BEAM
W
I-
w
I
-
I
U N STABL E
U
a
OPTICAL FLA T S
I IW
PMT
UNSTABLE -1
w
O J
O
I
I
I
2
I 3
I
4
I
5
I
8 , DRAG PARAMETER Figure 1. Marginal stability envelope and calibration data.
CY LINDRl CAL
LENS
Figure 2. Overhead view of the electrodynamic balance and
detectors.
a point on the marginal stability curve and the process repeated with a droplet of different size and/or by changing the frequency to move to another point on the stability curve. For each balance more than 40 such calibration measurements were made, and Figure l shows the results obtained for the older balance. The data points correspond to values of 6 based on the mean value of Co determined by stability measurements. The data scatter rather little about the lower stability limit. For the older balance we found Co = 0.85 (standard deviation of 0.017), and for the new balance Co = 0.79 (standard deviation of 0.020). The differences are due to different sizes and configurations of viewing ports, holes Experimental Techniques in the electrodes, etc. David7 reported a theoretical value of 0.8768 based on an approximate analytical solution for Determination of the Balance Constant. Two difthe dc field, and Philip, Gelbard, and Arnoldz1 used a ferent electrodynamic balances were used to suspend single numerical technique to obtain Co = 0.80 for the hypercharged droplets in the path of a vertically polarized laser boloidal configuration, not taking into account holes in the beam. Both were of the bihyperboloidal electrode type electrodes. The experimentally determined constants are discussed in detail by Davis.17 The older balance was used in reasonable agreement with the theoretical estimates, and and described by Davis and Ray,l* and special features of we conclude that the error introduced into the calculation the new balance have been presented recently by Taflin of the droplet charge from uncertainty in the value of the et al.19 balance constant is less than 3%. Because we shall determine the charge on the microLight-Scattering Measurement. Figure 2 is an ovdroplet from voltage and size measurements, it is essential erhead view of the newest balance showing the optical that the balance constant in eq 2 be determined as accusystem used to determine the droplet size. Two lightrately as possible. We used the method developed by scattering techniques were employed: (i) phase function Davis20 based on the marginal stability limit (or spring measurements (intensity of the scattered light as a function point) of a trapped microsphere. The technique requires of angle) and (ii) optical resonance measurements. Figure measurement of the ac amplitude, V,,, and frequency, f , 2 shows the location of the detectors used for these meaat which a balanced microsphere will become unstable as surements. the ac field strength is increased. The stability is governed For the new balance the phase function was obtained by an ac field strength parameter, P = 8gV,,/(C~oR2Vdc), periodically during a run by means of a 512-element linear and an aerodynamic drag parameter, 6 = 127rap/(mDfl), photodiode array connected to a microcomputer. With the where Q = 27rf. Ray/Davis balance a photomultiplier tube was driven A microdroplet is first trapped and balanced by adround the balance chamber by means of a stepper motor justing the dc voltage. Then either the ac voltage is into record the intensity of the scattered light through a creased or the frequency is decreased until violent oscilwindow in the ring electrode. The photodiode arrays lation ensues due to reaching the marginal stability limit permitted rapid data acquisition, for the 512 pixels could of the microsphere. This oscillation is readily distinguished be read in about 40 ms. As discussed by Davis and Perfrom the low-amplitude oscillation that occurs when the iasamy,22the number of peaks of the phase function in a dc field does not balance vertical forces. From knowledge given angular range is very nearly proportional to the size of zo and observation of V,,, V,,, and f , the product PC, of the droplet, so the radius can be determined by this was calculated. The drag parameter was calculated from method to within about &I%. the droplet density, gas viscosity, and the measured size A photomultiplier tube (PMT) was mounted at 0 = 90' and frequency. The constant C, was then adjusted to yield (right angle to the laser beam) on the new balance to record the intensity continuously as a function of time for the It is the purpose of this paper to introduce new experimental tools to investigate droplet fission and to provide precise measurement of mass loss. We have used optical resonance spectroscopy to measure the size continuously with a precision of better than one part in lo4. A t the outset of the research it was thought possible to use the new techniques to measure the surface tension of microdroplets, but that objective is now in doubt because the droplets did not burst at the Rayleigh limit. Improvement in our understanding of the mechanism of the droplet burst will permit this technique to be used for surface tension measurement.
(17) Davis, E. J. ZSA Trans. 1987, 26, 1. (18)Davis, E. J.; Ray, A. K. J . Colloid Interface Sci. 1980, 75, 566. (19) Taflin, D. C.; Zhana, S. H.; Allen, T.; Davis, E. J. AlChE J . 1988, 34, 1310. (20) Davis. E. J. Langmuir 1985, I , 379
(21) Philip, M. A,; Gelbard, F.; Arnold, S. J. Colloid Interface Scz. 1983, 91, 507. (22) Davis, E. J.; Periasamy, R. Langmuir 1985, 1 , 373.
Langmuir, Vol. 5, No. 2, 1989 379
Electrified Droplet Fission evaporating droplet. The “optical resonance spectrum” was used to obtain the size to within f0.01% by the methods described by Taflin and Davis.23 For the Davis/Ray balance used in the ionized gas experiments the optical resonances were recorded using a P M T set a t 6 = 93.70. The phase functions and resonance spectra can be interpreted by means of Mie theory for electromagnetic scattering from a dielectric sphere. For a sphere of radius a and refractive index m illuminated with a vertically polarized laser beam with wavelength A, the intensity of the light scattered in the horizqntal plane is given by
(6) where P,’(cos 6) is the associated Legendre function and 6 is the scattering angle measured from the back side of the sphere. Thus, the phase function is the intensity Il considered as a function of 6 only. Ashkin and DziedzicZ4were the first to the first to recognize that resonances can be used to size microspheres with high precision. Resonances correspond to zeros of the denominators of the coefficients a, and b,, where a, and b, are functions of the optical size a (a= 2aa/X) and the refractive index given by a, =
j n ( a ) [majn(ma)I’ - m2jn(ma)[ajn(a)l’ /z,(~)(
a)[maj,( ma)]’- m2j,(ma)[ah,(2)(a)]’
I
I
I
I
I
I
I
I
I
I
350
375
160
’“1 n
T I M E = 14:31:16
(7)
j,(a)[maj,(ma)l’ - j,(ma)[aj,(a)l’ b, = (8) / ~ , ( ~ )[maj,(ma)]’ (a) - j,(ma) [ah,(2)(a)]’ The prime indicates differentiation with respect to the argument, either ma or a. A comparison of the experimental intensity versus time data with Mie theory permits precise and continuous determination of the size as a function of time. Since it is a continuous record, it is an excellent method for measuring the size before and after an explosion. Chylek et al.= showed that it is possible to measure the size and refractive index to 4 parts in lo5,and in this study we have measured the size routinely to a t least one part in lo4. Because of the great sensitivity of resonances to size, we have used phase function data as a coarse estimate of size to begin the computations needed to compare experimental resonances with theory.
Results Explosions in an Un-Ionized Gaseous Medium. Experiments were performed on a variety of liquids covering a wide range of evaporation rates: 1-bromododecane, 1,8-dibromooctane, dibutyl phthalate (DBP), 1-dodecanol, heptadecane, and hexadecane. Dry nitrogen flowed through the balance to remove vapor and prevent saturation of the balance chamber. The aerodynamic drag caused by the flow was taken into account in relating the droplet charge and mass and in suspending dc voltage by using eq 4. Figure 3 shows typical phase functions obtained with the photodiode array for an evaporating droplet of heptadecane. These data were obtained with the system shown in Figure 2, and with this system only the middle 256 diodes were used because the diode array extended (23) Taflin, D. C.; Davis, E. J. Chem. Eng. Commun. 1987, 55, 199. (24) Ashkin, A.; Dziedzic, J. M. Phys. Rev. Lett. 1977, 38, 1351. (25) Chylek, P.; Ramaawamy, V.; Ashkin, A.; Dziedzic, J. M. Appl. Opt. 1983, 22, 2302.
150
175
200
225
250
275
300
325
P I X E L NUMBER
Figure 3. Phase functions obtained with the photodiode array for an evaporating droplet of heptadecane.
beyond the cylindrical lens, so the ends of the array did not detect the scattered light. The angle range covered by the 256 pixels was 34.35-55.65’. Figure 4 shows a typical optical resonance spectrum obtained for heptadecane with 6 = 90’ and X = 632.8 nm. Also shown in the figure is the spectrum computed from Mie theory. The ordinate on the Mie theory plot is the size parameter a. Note that there is close correspondence between the experimental data for intensity versus time and the theoretical resonances except in the region 178.0 I a I 179.4. At a = 179.4 the experimental optical resonance spectrum has a well-defined discontinuity. This corresponds to an explosion, and the resonance discontinuity was accompanied by a sudden increase in Vdc. The residual droplet was quickly recaptured by adjusting the dc voltage, and the optical resonance spectrum continued to match theory after “skipping” approximately one resonance due to the slight change in radius. The radius abefore the explosion was calculated to be 18.07 vm, and the radius a+ after the explosion was 17.93 pm. Thus the fractional change in mass was 2.34%. The charge was calculated by means of a plot of a2 versus Vd,/a based on eq 4. Figure 5 displays such plots for a dodecanol droplet and for the heptadecane (HPD) experiment corresponding to Figure 4. The data prior to an explosion are quite smooth, as seen in Figure 5, but after explosion the droplet had to be rebalanced. Usually another explosion occurred after a relatively short time, so the voltage data subsequent to an explosion are noisier and more sparse. For this reason the determination of the charge on the droplet after the explosion is less accurate than the initial charge, and the fractional loss of charge is less accurate than the fractional mass loss. Two approaches were used to fit the data to a straight line: (i) direct calculation of the slope and intercept for
Taflin et al.
380 Langmuir, Vol. 5, No. 2, 1989
TIME
-
MIE THEORY
RESONANCES
600
I
-
I
I
I
I
I
-
I.
DODECANOL
00
HEPTADECANE
intercept, we obtain a slope of 866 pm3/V after the explosion. The fractional change in charge for these data is 0.19 and 0.12, respectively. In the results reported below we have used the best estimate of the intercept in determining the charge state from the slope of the a2 versus V,Ja plot. From the analysis of all of the available data, we determined that prior to an explosion the maximum probable error in the slope is 2.5%, and the maximum probable error in the slope after the explosion is 5%. It should be pointed out that the size parameters at which these "matches" between optical resonance spectra and Mie theory were obtained are quite large. For a >lo0 the spectrum becomes extremely sensitive to slight changes in scattering angle, so that an unambiguous match becomes very diffi~ult.'~This problem was resolved by collecting the scattered lieht over a lareer aneular ranee. in this case A0 = 3.1°,and'thencompar7ng t h i d a t a wchMietheory results averaged over the range used. The data of Figure 4 were obtained with A0 = 3.1" and for other data A0 = 0 . 7 O . The effect of the larger aperture is to smooth out features which are sensitive to angle, whereas those sensitive to size (e.g., the resonance spikes) are retained. With the larger aperture an unambiguous match is much easier to achieve. This improvement in technique increased the precision of the size measurement to 4 parts in lo5. Table I summarizes the results obtained for l-bromododecane (DBP), 1,8-dibromooctane, dodecanol, heptadecane, and hexadecane evaporating in nitrogen at room temperature. The evaporation rates for these chemicals vary over 3 orders of magnitude, bromododecane evaporating the most rapidly and DBP evaporating the least rapidly. The droplets were usually positively charged, but in several experiments a negatively charged droplet was generated. For some of the experiments run using the small aperture between the droplet and the PMT it was not possible to determine accurately the size after the explosion because there was not excellent agreement between the experimental and theoretical resonances following the explosion. The organic droplets lost between 0.98% and 2.3% of their mass and between 9.5% and 18% of their charge with each explosion. There appears to be little correlation
-5001-/ I BEFORE EXPLOSION AFTER EXPLOSION
E, 400 N 0
300
,/* , ,
-
2o01
0.2
/
,
0.3
,
7'
r.' I
,
,
0.4 0.5 V d e l D , VOLTSlpm
-
,
1
0.6
Figure 5. Plot of az versus V,ja used to obtain the convective
velocity and the droplet charge for heptadecane and dodecanol microdroplets before and after explosion. The lines represent the least-squares fit of the data. each set of data by the method of least squares and (ii) calculation of the slope based on the best estimate of the intercept from data prior to the first explosion. To illustrate the sensitivity of the calculated charge loss to the method of data analysis, we shall consider the representative data of Figure 5. Prior to the explosion of the heptadecane droplet, the slope was found to be 976 pm3/V and the intercept was -55.0 pm2, and after the explosion, the slope and intercept were 823 pm3/V and -46.0 pm2, respectively. If we refit the data after the explosion to yield an intercept of -55.0 pm2, the calculated slope is 852 pm3/V. This change in slope is barely perceptible if we plot the line through the data, but the fractional change in charge, Aq/q., determined from these slopes is 0.13 and 0.16, respectively. For the dodecanol droplet, the initial slope and intercept are 989 pm3/V and 43.7 pm2, respectively, and the best fit of the data after the explosion yields 804 pm3/V and 60.0 pm*. Refitting the data to the same
Langmuir, Vol. 5, No. 2, 1989 381
Electrified Droplet Fission Table I. Droplet Explosion Data
30.5
bromododecane 21.74
-9.47
15.6
19.237 17.490 15.366 13.861
-10.03 -8.63 -7.10 -6.09
25.3 24.9 24.8 24.9
3.24
19.0
0.715
0.005 20 0.006 35 0.005 66 0.007 50
0.015 5 0.0189 0.016 0.022 3
0.14 0.18 0.14 0.18
0.863 0.856 0.854 0.856
34.09
dibutyl phthalate 9.960
0.747 29.14
dodecanol 17.78 16.13
8.13 7.10
21 21.5
16.583 14.560 32.58
6.15 5.06 17.11
14.8 14.8 15.1
18.072 16.363 17.047 15.963 14.536
7.51 6.51 -7.09 -6.41 -5.51
17.1 17.3 18.1 18.1 17.7
0.006 7
0.020
0.13 0.17
0.849 0.859
0.005 25 0.005 01
0.01 6 0.01 5
0.18 0.18 0.14
0.734 0.734 0.742
0.007 86 0.007 03 0.003 29 0.005 01 0.004 61
0.02 3 0.02 1 0.00 98 0.01 50 0.01 38
0.13 0.10 0.095 0.14 0.14
0.783 0.787 0.805 0.805 0.796
27.44
hexadecane
27.9
heptadecane
a Estimated
0.12 34.0a
dibromooctane
surface tension.
between mass loss and charge loss, and for all of the experiments reported in Table I the droplets burst before they reached the Rayleigh limit. The last column of the table lists the ratio of the charge just prior to explosion to the Rayleigh limit charge calculated based on the measured surface tensions. The reported ratios scatter about the mean value of 0.80. We have also calculated the "apparent" surface tension by means of eq 1, using the measured radius, a_,and charge, q-, a t the instant of the explosion. The calculated surface tensions range from 13.3 to 25.3 mN/m, and the table lists the surface tensions that we measured by the Wilhelmy plate method. All of the calculated surface tensions are below the measured values. With only a few exceptions, the lost charge could be contained on a single daughter droplet having a charge below the Rayleigh limit. This result is in conflict with the theory of Elghazaly and Castle,26who predicted that for all values of fractional mass loss less than 11.1% the daughter drop must have a charge exceeding the Rayleigh limit. The theory of Roth and Kelly,n which assumes two to seven "sibling droplets" are formed, is not inconsistent with our results, but with our experimental setup it was not possible to see how many sibling droplets were formed. The expulsion of a satellite droplet was rarely observed in our experiments, for they were expelled with relatively high velocity. The mass loss results are consistent with those of earlier investigators, who found very little loss of mass, but they are in conflict with Abbas and Latham,5 who reported losses of 2O-30%. On the other hand, Rhim et al. observed no mass loss for droplets of nearly 1mm in diameter, but it is unlikely that they would have been able to measure size changes as small as the typical value of 0.6% encountered here. Several explanations can be considered for this early explosion. Dawson2*derived the effect of a dc field on a charged droplet in altering the limiting value of charge, concluding that a field of the magnitude required to suspend the particle against gravity would have less than a
2.5% effect for droplets smaller than 250 pm. The ac field was not accounted for, but it is possible that a resonance associated with the ac frequency and the droplet oscillation could lead to premature fissioning of the droplet. Dawson argued that the deviation from the Rayleigh limit is due to drop distortion in the electric field. That distortion would have a significant effect on the light scattering, but a wavelike disturbance would also alter the light scattering. When we decreased the signal damping of the detector circuit we observed sudden oscillations of the intensity just before rupture was observed in the microscope. A second possible explanation of premature explosion is that random thermal effects created a random accumulation of charge over a small portion of the drop, which then bulged out and ruptured. This could occur if the droplet is "metastable", that is, if infinitesimal distortions damp out but a finite distortion grows. CahnZ9pointed out that any droplet which has more than 60% of the Rayleigh limit of charge is metastable with respect to splitting into two equal spheres, since two equal spheres with the same total volume and charge have a lower energy than the original sphere. Thermal fluctuations of droplet shape are quite small, since they do not show up in the light scattering. But fluctuations of surface charge density may be larger. In the following statistical mechanical analysis we examine the magnitude of expected fluctuations to determine if they will alter the point at which a droplet explodes. Effects of Thermal Fluctuations on Surface Charge. An arbitrary distribution of surface charge o on a spherical surface of radius a can be represented by
(26) Elghazaly, H. M. A.; Castle, G. S. P. IEEE Trans. Ind. Appl. 1986, ZA-22, 892. (27) Roth, D. G.; Kelly, A. J. IEEE Trans. Znd. Appl. 1983, IA-19,771. (28) Dawson, G. A. J. Geophys. Res. 1970, 75, 701.
where
m
4 = -[I
ha2
n
+ nC= l m=-n C
f n m p n m ( c8)oim@] 0~ (9)
where P,"(cos 8) is the Legendre function and @ is the azimuthal angle. The electrostatic energy E of this configuration is q2
m
n
+ C C An"(fn")21 8moa n = l m=-n
E = -[I
(29) Cahn, J. W. Phys. Fluids 1962,5, 1662.
(10)
382 Langmuir, Vol. 5 , No. 2, 1989 Anm=
Taflin et al.
1
(2n
(11)
+ l)(n + 1+ nK)
K is the dielectric constant of the droplet. Now q is made up of N elementary charges. If N is large enough, this continuum approximation can be retained for the energy, but since there are only 2N degrees of freedom (two angles for each charge), for statistical mechanical purposes the sum in eq 9 can contain only 2N - 1 terms. The 2N - 1 coefficients fam, plus the total charge, contain all the information about the system. The canonical partition function for this system takes the form of a classical configuration integral:
EXPLOSION
0 RAYLEIGH L I M I T
s,
DROPLET CHARGE
I
o
P 200'
400 l
l
Q I 600
I
I
800
I
I 1000
TIME, MINUTES
Figure 6. Number of elementary charges on a droplet of DBP and the theoretical Rayleigh limit of charge as functions of time for an experiment in a radioactively contaminated balance
The integration is performed over all 2N - 1 coefficients f,". Here it is assumed that each spherical harmonic has equal a priori probability. The result is Q(N.a,T)= ~ N!( & 2 ; ; k T ) ( 2 N - 1 ) ' 2 exp --8mOak q2 T II(A,m)-1/2(13)
)
(
Now the average energy, ( E ) ,and energy fluctuation, uE, can be obtained as
a In Q ( E ) = k P ( - )d T
q2 N,a
qkT
= -8Tc0a + e
(14)
which expresses the equipartition of energy among the spherical harmonic modes. It has been assumed that N >> 1and N = q / e , where e is the unit of elementary charge. A criterion is needed for the minimum surface area over which the charge must accumulate to cause a rupture. Since it is observed experimentally that at least 1.5% of the mass is expelled, it is reasonable to assume that the surface area corresponding to a droplet of this mass is required; thus AAIA = (Am/m)213= 0.06. A disturbance of the form 4 a(@) = -(I + C cos5 e) (16) ha2 results in an accumulation of charge at one pole of the droplet (and a depletion at the other pole), which covers about 6% of the total surface area. The energy of this disturbance AE is, by use of eq 4
AE= 3 + l6 49(2 + K ) 567(4 + 3K)
+ 43659(664+ 5K)
1
(17) Setting this equal to uE yields the constant C, which is the maximum fractional increase in local surface charge density for this typical energy fluctuation. For a droplet at the Rayleigh limit and K = 2 the result is Aa/u = C = 0.0095a-5/8 ( a in pm) (18) which, for a 10-pm drop, yields a 0.23% increase. It can be concluded that random thermal fluctuations in charge density are too small to cause a significant departure from the Rayleigh limit of charge, except possibly at very small sizes (a < 0.1 pm). It is unlikely that such small random fluctuations in surface charge lead to premature explosion, but the fluctuations can accentuate the instability associated with other modes.
chamber.
Explosions in an Ionized Gaseous Medium. If premature fissioning of the droplets was due to the presence of ions that normally exist at low concentration in the surrounding gas, introduction of an ion source could cause charge loss at even lower surface charge densities than encountered above. To examine the effects of ionization of the surrounding gas on droplet explosion, we performed a series of experiments with DBP evaporating into air in a balance slightly contaminated with C14-taggedDBP. p emission associated with radioactive decay of the C14 led to partial ionization of the gas. Figure 6 is a plot of the number of elementary charges on the droplet versus time. Also shown in the figure is the calculated Rayleigh limit of charge based on the measured droplet radius. Due to the generation of bipolar ions by emission, the droplet lost charge as ions of opposite charge were attracted to the droplet surface, but simultaneously the droplet radius decreased due to slow evaporation. The net effect was that the droplet slowly approached the Rayleigh limit, as indicated in the figure. At approximately 830 min the droplet exploded, indicated by the discontinuity in the data, but it had not yet reached the Rayleigh limit. A fragment was trapped after the fission, and it was found that the mass and charge losses were 66.6% and 63.1 %, respectively. Observation of the microdroplet through the microscope indicated that the droplet fissioned into two or more large fragments rather than expelling a small satellite droplet. This is a rather different result from the experiments in an unionized atmosphere. It should be pointed out that to produce DBP droplets of the desired size and initial charge we used a solution of DBP in toluene (90 wt 70toluene) to generate the microdroplet. The volatile toluene quickly evaporated. Figure 7 shows a typical resonance spectrum for DBP evaporating in the contaminated balance. Also shown is the Mie theory prediction. There are gaps in the oscillograph tracing, for from time to time during the long experiment the P M T detector used with the contaminated electrodynamic balance chamber was rotated to measure the phase function. During 'chat time no resonance spectrum was obtained. Again, there is quite good correspondence between theory and experiment at early times, but as the run proceeded the spectrum became increasingly noisy. Finally, the resonances could hardly be identified, and suddenly the droplet fissioned into large fragments. The droplet which we were able to trap after the explosion was substantially smaller than the original droplet, and following the explosion the noise in the light-scattering signal decreased substantially. Later, the noise increased, and the droplet exploded again. The noise in the light-
Electrified Droplet Fission
Langmuir, Vol. 5, No. 2, 1989 383 DBP RESONANCES
TIME
I
.
-
M I E THEORY
DIMENSIONLESS SIZE. a = h a / X Figure I. Exoerimental and theoretical resonance soectra for a DBP droplet evaporating in a radioactively contaminated balance chamber.
Table 11. DBP Expasion Data in a Radioactively L.~,
fim
10.60 10.00 9.21 9.04 7.83 6.40
.~xC
0-
1013.1
3.59 2.93 2.71 3.07 2.42 1.80
Contaminated Balance .".-I
mN/m
Aa/o.
19.4 15.4 16.7 22.8 21.8 22.0
0.0066 0.020 0.369 0.306 0.018 0.095
Am/m 0.020 0.059 0.750 0.666 0.053 0.260
Aq/q-
q-/qR.
0.134 0.010 0.585 0.631 0.219 0.375
0.719 0.707 0.668 0.777 0.764 0.770
= 34.09 m N / M .
scattering signals may be attributed either to oscillations of the droplet shape or to interference due to contamination on the microdroplet surface. The evaporation of a relatively large amount of toluene used as a volatile solvent for the DBP could result in the formation of insoluble contaminants in the DBP microdroplet. Experimental results obtained with the radioactively contaminated balance varied much more widely than those presented in Table I. Table I1 lists the results for several DBP explosions in the radioactively contaminated balance. As in the unionized gas, the droplet fissioned before the Rayleigh limit was reached (q./qRL averaged 73% for these data compared with a mean of 80% for the data of Table I). The variations in mass and charge loss were much greater for explosions in the ionized gas than in the unionized gas.
Discussion of Results There are at least two plausible explanations for droplet explosion prior to reaching the Rayleigh limit. Contamination of the surface could lead to regions of low surface tension, which would reach the Rayleigh limit before a pure component droplet would do so. As this effect would be difficult to reproduce in separate experiments, one would not expect the rather small scatter observed for several experiments with the same chemical species. Dawson" reported poor reproducibility because of the difficulty of eliminating surface contamination. Since the droplets generally lost about 90% of their mass by evaporation before an explosion would occur in our experiments, any nonvolatile impurities would concentrate by a factor of 10. The peculiar behavior of the DBP microdroplets in the ionized gas might be partly due to the buildup of contaminants caused by the evaporation of toluene at the outset of the experiment. The fact that fissioning occurred a t approximately the same fraction of the Rayleigh limit in unionized and ionized gaseous media suggests that surface tension variations due to surface contamination
was not the key factor in premature explosion. The data of Rhim and his eo-workers suggest that the ac field affected the droplet stability. A small perturbation in the charge distribution and/or droplet shape could be amplified by the ac field due to a resonance phenomenon. The weak external dc field could also contribute to instability. With our experimental setup we could not examine the power spectrum of the oscillations observed prior to droplet fission, but if those oscillators corresponded to the harmonics of the ac frequency, as they did in the JPL experiments, that would be a strong indication of the effect of the ac field. On the basis of Rayleigh's stability theory, it does not seem plausihle that a simple resonance phenomenon caused the premature explosion, because the natural frequencies of oscillation predicted by Rayleigh are much greater than the ac frequencies used here. Rayleigb's result for the natural frequency f. of oscillation (the nth mode) of a droplet with a charge less than the Rayleigh limit is
where n 2 2. For conditions typical of our experiments (y = 27.44 mN/m, pL = 773 kg/m3, a = 32.58 pm, and q = 1.711 x 10-'2 C) the frequency predicted by using eq 19 is f2 = 9657 Hz, which is very much greater. than any frequency used herein. For fz = 60 Hz the droplet could be expected to become resonant a t a charge only 0.0009% lower than the Rayleigh limit. Thus, although resonance undoubtedly occurs, the resonant mode does not appear to be the type analyzed by Lord Rayleigh.
Conclusions The method of optical resonance spectroscopy has been used to measure precisely for the first time the mass loss associated with the explosion of a charged, evaporating droplet. It was found that about 1.C-2.3% of the mass and l(t18% of the charge were lost from droplets of radii 1&33 mm. This charge loss is below that reported by previous investigators and is also in conflict with the theory of Elgbazaly and Castlez6 for single-sibling breakup. The theory of Roth and Kelly for multisibling breakup is consistent with these results. The rate of evaporation has no observable effect on the critical limit of charge for the wide range of evaporation rates studied herein. The organic liquids used all exploded before the expected Rayleigh limit of charge was reached (the mean value of the ratio q-/qRLwas 0.80).Experiments
Langmuir 1989,5, 384-389
384
performed with DBP in a radioactively contaminated electrodynamic balance yielded similar results except that mass losses as large as 75% and charge losses as large as 63% occurred. The mean value of q-/qRL was 0.73 for these data, which is not statistically different from the previous result. It is likely that the electric fields used to suspend the droplets affect the stability phenomenon, that random thermal fluctuations in local surface charge density
can contribute only in a minor way to the instability, and that surface contamination can alter the fragmentation of the exploding droplet.
Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, and to the National Science Foundation, Grant No. CBT-8611779, for their support of this research.
State of Water and Surfactant in Lyotropic Liquid Crystals Norbert0 Casillas,? Jorge E. Puig,? Roberto Olayo,J Timothy J. Hart,§ and Elias I. Frames* School of Chemical Engineering, Purdue Uniuersity, West Lafayette, Indiana 47907 Received July 1, 1988. I n Final Form: Nouember 29, 1988 Differential scanning calorimetry (DSC) and 'H NMR spectroscopy were used to study various lyotropic liquid crystalline phases of sodium bis(2-ethylhexyl)sulfosuccinate (Aerosol OT or AOT) and water. At 25 "C the lamellar liquid crystalline phase contains between surfactant bilayers bulklike water ("water l"),which has a melting point Tlm= 0 "C and an enthalpy of melting AHlm= 80 cal/g. Close to the bilayers, the lamellar liquid crystalline phase contains interfacial water, "water 2", which melts at TZm= -4 f 2 "C and has AH2" = 35 f 8 cal/g of water. After most of water 1 and 2 freezes, the surfactant and the remaining liquid water undergo a phase transition to a nonlamellar liquid crystalline phase. In this phase, "water 3" melts at T3"' = -9 1 "C and has AH3m = 12 f 5 cal/g of water. When water is in ultrathin (510 A) films in lamellar liquid crystal, then water 1 is absent and water 2 can be supercooled to -45 "C or lower. The surfactant has substantial rotational mobility even when the water is frozen at temperatures from -35 to -10 "C.Similar phase and thermal behavior is observed when initially isotropic aqueous micellar solutions of AOT are frozen or melted.
Introduction Combining long-range order with considerable local translational and rotational mobility, lyotropic liquid crystals find many applications in foam and emulsion stabilization1v2and enhanced oil r e c o ~ e r y . ~It?is ~ important, therefore, to understand and document their thermodynamic and molecular motional state. The phase behavior and the state of water a t room temperature in these systems are often probed by calorimetry or electron cryomicroscopy, which invariably involve freezing portions of the In this paper, we report DSC and NMR results on the AOT/water binary system, the phase behavior of which has been studied extensively.+l8 AOT is a pure doubletailed anionic surfactant which can form with water a t 25 "C a lamellar phase (GI, an inverse viscous-isotropic phase (V2), or an inverse hexagonal (M,) liquid crystalline phase."18 The single G phase forms from ca. 20 to 75 wt 9'0 AOT and is a t equilibrium with an isotropic micellar solution from 1.4 to 20 wt 70AOT.15 The water spacing in the G phase ranges from ca. 10 to over 100 A.13The V2 and the M2 phases form between 75 and 100 wt 70 AOT.11313-15In this paper it is shown that after freezing of the water the resulting phase microstructures differ from the original lamellar or isotropic microstructures. *Author t o whom all correspondence should be addressed ((317) 494-4078). Facultad d e Ciencias Quimicas, Universidad d e Guadalajara, Guadalajara, Jal., 44430 Mexico. Departamento de Fisica, Universidad Autonoma Metropolitana-Iztalapa, Apdo. Postal 55-534, Mexico City, 09340 Mexico. Now with Texaco USA, Bellaire, Texas 77401.
The thermal behavior a t four concentrations (15,30,40, and 70 wt 70)of AOT/D,O was studied by Czarniecki et a1.16using DSC and adiabatic calorimetry. They detected three endothermic transitions at T I = 276.3 K,T, = 271.33 K,and T3 = 268.40 K. The first transition (starting a t 3 "C), which is absent in the 70 wt 70sample, was attributed to melting of free (unbound) D20. The second transition was attributed to melting of "bound" D20. In their Abstract, Czarniecki et a1.16 report that the third transition (T3= 268.40 K)is due to melting of the surfactant chains. In their Results and Conclusions, they report, however, (1)Friberg, S. J . Colloid Interface Sci. 1971, 37, 291. (2) Fribere. S.: Larsson. K. In Adu. Lia. Crvstals.: Brown. G.. Ed.: Academic Press: New York. 1976: Vol. 2. D 175. (3) Natoli, J.; Benton, W.'J.; Miller, C.'A.; Fort, T., Jr. J . Dispersion Sci. Technol. 1986, 7, 215. (4) Oswald, A. A.; Huang, H.; Huang, J.; Valint, P., Jr. U.S. Patent 4,434,062, 1982. (5) Ter-Minassian-Saraga, L.; Madelmont, G. J . Colloid Interface Sci. 1982, 85, 375. (6) Kodama, M.; Seki, S. Prog. Colloid Polym. Sci. 1983, 68, 158. ( 7 ) Blum, F. D.; Miller, W. G. J . Phys. Chem. 1982, 86, 1729. (8) Zasadzinski, J. A. N.; Schneider, M. B. J. Phys. (Les Ulis, Fr.) 1987, 48, 2001.
(9) Rogers, J.; Winsor, P. A. Nature (London) 1967, 216, 477. (10) Gilchrist, C. A,; Rogers, J.; Steel, G.; Vaal, E. G.; Winsor, P. A. J . Colloid Interface Sci. 1967, 25, 409. (11)Rogers, J.; Winsor, P. A. J . Colloid Interface Sci. 1969,30, 247. (12) Park, D.; Rogers, J.;Toft, R. W.; Winsor, P. A. J . Colloid Interface Sci. 1970, 32, 81. (13) Fontell, K. J . Colloid Interface Sci. 1973, 44, 318. (14) Hart, T. J. M.S. Thesis, Purdue University, 1982. (15) Franses, E. I.; Hart, T. J. J. Colloid Interface Sci. 1983, 94, 1. (16) Czarniecki, K.; Jaich, A,; Janik, J. M.; Rachwalska, M.; Janik, J. A,; Krowczyk, J.; Otnes, K.; Volino, F.; Ramasseul, R. J . Colloid Interface Sci. 1983, 92, 358. (17) Callaghan, P. T.; Soderman, 0. J . Phys. Chem. 1983, 87, 1737. (18) Alexopoulos, A. H.; Puig, J. E.; Franses, E. I. J . Colloid Interface Sei. 1989, in press.
0743-746318912405-0384$01.50/0 0 1989 American Chemical Societv
