pubs.acs.org/Langmuir © 2009 American Chemical Society
Catastrophic Drop Breakup in Electric Field Janhavi S. Raut,*,† Sathish Akella,† Amit Kumar Singh,† and Vijay M. Naik‡ †
Unilever Research India, 64 Main Road, Whitefield, Bangalore-560066, India and ‡Department of Chemical Engineering, Indian Institute of Technology, Bombay, India Received November 11, 2008. Revised Manuscript Received February 27, 2009
We report novel observations revealing the catastrophic breakup of water drops containing surfactant molecules, which are suspended in oil and subjected to an electric field of strength ∼105 V/m. The observed breakup was distinctly different from the gradual end pinch-off or tip-streaming modes reported earlier in the literature. There was no observable characteristic deformation of the drop prior to breakup. The time scales involved in the breakup and the resultant droplet sizes were much smaller in the phenomenon observed by us. We hypothesize that this mode of drop breakup is obtained by the combined effect of an external electric field that imposes tensile stresses on the surface of the drop, and characteristic stress-strain behavior for tensile deformation exhibited by the liquid drop in the presence of a suitable surfactant, which not only lowers the interfacial tension (and hence the cohesive strength) of the drop but also simultaneously renders the interface nonductile or brittle at high enough concentration. We have identified the relevant thermodynamic parameter, viz., the sum of interfacial tension, σ, and the Gibbs elasticity, ε, which plays a decisive role in determining the mode of drop breakup. The parameter (ε + σ) represents the internal restoration stress of a liquid drop opposing rapid, short-time-scale perturbations or local deformations in the drop shape under the influence of external impulses or stresses. A thermodynamic “state” diagram of (ε + σ) versus interfacial area per surfactant molecule adsorbed at the drop interface shows a “maximum” at a critical transition concentration (ctc). Below this concentration of the surfactant, the drop undergoes tip streaming or pinch off. Above this concentration, the drop may undergo catastrophic disintegration if the external stress is high enough to overcome the ultimate cohesive strength of the drop’s interface.
The movement and deformation of fluids under the influence of electrical forces was observed and documented by William Gilbert more than 400 years ago.1 The electrospraying of water and blood was also reported more than 250 years ago by the French physicist Jean-Antoine Nollet.2 And yet the phenomenon of electric-field-driven deformations and the breakup of fluid drops is so rich that it has continued to fascinate those interested in understanding the mysteries of nature, such as the bursting of rain drops in a thunder storm, and pursuing some of the oldest unresolved problem in experimental and theoretical physics.3,4 The phenomenon has also inspired scientists interested in technological applications of practical value such as the electrostatic spraying of paint and insecticide, electrostatic printing, the production of ultrafine powders, microencapsulation, enhancement of mass transfer in chemical process equipment, and the development of mass spectrometers for the analysis of giant biomolecules.5-10 There *Corresponding author. E-mail:
[email protected]. (1) Gilbert, W. De MagentaCourier Dover Publications: New York, 1991; first published in Latin in 1600, translated by Fleury Mottelay. (2) Grimm, R. L. PhD Thesis. Fundamental Studies on the Mechanisms and Applications of Field-Induced Droplet Ionization Mass Spectroscopy and Electrospray Mass Spectroscopy, California Institute of Technology, 2006. (3) Duft, D.; Achtzehn, T.; Muller, R.; Huber, B. A.; Leisner, T. Nature 2003, 421, 128. (4) Collins, R. T.; Jones, J. J.; Harris, M. T.; Basaran, O. A. Nat. Phys. 2008, 4, 149–154. (5) Fenn, J. B.; Mann, M.; Meng, C. K.; Wong, S. F.; Whitehouse, C. M. Science 1989, 246, 64–71. (6) Grimm, R. L.; Beauchamp, J. L. J. Phys. Chem. B 2005, 109, 8244– 8250. (7) Grimm, R. L.; Beauchamp, J. L. J. Phys. Chem. B 2003, 107, 14161– 14163. (8) Bailey, A. G. Phys. Bull. 1984, 35, 146. (9) Feng, J. Q.; Depaoli, D. W.; Tsouris, C.; Scott, T. C. J. Appl. Phys. 1995, 78(), 2860–2862. (10) Scott, T. C. AIChE J. 1987, 33(), 1557–1559.
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are two distinct sources of forces underpinning the deformation and disintegration of fluid-fluid interfaces and drops due to electric fields. Fluid-fluid interfaces at equilibrium assume a form having a minimum energy for given boundaries, and free drops become spherical in shape under the influence of interfacial tension, in the absence of any other forces. If the fluids or their drops carry a net electrical charge, then the charge resides on the exterior surface as a result of Coulombic repulsion. If the charge becomes excessive, then the interface becomes unstable, and drops rupture into smaller drops. This was theoretically predicted by Lord Rayleigh as early as 1882.11 There could be several reasons for the net charge on a freely suspended drop to become excessive and cross the Rayleigh limit, such as decreasing size due to evaporation, while the absolute charge is unchanged. This phenomenon, for example, is exploited in the technique of electrospray ionization mass spectrometry.5 The destabilization and rupture of charged drops has been studied extensively both theoretically and experimentally over the years.3,11-21 Even if the fluids or their drops are chargeneutral, externally imposed electric fields can induce free or (11) Rayleigh, Lord Phil. Mag. 1882, 14, 184–186. (12) Fong, C. S.; Black, N. D.; Kiefer, P. A.; Shaw, R. A. Am. J. Phys. 2007, 75, 499–503. (13) Shrimpton, J. S. IEEE Trans. Dielectr. Electr. Insul. 2005, 12, 573– 578. :: (14) Achtzehn, T.; Muller, R.; Duft, D.; Leisner, T. Eur. Phys. J. 2005, D34, 311–313. (15) Lopez-herrera, J. M.; ganan-calvo, A. M. J. Fluid Mech. 2004, 502, 303–325. (16) Smith, J. N.; Flagan, R. C.; Beauchamp, J. L. J. Phys. Chem. A 2002, 106, 9957. (17) Grimm, R. L.; Beauchamp, J. L. Anal. Chem. 2002, 74, 6291. (18) Bologa, A.; Bologa, A. J. Electrost. 2001, 51/52, 470–475. (19) Okuda, H.; Kelly, A. J. Phys. Plasma 1996, 3, 2191–2196. (20) Gomez, A.; Tang, K. Phys. Fluids 1994, 6, 404–414. (21) Hanson, D. N.; Schweizer, J. W. J. Colloid Interface Sci. 1971, 35, 417–423.
Published on Web 3/31/2009
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bound charges in them depending upon whether they are conductors or dielectrics. These charges accumulate at the phase boundaries. Therefore, even charge-neutral droplets can be destabilized and made to deform or rupture by externally imposed electric fields and their gradients. The earliest systematic experimental studies of this phenomenon were reported by Zeleny in 1917.22 The first insightful theoretical electrodynamic models of the phenomenon were presented by Taylor in 1964/1966.23,24 Since these early papers, the field has been enriched by a number of important theoretical, simulation, and experimental studies.3,4,6,7,10,25-37 Although many of these studies were carried out with Newtonian fluids, a few investigations involved systems containing surfactants38-40 and systems containing dissolved polymers that made the fluids significantly non-Newtonian in rheological behavior.41,42 Although there are many outstanding questions and despite the diversity of operational conditions as well as the choice of fluids, a few generic observations can be made regarding the deformation and breakup of net-chargefree droplets in an electric field on the basis of these studies: (i) The theory for leaky dielectrics originally articulated by Taylor reasonably explains the behavior of droplets deforming in an electric field, and it appears to be more general than originally envisioned.33 (ii) There are only two principal modes of drop breakup: end pinch-off and tip streaming.27,33,37,39 (iii) Typically in Newtonian systems in the absence of surfactants, the end pinch-off mechanism is observed whereby the drop deforms and breaks into several droplets after forming bulbous ends. Sherwood27 also attributes the formation of bulbous ends to nonzero conductivities of the drop resulting in tangential stresses. (iv) In the presence of surfactants, the drop deforms into a spindlelike shape, and breakup is via the tipstreaming mode. This effect is seen in a specific surfactant concentration regime. Beyond this concentration, the drop breakup again goes back to the end pinch-off mode.39 (v) The tip-streaming mode has also been reported in cases where at least one of the two fluids has appreciable viscoelastic properties.42 In this letter, we report observations of a (hitherto unreported) third mode of drop breakup in electric fields. We have observed, for the first time, that with a proper choice of operating conditions and the addition of an appropriate surface-active agent to the fluid system the drops can be ruptured via a catastrophic mode wherein the drop shatters instantaneously into very tiny drops (radius ∼1 μm) without undergoing tip streaming or bulbous end formation. (22) Zeleny, J. Phys. Rev. 1917, 10, 1–6. (23) Taylor, G. Proc. R. Soc. London 1964, A 280, 383–397. (24) Taylor, G. Proc. R. Soc. London 1966, A 291, 159–166. (25) Macky, W. A. Proc. R. Soc. London 1931, A 133, 565–587. (26) Mason, S. G.; Allan, R. S. Proc. R. Soc. London 1962, A 267, 45–61. (27) Sherwood, J. D. J. Fluid Mech. 1988, 188, 133–146. (28) basaran, O. A.; Scriven, L. E. Phys. Fluids A 1989, 1, 799–809. (29) Sherwood, J. D. J. Phys. A 1991, 24, 4017–4053. (30) Vizika, O.; Saville, D. A. J. Fluid Mech., 1992, 239, 1-21. (31) Basaran, O. A; Patzek, T. W.; Brenner, R. E.Jr.; Scriven, L. E. Ind. Eng. Chem. Res. 1995, 34, 3454–3465. (32) Ganan-Calvo, A. M. Phys. Rev. Lett. 1997, 79, 217–220. (33) Saville, D. A. Ann. Rev. Fluid. Mech. 1997, 29, 27–64. (34) Feng, J. Q. Proc. R. Soc. London 1999, A 455, 2245–2267. (35) Ha, J-W.; Yang, S.-M. Phys. Fluids 2000, 12, 764–772. (36) Eow, J. S.; Ghadiri, M.; Sharif, A. Colloids Surf., A 2003, 225, 193– 210. (37) Lac, E.; Homsy, G. M. J. Fluid. Mech. 2007, 590, 239–264. (38) Ha,, J.-W.; Yang,, S.-M. J. Colloid Interface Sci. 1995, 175, 369-385. (39) Ha,, J.-W.; Yang,, S.-M. J. Colloid Interface Sci. 1998, 206, 195-204. (40) Ha,, J.-W.; Yang,, S.-M. J. Colloid Interface Sci. 1999, 213, 92-100. (41) Xi, K.; Krause, S. Macromolecules 1998, 31, 3974–3984. (42) Ha, J.-W.; Yang,, S.-M. J. Fluid Mech. 2000, 405, 131-156.
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These observations of catastrophic drop breakup in an electric field were made for various oil-water-surfactant systems, wherein the drop comprised the higher-permittivity aqueous phase and the surrounding medium comprised the lower-permittivity oil phase. The experiments were carried out using the setup shown in Figure 1. A rectangular cuvette with two parallel plate stainless steel (SS316) electrodes was used to hold the continuous, low-permittivity fluid phase. The distance between the electrodes was kept at 1 cm. A dc field was applied across the electrodes using a variable-voltage (0-10 kV) power supply system. In the results reported, castor oil (viscosity ∼1000 mPa s) was used as the continuous phase. The suspended drop or dispersed phase comprised deionized water (Milli-Q 18.2 MΩ cm) with a zwitterionic surfactant, 3-(N,N dimethylmyristylammonio)-propanesulfonate (henceforth referred to as ZW1), dissolved in it at varying concentrations. At the start of the experiment, a drop of ∼5 μL of the aqueous phase was dispensed into the continuous fluid using a micropipette. The electric field was switched on to a preset voltage. The drop deformation and breakup were recorded continuously by a CCD camera using the optical bench of a Kruss G10. Figure 2 shows a series of images describing the drop breakup modes at three different surfactant levels. The dc field strength was maintained at 2 105 V/m in all of the experiments. At a low surfactant concentration of 200 ppm, the drop breakup took place via the tip-streaming mode (Figure 2a). On increasing the surfactant concentration to 2000 ppm, the drop breakup was seen to take place via the end pinch-off mode (Figure 2b). On further increasing the surfactant concentration (20 000 ppm), a catastrophic breakup phenomenon was observed whereby the drop shattered into very small droplets within 1-5 s (Figure 2c). As seen in the Figure, the catastrophic mode of drop breakup was distinctly different from the other two modes. There was no initial gradual drop deformation with respect to the field direction before breakup. The drop was seen to undergo sudden and complete disintegration into a fine mist on time scales much smaller than those of the other two modes, as seen in the micrographs. At higher field strengths of 5 105 and 7.5 105 V/m (Figure 3), the drop breakup was seen to be much faster (∼100 ms). In some of the experiments, the drop was seen to throw out liquid in fine jets (cf. Figure 3a). This was similar to the Rayleigh discharge phenomenon reported in the literature.16,17 The dominant mode of drop disintegration, however, still appears to be the catastrophic one. The drop breakup was also investigated in low-frequency ac fields. At the line frequency of 50 Hz, with the rms ac voltage matched, there was no significant difference observed in the drop dynamics between ac and dc fields (Supporting Information). The drop dynamics may be different at high ac frequencies. However, we have not studied the phenomenon at high ac frequencies. The universality of the catastrophic mode of drop breakup was investigated using different oils and surfactant systems. The phenomenon was observed for palm oil, soya oil, rice bran oil, and mixtures thereof as well as fatty acids such as oleic acid and ricinoelic acid as the continuous phase and nonionic and ionic surfactants such as glycerol monostearate, Tween80, sodium dodecyl sulfate (SDS), and N-cetyl-N,N,N,-trimethyl ammonium bromide (CTAB) as well as sodium/ammonium soaps formed in situ at the interface (Supporting Information). The phenomenon was broadly found to be restricted to surfactants producing low Langmuir 2009, 25(9), 4829–4834
Letter
Figure 1. Experimental setup for the study of drop breakup in an electric field. oil-water interfacial tensions (e 0.5 mN/m). However, this criterion alone was found to be insufficient. We hypothesize that in addition to the interfacial tension (IFT) the interfacial elasticity also plays an important role in this phenomenon. The role of interfacial elasticity at high concentrations is apparent from Figures 2 and 3. As seen in these Figures, for systems showing catastrophic drop breakup, the water drop after being dispensed in the oil did not recover to a spherical shape in the typical ∼60 s time span before the electric field was applied. This indicates an increase in the surface rigidity at the high surfactant concentration, slowing down the relaxation of the drop to its equilibrium shape. The interfacial tension (σ) values as a function of surfactant concentration (c) for the oil-water-ZW1 system are plotted in Figure 4a. The oil-water interfacial tension (IFT) values were measured using a spinning drop tensiometer (Kruss), where the interfacial tension was calculated from the equilibrium deformed shape of the suspended oil drop at different RPM (4000-6000) values. The interfacial data was fitted to the Frumkin equation, ( σ ¼ σ0 þ
2 ) Γ Γ þ Γ¥ H RTΓ¥ ln 1 Γ¥ Γ¥
c ¼ aL
Γ Γ exp -2H ðΓ¥ -ΓÞ RTΓ¥
ð1Þ
ð2Þ
In the above set of equations, σ is the interfacial tension, and Γ is the surface excess. Γ¥, aL, and H denote the saturation adsorption, surface activity, and surface interaction parameter, respectively. Because the equation form is not explicit in the concentration term, c, an iterative procedure was used to obtain a fit to the experimental data. The fit in Figure 4a obtained using Γ¥ = 1.4 10-6 mol/m2, aL = 0.1 mol/L, and H = 3700 agrees well with the measured interfacial tension data. The corresponding surface excess is also plotted in Figure 4a. As seen in Figure 4a, beyond a certain high concentration, the experimentally measured points had an inflection, and the fit was no longer good beyond this concentration. The inflection seen in the measured values at the high concentration is similar to that seen at the critical micellar concentration (cmc) for typical surfactant systems. Whereas the cmc value of this system is not known, we hypothesize that the inflection is a consequence of the onset of micelle formation. The calculated surface excess also flattens out in this region. In addition to lowering the interfacial tension, the surfactant molecules also alter the interfacial elasticity. The interfacial elasticity Langmuir 2009, 25(9), 4829–4834
Figure 2. Breakup of a water drop in oil: effect of surfactant concentration. Oil - castor oil; surfactant - 3-N,N-dimethyl palmityl ammonio propane sulfonate. Surfactant concentrations: (a) 200, (b) 2000, and (c) 20 000 ppm. Field = 2 105 V/m dc. defined as ε ¼
dσ d ln A
ð3aÞ
is an equilibrium state variable, which can be computed from equilibrium IFT versus concentration data43 using an (43) Kralchevsky, P. A.; Danov, K. D.; Brose, G.; Mehreteab, A. Langmuir 1999, 15, 2351-2365. (44) Taylor, G. Proc. R. Soc. London 1950, A 201, 192–196. (45) Akella, V. S.; Franklin, D. C.; Naik, V. M.; Raut, J. S.; Singh, A.; Suresh, S. J.; Venkataraghavan, R. WO/2007/131917, 2007.
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Figure 3. Breakup of water drop in oil: effect of field strength on oil-water-ZW1 at a surfactant concentration of 20 000 ppm. (a) 5 105 V/m dc; (b) 7.5 105 V/m dc.
equivalent expression, ε ¼ -Γ
dσ dΓ
ð3bÞ
This parameter has been variously named the Gibbs elasticity (εG), areal elasticity, compressional modulus, and film elasticity.46 The interfacial elasticity can also be measured using dynamic methods such as the oscillating drop, and capillary/ longitudinal waves. Such values of the interfacial elasticity are referred to by different names such as the dilational modulus (εD) and the Marangoni elasticity.46,47 εD can also be computed from dynamic surface tension measurements48and can be expected to approach εG for very high frequencies of oscillations or for observations over extremely short time scales immediately after an equilibrium state is perturbed. Figure 4b shows the equilibrium interfacial elasticity, ε, obtained using eq 3b and plotted as a function of surfactant concentration, c. As seen in the Figure, with increasing concentration the surface elasticity goes through a maximum and then drops steeply. This can be seen explicitly or derived from data reported in the literature for many surfactant systems.48-50 In the sections to follow, we further elaborate on the role of this surface elasticity behavior in the mode of drop breakup. When a drop is subjected to an electric field E0, the discontinuity in the permittivity and conductivity values across the interface imposes a Maxwell stress at the interface, which is proportional to (∼εcE20),44 where εc is the permittivity of the continuous phase. The drop responds by deforming to balance the electrical and interfacial forces. Consider a cylindrically stretched drop with fluctuations in the extent of stretching at different locations on its surface due to thermal or any other sources of disturbance at a given instant in time (cf. Figure 5a, locations 1/10 and 2/20 have different extents of stretch). The instantaneous surface energy at any location is given by g = (46) Lucassen-Reynders, E. H.Surface Elasticity and Viscosity.In. Anionic Surfactants Physical Chemistry of Surfactant Action; Lucassen-Reynders, E. H., Ed. Marcel Dekker: New York, 1981. (47) Pugh, R. J. Adv. Colloid Interface Sci. 1996, 64, 67–142. (48) Huang, D. D.; Nikolov, A.; Wasan, D. T. Langmuir 1986, 2, 672–677. (49) Lucassen, J.Dynamic Proterties of Free Liquid Films and Foams.In. Anionic Surfactants Physical Chemistry of Surfactant Action; LucassenReynders, E. H., Ed.; Marcel Dekker: New York, 1981. (50) Kitchener, J. A. Nature 1962, 194, 676–677.
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Figure 4. (a) Oil-water interfacial tension and calculated surface excess. (b) Calculated surface elasticity as a function of surfactant concentration. The vertical bands in both plots show the range of the experimentally observed transition from the noncatastrophic to catastrophic mode of drop breakup. The system consists of watercastor oil-3-N,N-dimethyl myristyl ammonio propane sulfonate.
Aσ, where g is the energy per adsorbed molecule and A (= 1/Γ) is the interfacial area occupied by one adsorbed molecule. Differentiating and substituting for dσ from eq 3b, we obtain dg ¼ A dσ þ σ dA ¼ ε dA þ σ dA ¼ ðε þ σÞdA dg ¼ε þσ dA
ð4Þ
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Figure 5. (a, b) Stability of the interface to fluctuations, showing increased stability on the left-hand side of the (ε + σ) maximum. Strainhardening behavior of the interface above the ctc (d) and ductile behavior below the ctc (e) for a stretching drop (c).
Figure 5b shows plots of ε, σ, and (ε + σ) as functions of 1/Γ. The plots of ε as well as (ε + σ) also show maxima at a certain surface excess. We refer to this interfacial concentration or the corresponding equilibrium bulk concentration of the surfactant as the critical transition concentration (ctc). When the surfactant concentration range is less than the ctc such that points 1 and 2 in Figure 5a are on the left-hand side (LHS) of the maximum of the (ε + σ) curve, as seen in Figure 5b, then dg dg < dA 1 dA 2
ð5Þ
Therefore, the incremental energy for stretching the already stretched region (2) is higher than that for region 1 (Figure 5a, b). Hence, region 2 will resist further stretching, and the amplification of perturbations in stretch will be opposed. As a result, the drop will undergo uniform stretching. However, if points 10 and 20 are on the right-hand side (RHS) of the maximum of the (ε + σ) versus 1/Γ curve, then dg dg > dA 1 dA 2
ð6Þ
In this case, the incremental energy penalty for stretching the already stretched region (20 ) is lower than that for the lessstretched region (10 ). Hence, the perturbations will grow. This will lead to progressive thinning and subsequent pinching beyond location 20 . This condition is qualitatively similar to the Considere criterion for strain localization and necking (51) Mckinley, G. H.; Hassager, O. J. Rheol. 1999, 43, 1195–1212.
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when amorphous glassy materials are subjected to uniaxial elongation.51 The analysis clearly shows that at low concentrations of surfactant the system is amenable to the end pinchoff or tip-streaming modes of failure as reported in the literature; however, at high enough concentrations when the surface elastic modulus becomes important, the interfacial fluctuations due to external impulses or stresses are resisted, and the drop will resist strain localization or interfacial instabilities. Furthermore, when the original surfactant concentration is greater than the ctc (i.e., when the original state of the interface is located on the left of the (ε + σ) maximum (in Figure 5b), the drop not only shows uniform stretching under applied stress but also shows strain hardening or “brittle” behavior as elaborated on below. The surface energy of the drop can be represented by G = aσ, where a is the surface area of the drop, σ is the interfacial tension, and G is the total surface energy of the interfacial area under consideration. Let the drop be in a state (I) (Figure 5c) to begin with, having an interfacial area aI and a number of adsorbed molecules M. If the drop is uniformly stretched to state II, then dG ¼ a dσ þ σ da
ð7Þ
Substituting a = MA, where A = area per molecule, we have dG ¼ MðA dσ þ σ dAÞ ¼ Mðε þ σÞdA
ð8aÞ
Since M is constant at aIΓI, dG ¼ aI ΓI ðε þ σÞdA DOI: 10.1021/la803740e
ð8bÞ 4833
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Thus, Z
II
GII - GI ¼ aI ΓI
ðε þ σÞ dA
I
Z
GII - σI aI ¼ aI ΓI
II
I
G/II ¼
GII ¼ σ I þ ΓI aI
ðε þ σÞ dA
Z
II
ð9Þ
ðε þ σÞ dA
I
G*II is the interfacial energy of the uniformly stretched drop in state II (Figure 5c). Figure 5d shows the plots of G*II at different extents of strain (aII - aI)/aI. The values of G*II were calculated for two different locations of state I along the (ε + σ) curve. As seen from the Figure, if state I corresponds to point 10 on the right-hand side of the maximum, then the interface is more ductile, whereas if state I corresponds to point 1 on the left-hand side of the maximum, then the plot of G*II is concave up, indicating strain-hardening behavior. When such a drop displaying strain-hardening or non-yielding interfacial behavior is subjected to an electric field, there can be a transient buildup of electrical charge and electrodynamic energy that is high enough to “rupture” the interface. The vertical bands in Figure 4 indicate the experimentally observed region of transition from a noncatastrophic to catastrophic mode of breakup. As expected, the transition from noncatastrophic to catastrophic breakup occurs close to the maximum of the (ε + σ) versus 1/Γ curve. The limit of charge buildup on the interface, before it ruptures, is called the Rayleigh limit16,17 for a droplet and is given by qRayleigh ¼ 8πεc 1=2 σ1=2 R3=2
ð10Þ
The above equation is strictly valid for an interface formed by pure fluids where surface elasticity does not play a role on the typical time scales involved. In the presence of surfactants, however, interfaces can exhibit surface elasticity. The restoring force therefore would have a component of surface elasticity as well, and in this case, it would be more appropriate to substitute (ε + σ) for σ in eq 10. The drop would hence shatter when the interface of the liquid drop is located on the left-hand side of the (ε + σ) versus 1/Γ curve in Figure 5b and the causal electrical stresses exceed the interfacial forces holding the drop together. On the right-hand side of the maximum, the interface deforms freely in response to the electric field, especially at the location of already strained and stretched perturbations, and there would be no accumulation of charges. Thus, the maximum stress that the interface can withstand is given by the peak value of (ε + σ). The minimum field at which the catastrophic breakup can be achieved for a given oil-surfactant system can hence be estimated. The charge buildup on a spherical drop interface in the presence of an impressed field, E0, is given by qinduced ¼ 4πR2 E0 ðεd - εc Þ
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ð11Þ
where εd (= 80 8.85 10-12 farads/m) and εc (= 5 8.85 10-12 farads/m) are the electrical permittivity values of the water and oil phases, respectively. For a drop of radius R = 0.5 mm, when equating eqs 10 and 11 at the (ε + σ) peak value of 0.9 mN/m, the minimum field strength at which catastrophic breakup will occur can be estimated to be ∼3 104 V/m. This number is in fair agreement with the experimental results. In conclusion, we report our observations of a new mode of drop breakup in electric fields. We hypothesize that this mode of drop breakup will be exhibited by oil-water interfaces that have low interfacial tension combined with high surface elasticity We find that the sum of interfacial elasticity (same as Gibbs elasticity) and interfacial tension (ε + σ), is an important diagnostic thermodynamic parameter deciding whether the liquid drop will undergo catastrophic breakup or tip streaming when stretched by an external force field. The plot of (ε + σ) versus the interfacial surfactant concentration (or area per molecule) shows a maximum at a critical transition concentration, the ctc of surfactant. Below this concentration, the perturbation or stretch in the drop surface will be favored by strain localization. The drop will stretch, show ductile “necking”, and undergo breakup by tip streaming or the end pinch-off mechanism. Above the ctc of surfactant, the deformations in the drop surface due to any short-time-scale perturbations owing to external stress impulses will be resisted, and there will not be any strain localization. The drop will be uniformly strained and undergo even strain hardening. The stored energy will be released catastrophically when the externally imposed stress is higher than the restoration stress, viz., (ε + σ), by a brittle failure as qualitatively conjectured by early investigators in the field.52 The preliminary understanding and set of rules developed can be used to obtain catastrophic drop breakups through the use of the right surfactant-cosurfactant systems at the right concentrations. The use of electrostatic drop breakup in the chemical processing of emulsions is currently handicapped because of either low throughput rates when the desired droplet size is small (e.g., using Taylor cones) or because of fairly large (∼10 μm) resultant droplet size and polydispersity in the size distribution. The phenomenon observed by us can open up the possibility of exploiting electrodynamic drop breakup for the bulk production of fine water-in-oil emulsions.45 Acknowledgment. We acknowledge Professor V. A. Juvekar and Professor R. Thaokar from the Indian Institute of Technology, Bombay, for insightful discussions on electric field effects on liquid-liquid interfaces. Supporting Information Available: Videos of the drop breakup modes. Catastrophic drop breakup in a 50 Hz ac field and for different oil/surfactant systems. This material is available free of charge via the Internet at http://pubs.acs. org. (52) Kitchner, J. A. Q. Rev. Chem. Soc. 1959, 13, 71–97.
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