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Langmuir 1988,4, 170-175
Deformation of Viscous Droplets in an Electric Field: Poly(propylene oxide)/Poly (dimethylsiloxane) Systems Tsuyoshi Nishiwaki, Keiichiro Adachi, and Tadao Kotaka* Department of Macromolecular Science, Faculty of Science, Osaka University, Toyonaka, Osaka 560, Japan Received June 24, 1987. I n Final Form: August 31, 1987 Deformation of droplets in an alternating electric field of 60 Hz was investigated for drops of poly(propy1ene oxide) (PPO) and water suspended in poly(dimethylsi1oxane)(PDMS). Under a low field strength E, the droplet deformed into an ellipsoid. The equilibrium shape and the rate of deformation were analyzed in terms of the degree of deformation D, defined as ( X - Y)/(X + Y), with X and Y denoting the major and minor axes of the ellipsoid, respectively. Equilibrium degree of deformation D, was proportional to bF/y, with b being the original radius of the drop and y the interfacial free energy. The time, t, dependence D ( t ) was determined for combinations of PPO and PDMS with varying viscosity. For the deformation process after sudden application of an electric field at t = 0, D(t) conformed to a single retardation equation. ) two constants a and j3 and the shear The retardation time T was expressed as 7 = (aql + @ q 2 ) ( b / ywith viscosities q1 and q2 of the drop and the medium, respectively. This equation held irrespective to the relative magnitude of q1 and q2. From data fitting, a and @ were determined to be 2.0 f 0.2 and 1.0 f 0.2, respectively. For the recovery process after sudden removal of the field at t = 0, D(t) was expressed by an exponential decay function with the retardation time 7 b shorter by a factor of 0.8 than 7. Under a strong field, drops became unstable and were elongated to a thin thread or bursted into fine droplets. We classified these burst behaviors into four classes. I. Introduction A droplet suspended in a dielectric medium deforms into an ellipsoid under a weak electric field, but it becomes unstable and bursts above a certain critical field strength.'-" This phenomenon has been long studied by several authors to clarify the equilibrium shape of the droplet. Recently, we measured the time dependence of drop deformation by using aqueous solutions of poly(vinylpyrrolidone) suspended in di-n-butyl phthalate solutions of polystyrene.'$ We analyzed the results considering the balance among the interfacial tension, the electric stress, and the viscous drag.' The retardation time T for the deformation was represented by 7 = (a71+ @ l l z ) ( b / T ) (1) where a and @ are constants, q is the shear viscosity, b is the radius of the drop before deformation, and y is the interfacial free energy. The subscripts 1and 2 represent the inside and outside of the drop, respectively. We confirmed that eq 1 was satisfactory when the viscosity of either the drop or medium phase is sufficiently higher than the other. Unfortunately, the previous data points scattered considerably. This might be due to a possible slight change in the concentration of the drop phase causing serious changes in q and y from that of the sample of a macroscopic size. Despite this difficulty, we tentatively determined CY and @ to be 1 within the experimental error. Therefore, it was needed to examine more precisely the validity of eq 1 and to determine the values of a and 0 preferably by using much simpler systems consisting of pure immiscible liquids as the drop and medium phases. (1) Moriya, S.; Adachi, K.; Kotaka, T. Langmuir 1986, 2, 155,161. (2) Moriya, S.;Adachi, K.; Kotaka, T. Polym. Commun. 1985,26, 235. (3)OKonsli, C. T.;Thacher, H. C. J . Phys. Chem. 1953, 57, 995. (4)OKonski, C.T.; Harris, F. E. J. Phys. Chem. 1957, 61, 1172. (5)Garton, C. G.; Krasucki, Z. Proc. SOC.London, Ser. A 1964,280, 211. (6)Torza, S.;Cox,R. G.; Mason, S. G. Philos, Tram. R. SOC.London, Ser. A 1971, No.269,295. (7)OKonski, C. T.;Gunter, P. L. J. Colloid Sci. 1964, 10, 563. (8)Taylor, G. I. Proc. R. SOC.London, Ser. A 1964, 280, 383. (9) Taylor, G. I. Proc. R. SOC.London, Ser. A 1934, 146, 501. (10)Taylor, G. I. Proc. R. SOC.London, Ser. A 1966, 291, 159. (11) Allan, R.S.; Mason, S. G. Proc. R. Soc. London, Ser. A 1964,267, 45.
0743-746318812404-0170$01.50/0
For this purpose, we have chosen poly(propy1ene oxide)
(PPO) and pure water for droplets and poly(dimethy1siloxane) (PDMS)for media. Both polymers have a low glass transition temperature and are a viscous liquid at room temperature. We compared the present results with the theory presented in our previous paper.' Since in the previous study we did not examine the systems in which the viscosities of the drop and medium phases are similar, we also measured 7 for such systems. The interfacial free energy y plays an important role in both D, and 7. In the previous study, to test eq 1, we estimated y from the values of D, for most of the systems. In this study, we attempted to measure y independently by a modified ring method.12 When the strength of the electric stress exceeds a certain critical value, the droplet bursts into a complex shape because the electric stress overwhelms the recovery force due to the interfacial t e n ~ i o n . ~ ~ There ~ ~ J are ' few theories for the burst behavior. In this work, we also investigated the factors affecting the bursting phenomena. 11. Theory Infinitesimal Deformation. First, we summarize the theories describing the behavior of a droplet in a weak electric field. Generally, the deformation of a droplet is governed by the electric stress, the hydrodynamic stress due to the flow inside and outside the drop, and the interfacial free energy, which acts to minimize the surface area of the Taylorg defined the degree of deformation D of an ellipsoidal drop by D = (X- Y ) / ( X + Y)
(2)
where X and Y represent the major and minor axes of the ellipsoid, respectively. From the balance of the three stresses mentioned above, we obtained a simple retardation equation for the time dependence of D as1 D(t) = D,[1 - exp(-t/~)] (3) where 7 is the retardation time given by eq 1 and D, the (12) See, for example: Wu,S. Polymer Interface and Adhesion; Marcel Dekker: New York, 1982.
0 1988 American Chemical Society
Langmuir, Vol. 4, No. I, 1988 171
Deformation of Droplets in an Electric Field Table I. Dielectric Constant K ,Electric Conductivity u, Shear Viscosity q, and Molecular Weight M K" 0: s q/Pas W M 0.66 0.3 PPG 6.36 3.2 X lo* 36.7 PPO-40 6.88 74.6 PPO-75 250 51.5 PPO-250 0.98 PDMS-1 9.62 PDMS-10 2.93 > q2,eq 1 is written as log 7 = log a
+ log ( b q 1 / 7 )
(8)
From the data for the drops in PDMS-1 shown in Figure 5, we determined a to be 2.0 f 0.2. Similarly for the
Nishiwaki et al.
174 Lungmuir, Vol. 4, No. 1, 1988 7
Figure 8. Retardation time %for the reemery pmeess w r for the deformation proeesa.
system with qI
