Droplet Deformation in Dc Electric Fields: The Extended Leaky

6194

Langmuir 2005, 21, 6194-6209

Droplet Deformation in Dc Electric Fields: The Extended Leaky Dielectric Model Nikolaos Bentenitis† and Sonja Krause* Department of Chemistry and Chemical Biology, Rensselaer Polytechnic Institute, Troy, New York 12180 Received November 9, 2004. In Final Form: February 28, 2005 The leaky dielectric model (LDM) was extended to large droplet distortions in dc electric fields. The resulting extended LDM (ELDM) reduces to the LDM for small droplet aspect ratios and to the pure dielectric model when the ratio of droplet and matrix conductivities equals the inverse ratio of their permittivities. The ELDM distinguishes between two types of phenomena possible at high electric fields: continuous deformation and hysteresis. For droplets deforming parallel to the electric field, the relationship that distinguishes between the two phenomena is a function of the droplet and matrix conductivities and viscosities but not of their permittivities. For droplets deforming perpendicular to the electric field, the relationship is a function of the droplet permittivities and conductivities but depends only slightly on the ratio of their viscosities. Some of the predictions of the LDM and the ELDM were compared with our own data and with data from the literature. For the systems that deformed parallel to the field direction, the ELDM not only predicted the data qualitatively but also predicted the data quantitatively when the experimental errors in its input parameters were taken into account, whereas the older LDM did not even predict the qualitative trend of the data. For the systems that deformed perpendicular to the field direction, however, the ELDM predicted the observed the aspect ratios in only one out of the four systems examined. In the other three systems, the LDM appeared to give reasonable predictions when either the ratio of the matrix/droplet viscosities was relatively small or the value of total charge relaxation time was relatively large. Thus, the applicability of the ELDM, as presently formulated, appears to be limited in the case of deformations perpendicular to the electric field.

1. Introduction Historical Overview. When an uncharged droplet is suspended in another liquid under a uniform external electric field, there is a discontinuity of the field at the droplet interface. In the classical electrostatic model introduced by Garton and Krasucki,1 the fluids are treated as perfectly insulating dielectrics and the electric stress vector has only a component normal to the interface. According to this pure dielectric model, PDM, the stress vector normal to the interface can be balanced by the interfacial tension and the droplet deforms in the direction of the electric field (prolate deformation). Experiments conducted in 1962 by Allan and Mason2 however, produced droplets which deformed perpendicular to the electric field direction (oblate deformation). To explain this phenomenon, G. I. Taylor3 proposed the leaky dielectric model (LDM) in 1966. According to his model, dielectric liquids are assumed to have small conductivities (leaky dielectrics) so, when an electric field is applied, free charge appears at the droplet interface, although the total charge on the droplet is zero. The action of the electric field on the charges sets the fluids in motion and toroidal circulation patterns are formed inside and outside the droplet; such circulation was observed by Taylor. Taylor described the physical problem mathematically by assuming that the equilibrium shape of the droplet was a slightly deformed sphere. His * Corresponding author. † Present address: Chemistry Department, Colgate University, 13 Oak Drive, Hamilton, NY 13346. (1) Garton, C. G.; Krasucki, Z. Proc. R. Soc. London, Ser. A 1964, 280, 211-226. (2) Allan, R. S.; Mason, S. G. Proc. R. Soc. London, Ser. A 1962, 267, 45-61. (3) Taylor, G. I. Proc. R. Soc. London, Ser. A 1966, CCXCI, 159-166.

solution predicted both oblate and prolate deformations, depending on the properties of the fluids, in agreement with the experiments by Allan and Mason. The properties of the fluids involved were their conductivity, permittivity, viscosity, and interfacial tension. Taylor focused on the direction of deformation (either parallel or perpendicular to the applied electric field) and not on its quantitative description. The first quantitative comparisons between the LDM and experiments were reported five years later by Torza et al.4 who also extended the LDM to alternating electric fields. The deformation and breakup of droplets in two component systems involving silicone oils of various viscosities, castor oil, and water were studied in steady and oscillatory (up to 60 Hz) fields. In steady fields, oblate deformations were observed in eight systems in qualitative agreement with the theory. One of the theoretical results of the Torza et al. model, which was confirmed experimentally by the authors, was that a droplet which assumed an oblate deformation at low frequencies could change to a prolate deformation as the frequency was increased. Although the qualitative aspects of the LDM model were corroborated, the quantitative agreement was unsatisfactory. The observed deformations were larger than the theoretically predicted ones by a factor of 2 in many of the systems. The measured deformation was always larger than that predicted theoretically, suggesting that the deviations were not due only to experimental errors. To account for the quantitative discrepancies between the observed and the predicted deformations, several attempts to extend the LDM have been made.5-7 All (4) Torza, S.; Cox, R. O.; Mason, S. G. Philos. Trans. R. Soc. London 1971, 269, 295-319. (5) Sozou, C. Proc. R. Soc. London, Ser. A 1972, 331, 263-272.

10.1021/la0472448 CCC: $30.25 © 2005 American Chemical Society Published on Web 06/01/2005

Droplet Deformation in Dc Electric Fields

Langmuir, Vol. 21, No. 14, 2005 6195

focused on the terms involved in the system of differential equations. For instance, temporal acceleration terms were included in the hydrodynamic equations by Sozou.5 However, none of the attempts was successful in eliminating the above-mentioned quantitative disagreements. Since none of the theoretical extensions appeared to resolve the disagreements between theory and experiment, further work was undertaken: Vizika and Saville8 “paid close attention to the measurement of the physical properties”.8 They followed a suggestion by Torza et al.4 that dielectric properties need to be determined very accurately. According to Vizika and Saville, the agreement between the Torza et al. model and the experiments by Vizika and Savillle improved over the Torza et al.4 study. Another comparison between theory and experiment was published in 1993 by Tsukada et al.9 who studied deformations of the castor oil/silicone oil system. Castor oil droplets in silicone oil exhibited prolate deformations while silicone oil droplets in castor oil exhibited oblate deformations in dc electric fields. In addition to experimental work, a model based on finite elements was created to calculate deformations in steady fields. Because of the flexibility of the finite element method (FEM), no geometry was assumed for the droplet, making it possible for the theory to be applied to large deformations. At small deformations, numerical results agreed with the predictions of the Torza et al. model. At large deformations, substantial differences appeared, with the finite element prediction being closer to the observed deformations. The studies by Torza et al., Vizika and Saville, and Tsukada et al. constituted the most comprehensive tests of the LDM. The qualitative agreement between theory and experiment was encouraging, while the lifting of the constraint of small droplet deformations in the FEM approach led to satisfactory quantitative agreement. In this paper we describe the extension of the LDM to large deformations using an analytical method. The resulting extended LDM (ELDM) maintains all of the assumptions of the LDM, except that of the droplet shape: Instead of assuming that the droplets are only slightly deformed and dictating the use of spherical coordinates to solve the associated electrohydrodynamic differential equations, the ELDM assumes a spheroidal droplet shape and requires the use of spheroidal (prolate and oblate) coordinates. Basic Equations. Torza et al.4 considered a droplet of radius r0 which carried no net electric charge and was suspended in a neutraly buoyant condition in another, immiscible fluid. Under the influence of an electric field, the droplet was distorted into an ellipsoid. The authors used the droplet deformation, D

D)

a-b a+b

(1)

where a and b are the axes of the ellipsoidal droplet parallel and perpendicular to the applied field, respectively. When the droplet is deformed in the direction of the electric field, 0 < D < 1, and when the droplet is deformed in the direction perpendicular to the electric field, -1 < D < 0. Another measure of droplet deformation is the aspect ratio of the ellipsoid (6) Ajayi, O. O. Proc. R. Soc. London, Ser. A 1978, 364, 499-507. (7) Bayents, J. C.; Saville, D. A. Drops Bubbles. Third Int. Colloq. 1989, 7-17. (8) Vizika, O.; Saville, D. A. J. Fluid Mech. 1992, 239, 1-21. (9) Tsukada, T.; Yamamoto, Y.; Katayama, T.; Hozawa, M. J. Chem. Eng. Jpn. 1993, 26, 698-703.

R)

a b

(2)

When the droplet assumes an oblate deformation, R < 1, whereas R > 1 when the droplet assumes a prolate shape. D has been commonly used to describe small droplet deformations, whereas R has been used when such deformations are large. Since the droplet deformation is a result of the competition of the applied dc electric field, E0, which tends to deform the droplet, and the interfacial tension, γ, which tends to keep the droplet spherical, a useful dimensionless parameter that expresses the magnitude of this interaction is the electrical capillary number Ce

Ce )

dr0E02 γ

(3)

where d is the permittivity of the droplet. The LDM explains the correlation between the electrical capillary number and the droplet deformation using the following additional quantities:

S)

m d

(4a)

R)

σd σm

(4b)

M)

µm µd

(4c)

where m is the permittivity of the medium, σd and σm are the conductivities of the droplet and the medium, and µd and µm are the viscosities of the drop and the medium, respectively. According to the LDM, the steady-state deformation and the electrical capillary number are connected by

D) Φ0 )

9 R-1 ) ΦC R + 1 16 0 e

(5a)

1 [S(R2 + 1) - 2 + (SR - 1)Ω] (5b) (2 + R)2 2M + 3 Ω)3 5M + 5

(5c)

Compared to the LDM, the pure dielectric model (PDM) of Garton and Krasucki1 only uses the ratio of the permittivities, S, to describe droplet deformations, ignoring the conductivity and the viscosity of the fluids. It can only predict deformations parallel to the electric field for which the steady-state aspect ratio is given by

Ce ) S-1 H22 )

[m1 - G] H 2

2

(6a)

2

R1/3 2 β R + + (R2 - 4) 2 R sinβ (1 + 2R )G - 1

(6b)

β ) cos-1R-1

(6c)

[

]

-1

m)1-S 1 R cosh-1R - 1 G) 2 2 R - 1 xR - 1

[

]

(6d) (6e)

Aspect Ratio Hysteresis. Figure 1 shows the aspect ratio, R, vs Ce plots for three different values of S, as

6196

Langmuir, Vol. 21, No. 14, 2005

Bentenitis and Krause

enormous decrease of the aspect ratio and the system moves to point D. A further decrease in E0, produces a smooth decrease in the aspect ratio past point D. The two different paths (A f B for increasing E0 and C f D for decreasing E0), constitute a hysteresis loop. This argument for hysteresis in the PDM was described earlier by Sherwood,10 who based his conclusion on a similar phenomenon that has been observed in the droplet deformation of ferrofluids in magnetic fields by Bacri and Salin.11,12 2. The Extended Leaky Dielectric Model (ELDM)

Figure 1. Aspect ratio, R, vs electrical capillary number, Ce, according to the PDM for three hypothetical system: (a) S ) (5)-1, (b) S ) (18.6)-1 and (c) S ) (50)-1. Ce is proportional to E02 for a particular droplet in a particular system. Note the logarithmic scale for the abscissa and the ordinate.

The LDM, as described above, allows spherical droplets to deform into ellipsoids, but it uses spherical coordinates for all calculations. This approach becomes untenable at reasonably large aspect ratios, as has been demonstrated experimentally.9,13,14 We have thus decided to extend the LDM by using prolate and oblate spheroidal coordinates in our calculations. In the following analysis, the center of the droplet coincides with the middle of the interfocal distance of a spheroidal coordinate system; we use a prolate coordinate system when the droplet elongates in the direction of the electric field and an oblate coordinate system when the droplet elongates perpendicular to the electric field. The prolate and oblate spheroidal coordinate systems can be formed by rotating the two-dimensional elliptic coordinate system, consisting of confocal ellipses and hyperbolas, about the major and minor axes of the ellipses, respectively. It is customary to make the z-axis the axis of revolution in each case. The interfocal distance is called 2c. (In the following, the notation suggested by Flammer15 is followed, apart from the symbol for the interfocal distance. (Flammer uses d instead of 2c).) The prolate spheroidal coordinates are related to the rectangular coordinates by the transformation

x ) cxξ2 - 1x1 - η2cosφ

(7a)

y ) cxξ2 - 1x1 - η2sinφ

(7b)

z ) cξη

(7c)

1 e ξ < ∞, -1 e η e 1, 0 e φ e 2π

(7d)

with Figure 2. Aspect ratio hysteresis according to the PDM for a hypothetical system for which S ) (50)-1. Ce is proportional to E02 for a particular droplet in a particular system. Note the logarithmic scale for the abscissa and the ordinate.

predicted by the PDM (eq 6). For the purpose of this discussion, since Ce is proportional only to E02, for a particular droplet in a particular system, with d, r0, and γ constant in eq 3, the abscissa is proportional only to E02. For S g (18.6)-1, the aspect ratio is single valued and the droplet exhibits “continuous deformation” as the field is increased. But for lower values of S, the aspect ratio is not single valued, and hysteresis is possible, as shown in Figure 2. As the value of E0 increases, the magnitude of the deformation increases smoothly, until point A is reached. At that point a small increase in the value of E0 should bring about an enormous increase of the aspect ratio and the system moves to point B. A further increase in E0 produces a smooth increase in the aspect ratio past point B. If E0 is then decreased, the droplet aspect ratio should decrease until point C is reached. At that point, a small decrease in the value of E0 should bring about an

and the oblate spheroidal coordinates by the transformation

x ) cxλ2 + 1x1 - η2cosφ

(8a)

y ) cxλ2 + 1x1 - η2sinφ

(8b)

z ) cλη

(8c)

0 e λ < ∞, -1 e η e 1, 0 e φ e 2π

(8d)

with

Both the prolate and the oblate spheroidal coordinate systems are systems of orthogonal curvilinear coordinates (10) Sherwood, J. D. J. Fluid Mech. 1988, 188, 133-146. (11) Bacri, J. C.; Salin, D. J. Phys., Lett. 1982, 43, 649-654. (12) Bacri, J. C.; Salin, D. J. Phys., Lett. 1983, 44, 415-420. (13) Xi, K.; Krause, S. Macromolecules 1998, 31, 3974-3984. (14) Bentenitis, N.; Krause, S.; Benghanem, K. Langmuir 2004, 21, 790-792. (15) Flammer, C. Spheroidal Wave Functions; Stanford University Press: Stanford, CA, 1957.


Langmuir, Vol. 21, No. 14, 2005 6197

and in each case, the coordinates (ξ, η, φ) or (λ, η, φ) form a right-handed system. Expressed in terms of the aspect ratio, R (the ratio of the semiaxis parallel to the z-axis to that perpendicular to the z-axis)

ξ)

R

xR

2

(9)

-1

for a prolate and

λ)

R

x1 - R2

(10)

for an oblate spheroid. In the limit that the interfocal distance 2c becomes zero, both the prolate and oblate spheroidal systems reduce to the spherical coordinate system. For a finite interfocal distance, the surface ξ ) constant becomes spherical as ξ approaches infinity; thus

cξ f r, η f cosθ, as ξ f ∞

(11)

where r and θ are spherical coordinates. Similarly for oblate spheroidal coordinates

cλ f r, η f cosθ, as λ f ∞

(12)

For the two coordinate systems, a subscript is affixed to the symbol h to denote the metric coefficient of the corresponding coordinate axis. For a prolate coordinate system hξ, hη, and hφ (also denoted as h1, h2, and h3) can be expressed as

hξ ) h1 ) c

(13a)

xξ2 - η2 x1 - η2

(13b)

hφ ) h3 ) cxξ2 - 1x1 - η2

(15)

and the following boundary conditions. 1. The electric potential is bounded at the droplet center and tends toward the value of the applied potential for large distances away from the droplet. 2. The normal component of the electric field is continuous at the droplet surface. 3. The tangential component of the electric field is discontinuous at the droplet surface. The Laplacian of V in prolate spheroidal coordinates is

∇2V )

1 ∂ 2 ∂V ∂ ∂V (ξ - 1) + (1 - η2) 2 ∂ξ ∂ξ ∂η ∂η c (ξ - η ) (16) 2

[ (

2

)

(

)]

The potentials Vd and Vm inside and outside the droplet can be written in the form Ξ(ξ)H(η). This allows the Laplace equation to be separated into two ordinary differential equations, whose integration gives, H(η) ) P1(η) ) η and Ξ(ξ) equal to a linear combination of

P1(ξ) ) ξ

(17)

1 ξ+1 -1 Q1(ξ) ) ξ ln 2 ξ-1

(18)

P1 and Q1 are Legendre polynomials of the first and second kind, respectively. Because of the first electrical boundary condition, we can further write that

(19)

for the outside of the droplet and

Vd ) b1P1(ξ)P1(η)

(20)

for the inside of the droplet. Applying the remaining electrical boundary conditions

(13c) b1 )

For an oblate coordinate system

xλ2 + η2 xλ2 + 1

(14a)

xλ2 + η2 hη ) h2 ) c x1 - η2

(14b)

hφ ) h3 ) cxλ2 + 1x1 - η2

(14c)

hλ ) h1 ) c

∇2Vj ) 0, with j ) m, d

Vm ) [a1P1(ξ) + a2Q1(ξ)]P1(η)

xξ2 - η2 xξ2 - 1

hη ) h2 ) c

let are symmetric about the direction of the electric field and satisfy the Laplace equation

Solution in Prolate Spheroidal Coordinates. One of the assumptions of the LDM, as developed by Taylor3 and Torza et al.,4 is that the electric and hydrodynamic fields couple only at the surface of the droplet. Therefore, independent solutions of the Laplace and the NavierStokes equations suffice for the calculation of the difference in the total normal and tangential stress at the interface. The same assumption is followed in the extended leaky dielectric model. In the absence of space charge and assuming that there is no transport of charge along the interface, the electric potentials inside (Vd) and outside (Vm) the drop-

P1Q′1 - P′1Q1 a ) b1xcE0 P1Q′1 - RP′1Q1 1

(21a)

P1P′1(R - 1) P1Q′1 - RP′1Q1

(21b)

a1 ) cE0

(21c)

a2 )

In the above expressions, the following convention in the terminology has been followed. Taking eq 21a for example, instead of using the expression P1(ξ0) to indicate the value of the polynomial P1(ξ) at ξ ) ξ0, only the symbol of the polynomial P1 is used. Similarly, Q ′1 is used instead of (∂Q1/∂ξ)|ξ)ξ0. In other words, for the value of any polynomial at the interface, the symbol of the polynomial, without an argument, is used. An argument is used only if the polynomial is a function of η. As first postulated by G. I. Taylor,16 the difference between the tangential component of the electric stress vector inside and outside of a droplet generates flows which are axisymmetric about the direction of the electric field. The radial and transverse velocity components of this flow field can therefore be expressed in terms of the Stokes (16) Taylor, G. I. Proc. R. Soc. London, Ser. A 1964, 280, 383-397.

6198



stream-function ψ. The stream-functions inside (ψd) and outside (ψm) the droplet satisfy the linearized form of the Navier-Stokes equation

Γ4ψj ) Γ2(Γ2ψj) ) 0 (j ) m, d )

(22)

with the following boundary conditions. 1. The solution is bounded in the whole space, including the droplet axis. 2. There is no flow across the interface. 3. The tangential velocities inside and outside the droplet coincide at the interface. This is the no-slip boundary condition. The operator Γ2 is given in prolate spheroidal coordinates17

[

]

2 1 ∂2ψ 2 2 ∂ ψ (ξ 1) + (1 η ) Γ ψ) 2 2 c (ξ - η2) ∂ξ2 ∂η2 2

Gn(ξ) )

1 [P (ξ) - Pn(ξ)] 2n - 1 n-2

(24a)

Hn(ξ) )

1 [Q (ξ) - Qn(ξ)] 2n - 1 n-2

(24b)

However, equation Γ4ψ ) 0 does not accept separable solutions in the form of products of Gegenbauer polynomials. This “nonseparability” of the solutions of Γ4ψ ) 0 has considerably impeded the development of theoretical models for flow around spheroidal particles. However, in a 1994 paper by Dassios et al.,18 a complete solution of Γ4ψ ) 0 is given. According to the authors ∞

∑ [gn(ξ)Gn(η) + n)2 hn(ξ)Hn(η)] (25)

The functions gn(ξ) and hn(ξ) are linear combinations of Gegenbauer polynomials of the first and second kind, given in ref 18. As Dassios et al. note, although the individual terms of the above expansion are not solutions of eq 22, the full series is. Since the functions gn(ξ) and hn(ξ) are different, eq 22 does not accept separable solutions. The above expansion exhibits a kind of separation, which Dassios et al. called semiseparation. Following the analysis of Dassios et al.18 and taking into account the first hydrodynamic boundary condition and the symmetry of the flow field around (η ) 0, ξ ) 1) ∞

ψ)

∑ gn(ξ)Gn(η)

(26)

n)2

For the flow outside the droplet, the third term of the series is the solution

ψm ) g3,m(ξ)G3(η) ) [c1H1(ξ) + c2H3(ξ)]G3(η)

ψd ) g3,d(ξ)G3(η) ) [d1G3(ξ) + d2G5(ξ)]G3(η)

(27)

H3 c1 ) - c2 ) c1xc2 H1

(29a)

G5 H1H′3 - H′1H3 c ) d1xc2 d1 ) H1 G3G′5 - G′3G5 2

(29b)

G3 d2 ) - d1 ) d2xd1 ) d2xd1xc2 G5

(29c)

The difference in the electrical and hydrodynamic stress vector between the inside and the outside of the droplet evaluated at the interface will be called stress for brevity. An important assumption of the LDM is that the electrical and hydrodynamic fields couple only at the droplet interface. The LDM assumes that the electrical tangential stress is balanced by the hydrodynamic tangential stress (tangential stress condition). The same assumption is followed by the extended leaky dielectric model. The tangential component of the electric stress vector, peξη, is (see ref 19)

peξη ) EξEη

(30)

1 ∂V Eξ ) h1 ∂ξ

(31a)

1 ∂V Eη ) h2 ∂η

(31b)

where

The steady part of the difference in the tangential component of the electric stress vector between the outside and the inside of the droplet is

∆peξη ) (peξη)m - (peξη)d ) 2 2 1 c dE0 (SR - 1)P′1P1b1x2P1(η)P′1(η) (32) 2 h1h2

This expression reduces to eq 20b in the Torza et al.4 paper. (The expressions in this paper can be translated into the expressions in ref 4 by substituting P1 f r, P1(η) f cosθ, b1 from eq 21a, S f 1/q, R f 1/R, m f 1, d f 2, h1 ) 1, and h2 ) r and evaluating the resulting expression at r ) b, the radius of the droplet.) The tangential component of the hydrodynamic stress vector is (see ref 17)

For the flow inside the droplet, more than one term of the (17) Goldstein, S. Modern Developments in Fluid Dynamics; Dover: New York, 1965. (18) Dassios, G.; Hadjinicolaou, M.; Coutelieris, F. A.; Payatakes, A. C. Q. J. Mech. Appl. Math. 1994, 52, 157-191.

(28)

inside the droplet. Applying the hydrodynamic boundary conditions at ξ ) ξ 0,

(23)

Equation Γ2ψ ) 0 admits separable solutions in the form of products of Gegenbauer polynomials of the first and second kind, Gn and Hn, which are defined in terms of Legendre polynomials:

ψ ) g0(ξ)G0(η) + g1(ξ)G1(η) +

series is needed. Yet, if one requires that the solution reduces to the solution in spherical coordinates as given by Taylor3 and Torza et al.,4 then

phξη ) µehξη ) µ

[ ()

( )]

h1 ∂ u h2 ∂ v + h1 ∂ξ h2 h2 ∂η h1

where µ is the shear viscosity of the fluid and

(33)


Langmuir, Vol. 21, No. 14, 2005 6199

1 ∂ψ v)h1h3 ∂ξ u)

(34)

1 ∂ψ h2h3 ∂η

(35)

This expression reduces to the coefficients of cos2θ in eqs 20a in the Torza et al.4 paper (see the comments in parentheses following eq 32). The normal component of the hydrodynamic stress vector is (see ref 17),

phξξ ) -p + µeξξ

(43)

v ∂h1 1 ∂u eξξ ) 2 +2 h1 ∂ξ h1h2 ∂η

(44)

Expanding eq 33,

∆phξη

[

]

h1h2 ∂(h1h2h3)-1 G3(η) + ) (1 - M)H′3 h22 µdc2 ∂ξ h2 (d1xG′′3 + d1xd2xG′′5 - MH′′3) G (η) (36) h1h3 3

[

]

At the interface, the electrical tangential stress between the outside and the inside of the droplet is balanced by the hydrodynamic tangential stress, in other words,

∆peξη + ∆phξη ) 0

c3

The difference between the pole and the equator in the normal component of the hydrodynamic stress, ∆∆pe phξξ is

∆∆pe phξξ ) -∆pe pm + ∆eppd + µm∆pe (ehξξ)m - µd∆pe (ehξξ)d (45)

(37)

Combining eqs 32, 36, and 37, the constant c2 is determined by

µdc2

with

)-

(SR - 1)P′1P1(f1/c) b 2 E 2 ) (1 - M)H′3f2 + (d1xG′′3 + d1xd2xG′′5 - MH′′3)f3 1x d 0 c2xb1x2dE02 (38)

By substituting ∆pe (ehξξ)m ) ∆pe (ehξξ)d, µm ) Mµd,

c3 p c3 )∆p + ∆∆pe phξξ µdc2 µdc2 e m c3 c3 p ∆e pd + (M - 1)∆pe (ehξξ)m (46) µdc2 c2 The difference in the pressure between the pole and the equator is

∆pe p )

where

f1 )

2 2 3 ∫-11c(ξ2 - η2)G3(η)P1(η)P′1(η) ) c15 (ξ - 7)

(39a)

∫

2

2

2

h2 1 2 3 G (η) ) ξ f3 ) -1 c(ξ2 - η2)G3(η) h1h3 3 15 7

∫

(

1

)

(47)

where (see ref 20)

∂(h1h2h3)-1 2 G3(η) ) - ξ f2 ) -1 c(ξ - η )G3(η)h2 ∂ξ 15 (39b) 1

pole (1 ∂p dη dp ) ∫0 ∫equator ∂η

(39c)

∂(Γ2ψ) h2 ∂p )µ ∂η ∂ξ h1h3

(48)

∂(Γ2ψ) h1 ∂p ) -µ ∂ξ ∂η h2h3

(49)

By carrying out the integrations Following the suggestions of Taylor,16 and Garton and Krasucki,1 we calculated the difference in the normal stresses between the pole and the equator of a spheroid. We then equated this difference to the product of the interfacial tension and the difference in the curvature between the pole and the equator. The magnitude of the normal component of the electric stress vector peξξ is (see ref 19)

peξξ ) [Eξ2 - Eη2] 2

∆peξξ

[

( )

d 1 ∂Vd ) (SR2 - 1) 2 h2 ∂ξ 1

2

( )]

∂Vd 1 - 2 (S - 1) ∂η h2

2

(41)

(50)

3 1 c j1 ) 2 G3 1x

(51a)

where

[

(40)

The steady part of the difference in the normal component of the electric stress vector between the outside and the inside of the droplet is therefore

c3 ∆∆pe phξξ ) M(j1 + j3) + (j2 - j3) µdc2

j2 ) 7 P4 ln

xξ2 - 1 ξ

+

)]

7 2 3 ξ d d 8 7 1x 2x

(

(51b)

and

( )

Q2 (H′3)2 ) 2ξ ξ

j3 ) 2H′3 -

(52)

The normal stresses due to the electric field and the viscous flow are balanced by the interfacial tension at the droplet interface

and the difference between the pole and the equator (symbol, ∆pe ) of the electric stress is given by

∆peξξ + ∆phξξ - γC ) 0

1 (∆pe ∆peξξ)ss ) (dE02) [(S(R2 + 1) - 2)]b1x2 2

where C is the total curvature at any point on the surface of the droplet. Taking the difference between the pole and

(42)

(53)

6200



deformation are the same as that of eq 57, but the expression for kR is

the equator

∆∆pe peξξ + ∆∆pe phξξ ) γ∆pe C

(54)

The difference between the total curvature at the pole and the total curvature at the equator of a prolate spheroid is

kR 1 ∆pe C ) R1/3(2R - 1 - R-2) ) r0 r0

kR ) -R-1/3(2R-1 - 1 - R2) and the expressions for j1, j2, and j3 are

(55)

[

(56)

(dE0 )b1x

1 [S(R2 + 1) - 2] + 2

[

2

]

c2x[Mj1 + j2 + (M - 1)j3] )

γ k r0 R

kR ) ΦCe

(57)

or

1 Φ ) b1x2 [S(R2 + 1) - 2] + 2

[

]

c2x[Mj1 + j2 + (M - 1)j3] (58) where Φ is the steady-state discrimination function for the ELDM. Solution in Oblate Spheroidal Coordinates. In oblate spheroidal coordinates the Laplacian is

∇2 )

1 ∂ ∂ ∂ ∂ 2 (λ + 1) + (1 - η2) 2 ∂λ ∂λ ∂η ∂η c (λ + η ) 2

2

[ (

)

)] (59)

(

and the operator Γ2 is

Γ2 )

[

]

2 1 ∂2 2 2 ∂ (λ + 1) + (1 η ) c2(λ2 + η2) ∂λ2 ∂η2

(60)

The solution of the Laplace equation (∇2V ) 0) and the Navier-Stokes equation (Γ4ψ ) 0) in oblate spheroidal coordinates is obtained by substituting ξ by iλ in eqs 19, 20, 27, and 28, respectively. The expressions for the constants b1, a2 and a1 are obtained by substituting ξ with iλ in P1 and Q1 from eqs 21a, 21b, and 21c and calculating the derivatives P ′1 and Q ′1 with respect to λ. Similarly, the expressions for the constants c1, d1, and d2 are obtained by substituting ξ with iλ in H1, H3, G3, and G5 from eqs 29a, 29b, and 29c and calculating the derivatives H ′1, H ′3, G ′3 and G ′5 with respect to λ. The constant c2 is given by eq 38 but f1, f3, and f3 are given by

f1 )

2 2 3 ∫-11c(λ2 + η2)G3(η)P1(η)P′1(η) ) c15 (λ + 7)

(61a)

∂(h1h2h3)-1 2 2 2 2 G3(η) ) - λ c(λ + η )G (η)h 3 2 -1 ∂λ 15 (61b)

f2 )

∫

f3 )

1 2 3 ∫-11c(λ2 + η2)G3(η)h1h2 3G3(η) ) 15 (λ + 7)

1

h

xλ2 + 1 λ

-

(63a)

)]

7 2 3 λ + d d 8 7 1x 2x

(

(61c)

The expressions for the discrimination function and the

(63b)

and 2 -1 1 (1 + 3λ ) cot λ - 3λ (H′3) ) j3 ) H′3 2 λ λ

Combining eqs 42, 50, 53, and 55 2

3 1 c j1 ) 2 iG3 1x

j2 ) 7i P4 ln

where r0 is the radius of the undeformed droplet and

kR ) R1/3(2R - 1 - R-2)

(62)

2

(64)

3. Methods Materials. A mixture of polysiloxanes and silicones (PDMS, CAS-RN 150678-61-8) was obtained from the General Electric Company, Waterford, NY; an aqueous KCl solution of 52 168 ppm (CSH), sold as conductivity standard for the calibration of conductivity cells, was obtained from Cole-Palmer Instrument Co, Vernon Hills, IL; castor oil (CasOil, CAS-RN 8001-79-4) from Eckerd Corporation, Clearwater FL, epoxidized linseed oil (EpoxLinsOil, CAS-RN 8016-11-3) from Union Carbide, Danbury CT, and the diglycidyl ether of bisphenol A (DGEBA, CAS-RN 1675-54-3) from Dajac Labs, Feasterville, PA. Measurement of Electrohydrodynamic Properties. The viscosity of the fluids was measured at different shear rates using a Brookfield LVDV-II+ cone and plate viscometer (Brookfield Engineering Laboratories, Stoughton, MA). The dielectric constants were measured at a frequency of 1 MHz and a voltage of 30 mV (peak-to-peak), using a Hewlett-Packard 4192A LF impedance analyzer (Hewlett-Packard, Palo Alto CA) and a calibrated dielectric sensor designed by Kranbuehl et al.21 The sensor consisted of two gold electrodes deposited in an interdigitated configuration on a nonconductive substrate. For the KCl solution, direct measurement of the dielectric constant was not possible due to the high conductivity of the solution. It was estimated using (see ref 22):

soln ) solv + 2δc

(65)

where soln is the desired dielectric constant of the KCl solution, solv is the dielectric constant of the solvent (76.55 for water at 30 °C),23 δ is a constant for the electrolyte (-5.5 for KCl),22 and c is its molar concentration. Ordinary conductivity meters were not appropriate for the leaky dielectric fluids, because the values of their conductivity was extremely low. An Emcee 1154 cell and precision conductivity meter (Emcee Electronics, Inc., Venice, FL) was used. This instrument satisfied the requirements set by the ASTM D-4308 standard for conductivity measurements. Although the interfacial tension of several polymeric fluid pairs has been measured with a Spinning Drop Tensiometer in our laboratory (see ref 24), the high viscosity of the fluids studied in this work precluded the use of this technique. The interfacial tension was instead estimated by measuring the deformation of (19) Stratton, J. A. Electromagnetic Theory; McGraw-Hill: New York, 1941. (20) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media; Prentice Hall: Englewood Cliffs, NJ, 1965. (21) Kranbuehl, D. E.; Delos, S. E.; Jue, P. K. Polymer 1986, 27, 11-18. (22) Hasted, J. B.; Ritson, D. M.; Collie, C. H. J. Chem. Phys. 1948, 16, 1-21. (23) Lide, D. R. CRC Handbook of Chemistry and Physics, 81st ed.; CRC: Boca Raton, FL, 2000. (24) Xi, K. Structure Formation in Polymer Blends in an Electric Field During Solvent Evaporation. Thesis, Chemistry Department, Rensselaer Polytechnic Institute, 1996.

Droplet Deformation in Dc Electric Fields the droplets in low dc electric fields. The method was recommended by Adamson25 who suggested the use of the PDM, however, the LDM is more appropriate to accurately predict droplet deformation in low dc electric fields. The interfacial tension, therefore, was estimated by the use of eq 5. The magnitude of the initial deformation, D, was plotted vs r0E02, and γ was obtained from the slope of the curve. Apparatus for the Application of Electric Fields. The apparatus used for the application of the electric field and the measurement of droplet deformation was described in a previous publication.14 Briefly, it consisted of a cylindrical slab of Teflon with a rectangular slit in the middle of the cylinder. Two stainless steel electrodes were pushed into each side of the slit and held within two cuts in the Teflon. The whole cell fit onto a glass Petri dish. A wire was soldered onto each of the cell electrodes, and the wires were connected to a high voltage supply. The matrix liquid was poured into the cell and the immiscible droplet was then injected into the matrix via a micropipet. The cell was placed on the stage of a microscope. The pictures from the microscope were captured by a digital camera, sent to a time code generator and then to a videocassette recorder and from there to a video monitor. Each image was then sent to a computer through a frame grabber that encoded the input as 8 bit gray scale images. The images were analyzed to obtain the length of the major and minor axes of the distorted ellipsoidal droplets with image analysis software. Error Propagation. In this work, the conductivities, permittivities, and viscosities of the fluids were measured repeatedly, and their precision error was estimated. Ratios of these quantities were then used as inputs to the ELDM that predicts droplet aspect ratios. The precision errors in the conductivities, permittivities and viscosities propagated to the estimation of S, R, and M and therefore to the theoretical calculation of the droplet aspect ratios. The Appendix summarizes the equations used to calculate the propagated precision errors and it shows that the values of the partial derivatives of R and Ce with respect to S, R, and M were necessary. These derivatives were calculated numerically using the method of automatic differentiation (AD).26 The AD method uses the chain-rule of differentiation to evaluate the derivatives with respect to the variables of functions defined by a high-level language computer program. An important feature of the AD is that it does not introduce truncation errors and yields results accurate to machine precision. The implementation of the AD method for Matlab, called ADMAT and distributed by its author free of charge,27 was used. The propagated precision error bounds for a prediction are shown in the graphs that follow with dashed lines. The ELDM is also compared with experimental data from the literature, but in such cases, the precision errors in the measurement of the electrohydrodynamic properties were unknown. We instead assumed a 5% error in the estimation of S, R, and M in order to give error bounds for the predictions of the ELDM. On the other hand, the dimensions of the droplets, including their aspect ratio, the interfacial tension and the magnitude of the applied electric field were assumed to be known exactly. Although this is not absolutely true, the associated precision errors were assumed to be negligible compared to the errors of the other measured physical quantities. However, since electric field experiments were repeated more than once and the aspect ratios measured showed variability, error bars based on the standard error of the mean of the measurements also appear on the graphs that follow. No error bounds were calculated for the predictions of the various FEM models described in this work, since the original computer code and associated programs were not available to us.

4. Results & Discussion Electrohydrodynamic Properties. Table 1 shows the values of the measured electrohydrodynamic properties (25) Adamson, A. W.; Gast, A. P. Physical Chemistry of Surfaces, 6th ed.; Wiley: New York, 1997. (26) Verma, A. Curr. Sci. 2000, 78, 804-807. (27) Verma, A. http://www.tc.cornell.edu/averma/AD (accessed June 2003).

Langmuir, Vol. 21, No. 14, 2005 6201 Table 1. Electrohydrodynamic Properties of Liquids Studied fluid

conductivity σ/pS m-1

permittivity /0

viscosity µ/cP

CSH CasOil EpoxLinsOil PDMS DGEBA

8.00 × 1012 38.8 ( 0.8 (1.46 ( 0.1) × 103 2.61 ( 0.02 (2.20 ( 0.57) × 103

68.9 4.41 ( 0.02 6.18 ( 0.03 3.53 ( 0.02 10.9 ( 0.1

0.81 ( 0.01 530 ( 3 339 ( 2 451 ( 2 (6.15 ( 0.35) × 103

of the fluids studied in this work along with 95% confidence limits for their values. The conductivity of CSH has no error limits because CSH was treated as a standard and the permittivity of CSH has no error limits either because its value was estimated. Table 2 shows the ratios of permittivity, S, conductivity, R, and viscosity, M, for systems studied in this work along with data for a few systems studied in the literature. It also gives the value of the interfacial tension γ and that of the total charge relaxation time, τe, where

τe )

d + m σd + σm

(66)

The calculation of τe requires the knowledge of the absolute value of both the droplet and the matrix conductivity. However, in several published studies, only the ratio of those conductivities was reported. In such cases, τe could not calculated. No published study includes precision errors for S, R, or M and therefore no confidence limits are included in Table 2. A 5% error for each of S, R, or M was assumed instead. Comparison of the Extended Leaky Dielectric Model with Previous Models. Figure 3 shows the connection between the extended leaky dielectric model developed in this work and the previous theoretical treatments of droplet deformation in dc fields. The Allan and Mason model (AMM,2) introduced in 1962, predicts the aspect ratio of a slightly deformed droplet under a dc field, assuming that both the droplet and the matrix are pure dielectrics and it can only predict deformations parallel to the electric field. The pure dielectric model (PDM1,10) removed the AMM assumption of slight deformation of the droplets and therefore included the AMM as a limiting case when R ≈ 1. The PDM is also restricted to deformations parallel to the electric field. The leaky dielectric model (LDM3,4) removed the assumption of the AMM of pure dielectric materials and included leaky dielectric materials. The AMM model is included in the LDM as the special case in which the ratio of the fluid conductivities equals the ratio of fluid permittivities (R ) S-1). This is because, when R ) S-1, the LDM predicts that no flow is generated and that the total stress difference is purely electric. When R ) S-1, the AMM model can be used instead of the LDM. The extended leaky dielectric model removes the assumption of small droplet aspect ratios included in the LDM, and therefore contains the LDM as a special case (R ≈ 1, see eqs 32 and 42). The ELDM also, and necessarily, reduces to the AMM when R ) S-1 and R ≈ 1. For the special case in which R ) S-1, the ELDM reduces to the PDM. Similarly to the reduction of the LDM to the AMM, when R ) S-1, the ELDM predicts that no flow is generated and that the total stress difference is purely electric. When R ) S-1, the PDM model can be used instead of the ELDM. Prediction of Aspect Ratio Hysteresis. Similarly to the PDM, the ELDM predicts that for certain combinations of electrohydrodynamic properties, there is a critical

6202



Table 2. Ratios of Electrohydrodynamic Properties of Liquids Studieda droplet/matrix oil28

water/silicone CSH/EpoxLinsOilb DGEBA/PDMSb castor oil/silicone oil28 vegetable oil/silicone oil9 water/castor oil8 CasOil/DGEBAb silicone oil (10 P)/castor oil8 silicone oil (125 P)/castor oil and Triton8 silicone oil/vegetable oils9

S 10-2

3.22 × (8.97 ( 0.04) × 10-2 0.324 ( 0.003 0.730 0.581 5.68 × 10-2 2.47 ( 0.03 1.62 2.04 1.72

R 108

(5.48 ( 0.38) × 109 (8.43 ( 2.18) × 102 10.0 62.9 104 (1.76 ( 0.46) × 10-2 10-2 10-4 1.59 × 10-2

M

τe/s

(4.19 ( 0.06) × 102 (7.33 ( 0.42) × 10-2 12.5 6.80 × 10-2 1.40 × 103 11.6 ( 0.7 1.40 8.00 × 10-2 14.7

8.31 × 10-14 5.80 × 10-5

103

0.353 6.05 × 10-5 0.353

γ/N m-1 0.0031 0.0028 0.0140 0.0040 0.0030 0.0168 0.0016 0.0046 0.0043 0.0030

S ) m/d, ratio of permittivities, R ) σd/σm, ratio of conductivities, M ) µm/µd ratio of viscosities, τe ) (d+m)/(σd+σm), total charge relaxation time. b This work. a

Figure 3. Relation among theoretical treatments for droplet deformation in dc fields. ELDM, extended leaky dielectric model; PDM, pure dielectric model; LDM, leaky dielectric model, AMM, Allan and Mason model.

Figure 5. Predictions by the ELDM of aspect ratio hysteresis for a hypothetical system with S ) 2.35, R ) 0.1, and M ) 100. Note the logarithmic scale for the ordinate.

Figure 4. Predictions by the ELDM of aspect ratio hysteresis for a hypothetical system for which S ) 0.1, R ) 30, and M ) 100.

electrical capillary number for which hysteresis is possible. An example of such hysteresis is shown in Figure 4 for a hypothetical system in which S ) 0.1, R ) 30, and M ) 100, deforming parallel to the electric field. A hysteresis loop, analogous to that shown in Figure 2 can be constructed for this system. The ELDM also predicts hysteresis for systems deforming perpendicular to the electric field. An example of such a hysteresis is shown in Figure 5 for a hypothetical system with S ) 2.35, R ) 0.1, and M ) 100. Although the PDM predicts aspect ratio hysteresis for systems deforming parallel to the electric field when S e (18.3)-1,10 one of the main conclusions of the ELDM is that the value of S does not play any role in the

Figure 6. Predictions by the ELDM of aspect ratio hysteresis for systems deforming parallel to the electric field. Area A: continuous deformation. Area B: possible hysteresis.

determination of the onset of aspect ratio hysteresis; it is rather the values of R and M that determine this onset. The predictions of the ELDM for systems deforming parallel to the electric field are shown in Figure 6. The ELDM separates these systems into two classes: systems that exhibit the possibility of aspect ratio hysteresis (area B in Figure 6) and systems that deform continuously as the field is increased or decreased (area A in Figure 6).


Langmuir, Vol. 21, No. 14, 2005 6203

electrohydrodynamic flow to the overall stress difference is proportional to the following expression:

M(j1 + j3) + (j2 - j3)

Figure 7. Predictions by the ELDM of aspect ratio hysteresis for systems deforming perpendicular to the electric field with (a) M ) 100 or (b) M ) 0. Area A: continuous deformation. Area B: hysteresis possible.

Figure 6 also shows that regardless of the value of M, when R g 21.7, systems always exhibit the possibility of hysteresis. In contrast to the case of systems deforming parallel to the electric field, the value of S does play a role in the determination of hysteresis for systems deforming perpendicular to the electric field. Area B in Figure 7 represents the combinations of S and R that produce hysteresis for two extreme cases of the value of M (0 and 100). Area A corresponds to systems that exhibit continuous deformation. Figure 7 also shows that, regardless of the values of S and M, if R g 0.6, the system always exhibits continuous deformation. Limitations of the Extended Leaky Dielectric Model. We showed that the solution for the stream function outside a droplet, ψm, is described by just one term of the infinite series of eq 26, whereas the solution for the stream function inside the droplet, ψm, needs more than one term from the infinite series in order to be determined. The approximation that was made in this work, was that the infinite series should be truncated so that it reduces to the Torza et al.4 solution when the droplet becomes spherical. The effects of this approximation appear in the calculation of the difference in the stresses between the inside and the outside of the droplet. That difference determines the magnitude of the droplet deformation. According to eq 50, the contribution of the

(50a)

Of j1, j2, and j3, only j2 is calculated from the stream function inside the droplet, ψd (see eq 51 for prolate coordinates and 63 for oblate coordinates). The values of j1 and j3 are calculated from the stream function outside the droplet, ψm (see eqs 51 and 52 for prolate coordinates and 63 and 64 for oblate coordinates). Since ψd is the stream function that was truncated, only j2 is approximate. The contribution of j2 to the overall stress difference is large when M, the ratio of the matrix viscosity to the droplet viscosity, is small. The accuracy of the ELDM is therefore reduced as the viscosity of the droplet becomes greater than that of the matrix. This effect is less important for systems in which droplets deform parallel to the electric field. In such cases, the contribution of the hydrodynamic stress difference to the overall stress difference is smaller than that of the electrical stress. However, the effect is pronounced in systems where the droplet is deforming perpendicular to the electric field, because, in such cases, the hydrodynamic stresses are contributing more to the overall aspect ratio than the electrical ones. It is therefore clear that, because of its inherent limitation, the ELDM may perform poorly for small values of M, when the droplets deform perpendicular to the electric field. Observations of Droplet Deformations. Figure 8 shows the steady state deformation of a CSH droplet in a matrix of EpoxLinsOil at two magnitudes of an applied dc electric field. The deformation of the droplet increased with the magnitude of the electric field up to the point at which the droplet started to break up. At the breakup point, the droplet first assumed a nonellipsoidal shape with pointed ends (Figure 8b). Immediately afterward, a stream of tiny droplets was ejected from the pointed ends and the drop broke up. This type of breakup has been described as the “tip-streaming” mechanism.4 Figure 9 shows the steady state deformation of a CasOil droplet in a matrix of DGEBA at various magnitudes of the applied dc electric field. After the CasOil was injected, it broke up into two droplets, one small and one large. Although CasOil was significantly less dense than DGEBA the droplets did not move toward the air-DGEBA interface during the experiment, probably because of the very high viscosity of DGEBA (6020 Pa‚s, Table 1). The large CasOil droplet deformed in the direction perpendicular to the electric field (oblate deformation) and its aspect ratio increased as the electric field increased. The small CasOil droplet did not change its deformation appreciably with the magnitude of the electric field. Although both droplets were subjected to an electric field of the same magnitude, their initial size was the determining factor in the magnitude of their deformation. As the magnitude of the electric field increased above 2.65 kV‚cm-1, the large droplet increased its deformation and a further increase in the electric field led to its breakdown into two smaller droplets (Figure 9, parts b and c). When the electric field was increased further the droplet that was created from the breakup of the originally large droplet also broke up (Figure 9d). Only one of the droplets formed from the breakup of the original droplet is visible in Figure 9, parts g and h; the other droplet broke up also. Although droplet breakup has been observed in this and other systems studied, it is important to note that the ELDM does not predict the critical electric field value at which breakup occurs.

6204



Figure 8. Optical micrographs of steady-state deformations of a droplet of CSH in a matrix of EpoxLinsOil under a dc electric field: (a) 1.00 and (b) 1.35 kV‚cm-1. Arrows indicate the direction of the electric field.

Figure 9. Optical micrographs of steady-state deformations of a droplet of CasOil in a matrix of DGEBA under a dc electric field: (a) 2.57, (b) 2.65, (c) 2.92, and (d) 3.12 kV‚cm-1. Arrows indicate the direction of the electric field.

Comparisons of Prolate Droplet Deformations with Theoretical Models. Figure 10 shows data obtained by Ha et al.28 for a water droplet in a silicone oil matrix under a dc electric field. It also shows the predictions of various models. The LDM (dotted line in Figure 10) incorrectly predicts a linear relation between R and Ce, and therefore fails to predict the trend of the data at higher aspect ratios and higher values of the Ce. Ha et al.28 used an FEM model introduced by Feng and Scott,29 to describe their experimental data. The Feng and Scott model (dashdot line in Figure) both correctly predicts the experimentally observed nonlinearity of R and Ce (qualitative agreement) and matches the experimental data closely (quantitative agreement). The Feng and Scott model was developed for conductive droplets in insulating matrices and, in this case, it succeeded in predicting the experimentally observed aspect ratios, probably because R ) σd/σm ) 108 (in other words because σd . σm and thus the matrix can be considered as a dielectric fluid). Similarly to the Feng and Scott model, the ELDM (solid line in Figure (28) Ha, J.-W.; Yang, S.-M. J. Fluid Mech. 2000, 405, 131-156. (29) Feng, J. Q.; Scott, T. C. J. Fluid Mech. 1996, 311, 289-326.

10) also agrees both qualitatively and quantitatively with the experimental data (the ELDM and the Feng and Scott model are indistinguishable up to R ) 1.3). However, the advantage of the ELDM over the Feng and Scott model is its broader applicability, since it was developed for arbitrary combinations of conductivity values. Figure 11 shows aspect ratio, R, vs electrical capillary number, Ce, data for droplets of CSH in a matrix of EpoxLinsOil under a dc field, obtained in this work. The predictions of the LDM, shown with a dotted line in the figure, once more disagree with the nonlinear relation between R and Ce exhibited by the experimental data. On the other hand, the predictions of the ELDM, shown with a solid line in the Figure, agree both qualitatively and quantitatively with the experimental data. Figure 12 shows a comparison of the predictions of the ELDM and experimental data for droplets of DGEBA in a matrix of PDMS deforming under a dc field, as obtained in this work. The ELDM deviates slightly from the experimental data for high droplet aspect ratios. Figure 13 shows data obtained by Ha et al.28 for a castor oil droplet in a silicone oil matrix under a dc electric field.


Figure 10. Aspect ratio, R, vs electrical capillary number, Ce, of droplets of water in a matrix of silicone oil under a dc field. Key: open circles, experimental data from Ha et al.;28 dotted line, predictions of the LDM; dash-dot line, predictions of the Feng and Scott29 model; solid line, predictions of the ELDM; dashed lines, upper and lower bounds of ELDM predictions based on presumed 5% error in S, 5% error in R and 5% error in M, as given in Table 2.

Figure 11. Aspect ratio, R, vs electrical capillary number, Ce, of droplets of CSH in a matrix of EpoxLinsOil under a dc field. Key: open circles and error bars,experimental data from this work; dotted line, predictions of the LDM; solid line, predictions of the ELDM; dashed lines, upper and lower bounds of ELDM predictions based on 7% error in S, 1% error in R and 1% error in M, as given in Table 2.

In this case, the experimental uncertainties in S, R, and M were not known, and a 5% error in each variable was assumed. As in the case of the systems described previously, the LDM does not agree either quantitatively, nor qualitatively with the experimental data. In this case, the Feng and Scott29 model, which was used by Ha et al. for comparison with experimental data, agrees qualitatively with the experimental data but performs poorly quantitatively. This is probably because, R ) σd/σm ) 10, in other words, the droplet is not enormously more conductive than the matrix, thus the matrix cannot be considered a pure dielectric and the Feng and Scott29 model should not be used. On the other hand, since the ELDM does not have a limitation on the ratio of the droplet/

Langmuir, Vol. 21, No. 14, 2005 6205

Figure 12. Aspect ratio, R, vs electrical capillary number, Ce, of droplets of DGEBA in a matrix of PDMS under a dc field. Key: open circles and error bars,experimental data from this work; dotted line, predictions of the LDM; solid line, predictions of the ELDM; dashed lines, upper and lower bounds of ELDM predictions based on 26% error in S, 1% error in R and 6% error in M, as given in Table 2.

Figure 13. Aspect ratio, R, vs electrical capillary number, Ce, of droplets of castor oil in a matrix of silicone oil under a dc field. Key: open circles, experimental data from Ha et al.;28 dotted line, predictions of the LDM; dash-dot line, predictions of the Feng and Scott29 model; solid line, predictions of the ELDM; dashed lines, upper and lower bounds of ELDM predictions based on presumed 5% error in S, 5% error in R and 5% error in M.

matrix conductivity, it agrees more closely with the experimental data. There is a small quantitative disagreement between the experimental data and the ELDM. However, this disagreement can be accounted for by incorporating the uncertainties of the S, R, and M ratios into the predictions of the ELDM. When these uncertainties are taken into account, the ELDM agrees quantitatively with the experimental data. Tsukada et al.9 studied systems similar to the ones of Ha et al.28 but they developed a specialized FEM model to describe their systems. Figure 14 shows data obtained by Tsukada et al.9 (for a vegetable oil droplet in a silicone oil matrix under a dc electric field) and the predictions of the LDM, the ELDM, and the Tsukada et al.9 model. The LDM, as usual, predicts a linear relation between R and

6206


Figure 14. Aspect ratio, R, vs electrical capillary number, Ce, of droplets of vegetable oil in a matrix of silicone oil under a dc field. Key: open circles, experimental data from Tsukada et al.;9 dotted line, predictions of the LDM; dash-dot line, predictions of the Tsukada et al.9 model; solid line, predictions of the ELDM; dashed lines, upper and lower bounds of ELDM predictions based on presumed 5% error in S, 5% error in R and 5% error in M.

Ce, and fails to predict the trend of the data for higher values of the aspect ratio. On the other hand, the Tsukada et al.9 model agrees qualitatively with the experimental data. The ELDM closely matches the predictions of the Tsukada et al.9 model. Part of the quantitative deviation of the ELDM from the experimental data can again be accounted for by assuming that the S, R, and M ratios were measured within a 5% precision. Once again, the ELDM exhibits a reasonable quantitative agreement with experimental data when such uncertainties are taken into account. Vizika and Saville8 studied several droplet/matrix systems under an electric field and compared their experimental results with the predictions of the LDM. They concluded that the LDM could not only forecast the direction in which a droplet would deform (prolate or oblate), but also accurately predict the magnitude of the droplet aspect ratio. However, for two systems such quantitative agreement was not exhibited. The experimental data for one of these systems, a water droplet in a castor oil matrix studied under a dc electric field, are shown in Figure 15. Clearly, the LDM does not agree with the experimental data even qualitatively, but the ELDM follows the data more closely. When experimental uncertainties are taken into account the ELDM agrees quantitatively with the data. Comparisons of Oblate Droplet Deformations with Theoretical Models. Figure 16 shows the predictions of the LDM and ELDM for a CasOil droplet in a DGEBA matrix under a dc field, along with experimental data obtained in the present work. In this case, the ELDM was reasonably successful in predicting the experimentally observed relation between aspect ratio and electrical capillary number for a droplet that deformed perpendicular to the field direction (oblate deformation). Figure 17 shows the aspect ratio vs electrical capillary number for a silicone oil (10 P) droplet in a castor oil matrix deforming in a dc field. Data were taken from Vizika and Saville.8 For this system, the ELDM predicts much higher deformations than the ones experimentally observed. Oddly enough, the ELDM exhibits an even larger dis-


Figure 15. Aspect ratio, R, vs electrical capillary number, Ce, of droplets of water in a matrix of castor oil under a dc field. Key: open circles, experimental data from Vizika and Saville;8 dotted line, predictions of the LDM; solid line, predictions of the ELDM; dashed lines, upper and lower bounds of ELDM predictions based on presumed 5% error in S, 5% error in R and 5% error in M.

Figure 16. Aspect ratio, R, vs electrical capillary number, Ce, of droplets of CasOil in a matrix of DGEBA under a dc field. Key: open circles and error bars are experimental data from this work; dotted line, predictions of the LDM; solid line, predictions of the ELDM; dashed lines, upper and lower bounds of ELDM predictions based on 26% error in S, 1% error in R and 6% error in M, as given in Table 2.

agreement with experimental data for a system of a silicone oil (125 P) droplet in a castor oil and Triton matrix deforming in a dc field (Figure 18). The common “denominator” in the last two systems is the small value of M: 1.40 in the first case and 0.08 in the second. For such systems, the ELDM may be limited in its predictive ability, as already indicated under “Limitations of the Extended Leaky Dielectric Model”. Figure 19 shows the experimental data obtained by Tsukada et al.9 for a silicone oil droplet in a vegetable oil matrix under a dc electric field. It also shows the predictions of the Tsukada et al.9 model along with the predictions of the LDM and of the ELDM. The Tsukada et al.9 model and the ELDM agree reasonably well with each other and they both disagree with the LDM. However,


Figure 17. Aspect ratio, R, vs electrical capillary number, Ce, of droplets of silicone oil (10 P) in a matrix of castor oil under a dc field. Key: open circles, experimental data from Vizika and Saville;8 dotted line, predictions of the LDM; solid line, predictions of the ELDM; dashed lines, upper and lower bounds of ELDM predictions based on presumed 5% error in S, 5% error in R and 5% error in M.

Langmuir, Vol. 21, No. 14, 2005 6207

Figure 19. Aspect ratio, R, vs electrical capillary number, Ce, of droplets of silicone oil in a matrix of vegetable oil under a dc field. Open circles, experimental data from Tsukada et al.;9 dotted line, predictions of the LDM; dash-dot line, predictions of the Tsukada et al.9 model; solid line, predictions of the ELDM; dashed lines, upper and lower bounds of ELDM predictions based on presumed 5% error in S, 5% error in R and 5% error in M. The dash-dotted line almost coincides with the right dashed line.

namic flow. Therefore, the effect of charge convection is more important for droplets deforming perpendicular to the electric field, since it is only then that the hydrodynamic stresses oppose the electrical ones. For droplets deforming parallel to the electric field, the effect is less significant. The relative significance of charge convection can be quantified by the electric Reynolds number, ReE, defined as the ratio of the time scales of the charge convection by flow and the charge relaxation by Ohmic conduction. According to Feng30 for a slightly deformed droplet

ReE ) U) Figure 18. Aspect ratio, R, vs electrical capillary number, Ce, of droplets of silicone oil (125 P) in a matrix of castor oil and Triton under a dc field. Open circles, experimental data from Vizika and Saville;8 dotted line, predictions of the LDM; solid line, predictions of the ELDM; dashed lines, upper and lower bounds of ELDM predictions based on presumed 5% error in S, 5% error in R and 5% error in M.

there is a striking feature in Figure 19: The LDM model follows the experimental data more closely than the ELDM and the Tsukada et al.9 model do. Both the ELDM and the Tsukada et al.9 model predict higher droplet deformations than the LDM does. In these cases, M is large, so that a small value of M cannot be at fault. There is, however the possibility, of a significant contribution to the aspect ratio from charge convection along the interface. The ELDM assumes that charge is conducted toward the interface and not convected along the interface. Feng30 considered the influence of charge convection on droplet deformation and breakup and according to his analysis, charge convection reduces the intensity of the electrohydrody(30) Feng, J. Q. Proc. R. Soc. London, Ser. A 1999, 455, 2245-2269.

τeU r0

|SR - 1| 9 γ C 2 10 (2 + R) (1 + M) µd e

(67a) (67b)

where τe is the total charge relaxation time and U the maximum fluid velocity. The electric Reynolds number can be large for fluid pairs with high permittivities, low conductivities, and for small droplets with low viscosity. Therefore, as the fluids become more viscous the electric Reynolds number decreases, and consequently, the charge convection effect diminishes. Following the suggestion by Feng,30 the importance of charge convection is based on the value of the total charge relaxation time τe. The possibility of charge convection was also addressed by Ha et al.31 For our case, charge convection cannot be ignored, since τe ) 0.353 s. It is therefore expected that both the ELDM and the Tsukada et al.9 model predict higher deformations than the experimentally obtained ones, but it is probably coincidental that the LDM predicts these deformations accurately, due to a fortuitous cancellation of errors. Table 3 summarizes the predictions of the ELDM for four droplet/matrix systems, whose droplets deformed perpendicular to the electric field. When either M is small (31) Ha, J.-W.; Yang, S.-M. Phys. Fluids 2000, 12, 764-772.

6208



Table 3. Quality of Agreement between Predictions of the ELDM and Experimental Data for Systems that Deformed Perpendicular to the Electric Field droplet/matrix

M

CasOil/DGEBAa silicone oil/castor oil8 silicone oil/castor oil and Triton8 silicone oil/vegetable oils9 a

τe/s

11.6 6.05 × 10-5 1.40 8 × 10-2 14.7 0.353

agreement good poor poor poor

This work.

or τe is long, the ELDM diverges from the experimentally determined aspect ratios even when experimental uncertainties are taken into account. Predictions of the ELDM for Possible Hysteresis and Continuous Deformation of the Systems Studied. For 9 of the 10 systems described in this paper, the ELDM predicts the possibility of hysteresis (area B in Figures 6 and 7) and for only one (castor oil/silicone oil, studied by Ha and Yang28) the possibility of continuous deformation (area B in Figures 6 and 7). For the systems studied in this work, we, and other workers, have not been able to observe hysteresis experimentally, but only droplet breakup. Ha and Yang28) reported that at high electric fields and for one of their castor oil/silicone oil systems, “the castor oil drop elongated into a thin thread with sharply pointed ends”.28 For this system, the ELDM predicts the possibility of continuous deformation. However, Ha and Yang28) also observed a similar elongation for a drop of glycerine-propanol solution of poly(vinylpyrrolidone) in a silicone oil matrix, for which the ELDM predicts hysteresis. It is therefore possible that droplet breakup could compete with hysteresis and continuous deformation under many conditions.

the ELDM for the two types of phenomena at high electric fields (hysteresis or continuous deformation). Appendix Procedure to Calculate Ce for a Specific Aspect Ratio. The relation between the droplet aspect ratio and the electrical capillary number for droplets deforming parallel to the electric field, can be found by following the procedure described below. This procedure, as well as a similar one for droplets deforming perpendicular to the electric field, has been included in two Matlab files, which are given in Supporting Information. 1. Start by picking a value for the aspect ratio, R, and calculate the following Legendre and Gegenbauer polynomials and their derivatives:

ξ0 )

The Torza et al.4 LDM, was extended to large droplet distortions in dc electric fields. The resulting extended LDM (ELDM) reduces to the LDM for small droplet aspect ratios and to the pure dielectric model when R ) S - 1. At high electric fields, the ELDM distinguishes between two types of possible phenomena: continuous deformation and hysteresis. For droplets deforming parallel to the electric field, the relationship that distinguishes between the two phenomena is a function of R and M but not of S. For droplets deforming perpendicular to the electric field, the relationship is a function of S and R but depends slightly on M. Comparisons of the LDM and the ELDM with experimental data showed the following. For systems that deformed parallel to the field direction, the LDM did not even predict the qualitative trend of the data. On the other hand, the ELDM not only predicted the data qualitatively but also quantitatively, when the experimental errors in its input parameters were taken into account. For the systems that deformed perpendicular to the field direction, the ELDM overestimated the observed aspect ratios in three out of the four cases compared. For those systems, either the value of M was relatively small or the value of total charge relaxation time was relatively large. Small values of M and large values of τe conflict with different assumptions of the ELDM. In such cases, therefore, the ELDM is expected to perform poorly. Unfortunately, no experimental data existed for testing the predictions of

xR

-1

P 1 ) ξ0 1 P2 ) (3ξ02 - 1) 2 1 P4 ) (35ξ04 - 30ξ02 + 3) 8 1 G3 ) - ξ0(ξ02 - 1) 2 1 G5 ) - ξ0(ξ02 - 1)(7ξ02 - 3) 8 Q0 )

5. Conclusions

R 2

1 ξ0 + 1 ln 2 ξ0 - 1

Q1 ) ξ0Q0 - 1 3 Q2 ) P2Q0 - ξ0 2 H1 ) -1 1 1 1 H3 ) - ξ0(ξ02 - 1)Q0 + ξ02 2 2 3 P′1 ) 1 1 G′3 ) - (3ξ02 - 1) 2 G′′3 ) -3ξ0 G′5 ) -P4 5 G′′5 ) - ξ0(7ξ02 - 3) 2 Q′1 ) Q0 - ξ0/(ξ02 - 1) H′1 ) 0 1 3 H′3 ) - (3ξ02 - 1)Q0 + ξ0 2 2 H′′3 ) -3ξ0Q0 + 1/(ξ02 - 1) + 3 2. Calculate the various constants obtained from the electrohydrodynamic boundary conditions:


Langmuir, Vol. 21, No. 14, 2005 6209

c1x ) -H3/H1

Φ ) ∆∆pe + ∆∆ph

d1x ) -G5/H1(H1H′3 - H′1H3)/(G3G′5 - G′3G5)

kR ) R1/3(2R - 1 - 1/R2)

d2x ) -G3/G5

Ce ) kR/Φ

f1/c )

Error Propagation. For a function f (x, y), for which the precision error is

2 2 (ξ - 3/7) 15 0

tνshf

2 f2 ) - ξ0 15

where shf is the standard error of the mean of f and tν is the Student t-statistic, the equation for the propagation of the standard error of the mean is

1 f3 ) (f1/c) 2 b1x ) (P1Q′1 - P′1Q1)/(P1Q′1 - RP′1Q1)

shf 2 )

1 c2xn ) (SR - 1)P1P′1(f1/c)b1x2 2 c2xd ) (1 - M)H′3f2 + (d1xG′′3 + d1xd2xG′′5 - MH′′3)f3 c2xn c2x ) c2xd

2

2 xj

2 yj

+ 2Fxy

∂f ∂f ss ∂x ∂y xj yj

Here sxj, syj are the standard errors of the means xj and yj and Fxy is the correlation coefficient between x and y. Assuming that all variables are measured independently from each other, Fxy ) 0 and the effective degrees of freedom of shf can be calculated from the Welch-Satterhwaite equation (see ref 32),

[ ] [ ] 2

∂f s 1 [∂x] 1 )

3. Calculate the difference in the electric “stress” between the pole and the equator:

νf

νx

shf 2

2

hf

2

2

∂f s 1 [∂y] + νy

2 yj

2

shf 2

where νx and νy are the degrees of freedom associated with the variables x and y.

1 ∆∆pe ) b1x2(S(R2 + 1) - 2) 2 3 j1 ) - c1x(1/G3) 2 7 j2 ) 7d1xd2x P4 log(1/R) + (ξ02 - 3/7) 8

(

2

[∂x∂f ] s + [∂y∂f ] s

)

j3 ) (H′3)2/ξ ∆∆ph ) (Mj1 + j2 + (M - 1)j3)c2x

Acknowledgment. This paper is based on work partly supported by the National Science Foundation under grant DMR-9521265. The authors thank the late Roland Lichtenstein for his help in the early development of the ELDM. The authors also thank Keith J. Nelson for the use of the conductivity apparatus and Toh-Minh Lu for the use of the permittivity apparatus in their laboratories. Supporting Information Available: Two Matlab files for the procedure described in the Appendix. This material is available free of charge via the Internet at http://pubs.acs.org. LA0472448

4. Apply the normal electrohydrodynamic boundary condition:

(32) Ku, H. H. J. Res. Natl. Bureau Stand., Ser. C 1966, 70C, 263273.

Droplet Deformation in Dc Electric Fields: The Extended Leaky

Recommend Documents