Development of a New Methodology To Study Drop Shape and

Development of a new methodology for the study of both shape and surface tension ... study drop shapes in the electric field and to determine its effe...
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Langmuir 2004, 20, 7589-7597

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Development of a New Methodology To Study Drop Shape and Surface Tension in Electric Fields A. Bateni,† S. S. Susnar,† A. Amirfazli,‡ and A. W. Neumann*,† Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario M5S 3G8, Canada, and Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta T6G 2G8, Canada Received March 5, 2004. In Final Form: May 17, 2004

Development of a new methodology for the study of both shape and surface tension of conducting drops in an electric field is presented. This methodology, called axisymmetric drop shape analysisselectric fields (ADSA-EF), generates numerical drop profiles in an electrostatic field, for a given surface tension. Then, it calculates the true value of the surface tension by matching theoretical profiles to the shape of experimental drops, using the surface tension as an adjustable parameter. ADSA-EF can be employed to simulate and study drop shapes in the electric field and to determine its effect on liquid surface tension. The method can also be used to measure surface tension in microgravity, where current drop-shape techniques are not applicable. The axisymmetric shape of the drop is the only assumption made in the development of ADSAEF. The new scheme is applicable when both gravity and electrostatic forces are present. Preliminary measurements using ADSA-EF suggest that the surface tension of water increases by about 2% when an electric field with the magnitude of 106 V/m is applied.

Introduction The application of electric fields to enhance or control industrial processes is increasing due to economic and environmental factors. Currently, charged or electrified drops play an important role in many technologies, such as chemical and physical separations,1-5 electrostatic painting and spraying,5-11 ink-jet printing,12 agricultural treatments,13-14 electro wetting,15-17 and electrostatic gas cleaning.18 The fact that an electrostatic field can deform, break up, or modify the surface properties of drops is of great importance from both practical and fundamental points of view. Further development of new technologies in any of the above areas requires a thorough understanding of the effects of the electric field on the drop * To whom correspondence should be addressed: A. W. Neumann, Professor, Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, Ontario M5S 3G8, Canada. Tel.: (416) 978-1270. Fax: (416) 978-7753. E-mail: [email protected]. † University of Toronto. ‡ University of Alberta. (1) Byers, C. H.; Amarnath, A. Chem. Eng. Prog. 1995, 63. (2) Pederson, A. J.; Ottosen, L. M.; Villumsen, A. J. Hazardous Mater. B 2003, 100, 65. (3) Nemoto, T.; Ishikawa, M.; Momota, H. Fusion Eng. Des. 2002, 63-64, 501. (4) Eow, J. S.; Ghadiri, M. Chem. Eng. Process. 2003, 42, 259. (5) Tang, K.; Smith, R. D. J. Am. Soc. Mass Spectrom. 2001, 12, 343. (6) Cloupeau, M.; Prunet-Foch, B. J. Electrost. 1990, 25, 165. (7) Hayati, I.; Bailey, A. I.; Tadros, T. H. F. J. Colloid Interface Sci. 1987, 117, 205. (8) Ku, B. K.; Kim, S. S. J. Electrost. 2003, 57, 109. (9) Bologa, A.; Bologa, A. J. Electrost. 2001, 51, 470. (10) Asano, K.; Yatsuzuka, K. J. Electrost. 1999, 46, 69. (11) Schmitt, C.; Lebienvenu, M. J. Mater. Process. Technol. 2003, 134, 303. (12) Twardeck, T. G. IBM J. Res. Dev. 1977, 21, 31. (13) Atungulu, G.; Nishiyama, Y.; Koide, S. Biosystems Eng. 2003, 85, 41. (14) Tuller, M.; Or, D. J. Hydrol. 2003, 272, 50. (15) Lee, J.; Moon, H.; Fowler, J.; Schoellhammer, T.; Kim, C. J. Sens. Actuators, A 2002, 95, 259. (16) Verheijen, H. J. J.; Prins, W. J. Langmuir 1999, 15, 6616. (17) Vallet, M.; Berge, B. Polymer 1996, 37, 2465. (18) Chang, J. S. J. Electrost. 2003, 57, 273.

Figure 1. Effect of an electrostatic field on the shape of sessile drops of water on Teflon coated silicon wafers. The figure shows that the shape of the drop is changed significantly when a 5-kV electric potential is applied to the capacitor with a 6-mm distance between the two plates. Note that the two drops are not of the same volume.

shape and properties such as drop stability, drop apex curvature, liquid surface tension, and contact angle. Generally, the electric field affects liquid drops or bubbles in two ways: the first effect is that the shape of the drop is changed in the electric field. This is a pronounced effect and can be easily observed experimentally (see Figure 1). Second, it is generally believed that the surface tension of liquids is changed when an electric field is applied. This is a relatively subtle effect and is more difficult to be detected or measured experimentally. The shape and stability of drops in the electric field have been investigated by many authors. Only a few employed analytical approaches.19,20 Due to the mathematical complexities of the problem, these studies were often based on simplifying assumptions such as assumed charge distribution or zero gravity. In recent years, numerical approaches were used frequently to predict drop shapes and stabilities in the electric field.21-32 In most of (19) Taylor, G. Proc. R. Soc. London, Ser. A 1964, 280, 383. (20) Warshavsky, V. B.; Shchekin, A. K. Colloids Surf., A 1999, 148, 283. (21) Borzabadi, E.; Bailey, A. G. J. Electrost. 1978, 5, 369. (22) Miksis, M. J. Phys. Fluids 1981, 24, 1967.

10.1021/la0494167 CCC: $27.50 © 2004 American Chemical Society Published on Web 08/06/2004

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these studies, the effect of the electric field on the drop surface tension was neglected. This assumption might be the source of the discrepancies between the predictions and the observations at high voltages. For sessile drop investigations, it was often assumed that either the contact angle or the contact line of the drop is unchanged when the electric field is applied.25,29 Moreover, in the most recent numerical studies25,27,28 it was assumed that the gravitational Bond number, G, is 0; that is, the gravitational effect is negligible compared to surface tension and electric field effects. These assumptions limit the application of the schemes just described for many realistic conditions. In addition to the above drop-shape studies, some theoretical and experimental work has been performed to detect the effect of an electric field on the surface tension of liquids;33-40 however, no general agreement can be found in the results. Most of the authors concluded that the surface tension is significantly changed in the electric field,35,36,39,40 while some believe that such change would be very small and negligible.33,38 The lack of a tool for high accuracy measurement of surface tension in the electric field is the main reason for the inconsistency in the results of these studies. Basaran and Scriven25 proposed earlier the idea of measuring surface tension from the stability limit of drops in the electric field; however, this idea has not been implemented to date. Considering these limitations, it was decided to develop new tools to (a) accurately measure the effect of the electric field on the surface tension of liquids, (b) simulate and study drop shapes in the electrostatic field, and (c) measure surface tension in microgravity conditions, where the drop is not deformed by gravity and the current drop-shape techniques are not applicable. A new methodology, called axisymmetric drop shape analysisselectric fields (ADSAEF), is presented for the study of surface tensions where both gravity and electric field forces are present. A secondary tool is also introduced, which can predict and simulate drop shapes in the presence of gravity and electric fields, using an iterative technique. This methodology is applicable to pendant and sessile drops, as well as bubbles. ADSA-EF consists of several numerical and experimental components. The objective of this article is to prove the concept for this methodology and to report on the development of a first version of ADSA-EF. Preliminary experiments are presented for the purpose of illustrating (23) Joffre, G.; Prunet-Foch, B.; Berthomme, S.; Cloupeau, M. J. Electrost. 1982, 13, 151. (24) Tsukada, T.; Sato, M.; Imaishi, M. J. Chem. Eng. Jpn. 1986, 19, 537. (25) Basaran, O. A.; Scriven, L. E. J. Colloid Interface Sci. 1990, 140, 10. (26) Wohlhuter, F. K.; Basaran, O. A. J. Fluid Mech. 1992, 235, 481. (27) Harris, M. T.; Basaran, O. A. J. Colloid Interface Sci. 1993, 161, 389. (28) Harris, M. T.; Basaran, O. A. J. Colloid Interface Sci. 1995, 170, 308. (29) Cho, H. J.; Kang, I. S.; Kweon, Y. C.; Kim, M. H. Int. J. Multiphase Flow 1996, 22, 909. (30) Notz, P. K.; Basaran, O. A. J. Colloid Interface Sci. 1999, 213, 218. (31) Adamiak, K. J. Electrost. 2001, 51-52, 578. (32) Moon, H.; Garrell, R.; Kim, C. J. J. Appl. Phys. 2003, 93, 5794. (33) Hayes, C. F. J. Phys. Chem. 1975, 79, 1689. (34) Morcos, I. J. Electrost. 1978, 5, 51. (35) Morimoto, Y.; Saheki, K. Jpn. J. Appl. Phys. 1979, 18, 1239. (36) Shchipunov, Y. A.; Kolpukou, A. F. Adv. Colloid Interface Sci. 1991, 35, 331. (37) Goodisman, J. J. Phys. Chem. 1992, 96, 6355. (38) Liggieri, L.; Sanfeld, A.; Steinchen, A. Physica A 1994, 206, 299. (39) Sato, M.; Kudo, N.; Saito, M. IEEE Trans. Ind. Appl. 1998, 34, 294. (40) Sanfeld, A. Philos. Trans. R. Soc. London, Ser. A 1998, 356, 819.

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the novel methodology, and, hence, the results of this investigation should be considered as early findings. ADSA-EF ADSA is a powerful methodology, originally developed for the study of drop shapes and properties in gravity, as the only external field. In this methodology, the image of the drop is acquired in an experiment and the drop profile is extracted using image-processing techniques. Assuming that the drop is axisymmetric, theoretical curves described by the Young-Laplace equation are fitted to the experimental drop profile, taking the surface tension and apex curvature as adjustable parameters. The values of surface tension, contact angle, apex curvature, drop volume, and drop surface area in an experiment can be determined from the theoretical profile that best matches the experimental drop shape. Several schemes of ADSA have already been implemented,41-45 which are applicable when only a gravitational field exists (in the absence of an electric field). These schemes have been widely used and were proven to be accurate and reliable. ADSA-EF, in addition to gravity, deals with the electrostatic field, which is variable from point to point, and its distribution is not known a priori. Thus, the electric field distribution must be calculated as a part of the ADSA-EF scheme. In essence, the equilibrium shape of a drop is determined by balancing the surface tension and the body forces such as gravity and electric field. Surface tension forces tend to make a drop spherical whereas gravity and electric field tend to elongate or flatten a drop depending on the direction of the body force. The mechanical equilibrium between the surface tension and body forces (i.e., the electric field and gravity) can be described mathematically by the so-called Young-Laplace equation21,29,31,43

(

γ

)

1 1 + ) ∆P0 + (∆F)gz + ∆Pe R1 R2

(1)

where γ is the surface tension, R1 and R2 are the two principal radii of curvature, ∆P0 is the pressure difference across the interface at the reference (e.g., the apex of the drop), ∆F is the density difference across the interface, g is the gravitational acceleration, z is the vertical distance of any point on the drop surface from the reference, and ∆Pe is the electrical pressure (i.e., the jump in the normal component of the Maxwell stress tensor across the interface).27-29 The second term on the right-hand side of eq 1 accounts for gravity, which is neglected when ADSAEF is employed for surface tension measurement in microgravity conditions. The value of ∆Pe at each point of the drop surface depends on the material properties, as well as the intensity of the electric field at that point; the latter is not known in advance. The distribution of the electric field can be calculated as the gradient of the electric potential. Therefore, the Young-Laplace equation needs to be solved in conjunction with the Laplace equation, (41) Rotenberg, Y.; Boruvka, L.; Neumann, A. W. J. Colloid Interface Sci. 1983, 93, 169. (42) Chen, P.; Li, D.; Boruvka, L.; Rotenberg, Y.; Neumann, A. W. Colloids Surfaces 1990, 43, 151. (43) Lahooti, S.; del Rı´o, O. I.; Cheng, P.; Neumann, A. W. In Applied Surface Thermodynamics; Neumann, A. W., Spelt, J. K., Eds.; Marcel Dekker: New York, 1996; p 441. (44) Amirfazli, A.; Graham-Eagle, J.; Pennell, S.; Neumann, A. W. Colloids Surf., A 2000, 161, 63. (45) Alvarez, J. M.; Amirfazli, A.; Neumann, A. W. Colloids Surf., A 1999, 156, 163.

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Figure 3. The boundaries and the integration domain of the electrostatic-field problem, that is, a conducting sessile drop formed on a lower plate of a capacitor. Taking advantage of the axisymmetric nature of the problem, the distribution of the electric field should be calculated over a two-dimensional domain.

Figure 2. Algorithm and the structure of the ADSA-EF methodology. The two key modules, that is, the electrostaticfield and the drop-shape modules, calculate the distribution of the electric field and simulate the shape of the drop, respectively. The optimization scheme (showed by the dotted ellipse) calculates the optimum values of the surface tension and the apex curvature by finding the best match between the numerical and the experimental drop profiles.

which describes the distribution of the electrostatic potential (V)21,27-29

∇2V ) 0

(2)

The electric pressure, ∆Pe, in eq 1 can then be calculated from the values of the electric field along the drop surface area (see next section). Figure 2 shows the algorithm and the main structure of ADSA-EF. A drop is formed in a parallel plate capacitor, and the electric field is applied. Using an image processing module, images of the drop are acquired and the experimental drop profile is extracted. The experimental profile is used to calculate the distribution of the electric field and to evaluate the initial values of the optimization parameters, that is, the drop apex curvature and the liquid surface tension. Using these values, a numerical shape of the drop is generated. Then through an iterative scheme the optimization parameters are adjusted to fit the numerical profile into the experimental one. Two main modules are needed for the development of ADSA-EF: (1) the electrostatic-field module, for the calculation of the electric field along the drop surface, and (2) the drop-shape module for the calculation of theoretical drop shapes in the electric field, given the distribution of the field along the drop surface. No analytical approach is known, so far, for either the drop-shape or the electrostatic-field module. Thus, numerical schemes have been developed for this purpose, which are described in the next sections. Electrostatic-Field Module This module calculates the distribution of the electrostatic field along the drop surface, given the capacitor geometries and the shape of the drop as input. The latter is obtained by extracting the drop profile from the experimental images (see Figure 2). When the scheme is used as a predictive tool (no experimental drop available), the shape of the drop will be calculated numerically by the drop-shape module (see the following). The calculated electrostatic fields will be provided as input to the other modules of ADSA-EF, which fit the numerical drop profiles to the experimental ones through an optimization scheme.

The electrostatic-field module first calculates the distribution of the electrostatic potential within the capacitor using eq 2. Then it determines the electric field at any point as the gradient of the electrostatic potential. Taking advantage of the axisymmetric nature of the problem, the governing equation can be expressed (in cylindrical coordinates) as a two-dimensional differential equation (see Figure 3)

∂2V ∂2V 1 ∂V )0 + 2+ x ∂x ∂x2 ∂z

over Ω

(3)

where x is the radial coordinate, z is the coordinate measured along the axis of symmetry, and Ω is the integration domain. The integration domain and the problem boundaries for a simple case where a conducting drop is formed inside a parallel plate capacitor are shown in Figure 3. When the drop consists of a conducting liquid, the drop surface can be considered as an equipotential area. Therefore, the boundary conditions for this case can be defined as (see Figure 3)

V ) V0 V)0

on STOP

(4-1)

on SBOTTOM and SDROP

(4-2)

En ) n∇V ) 0

on SSYM and SINF

(4-3)

where En is the normal component of the electric field at the boundary and n is the unit vector normal to the surface. A second-order finite difference scheme with a Cartesian grid has been developed to solve the dimensionless form of the system of equations just described (i.e., eqs 3 and 4). Linear interpolation techniques were used to approximate the values of the electric potential at the nodal points that do not coincide with the boundary (e.g., the nodal points next to the curved drop boundary).46,47 Implementation of the finite difference scheme results in a large system of linear algebraic equations. Several iterative methods (i.e., Jaccobi, Gauss-Seidel, and successive over-relaxation) were examined for solving the system of equations.47-49 It was found that the successive over-relaxation method, with the optimum acceleration parameter, outperforms the other iterative techniques. The stability of the scheme was examined by solving the problem for a wide variety of drop and capacitor geom(46) Fox, L. Numerical Solutions of Ordinary and Partial Differential Equations; Pergamon Press: New York, 1962. (47) Smith, G. D. Numerical Solutions of Partial Differential Equations: Finite Difference Methods, 3rd ed.; Oxford University Press: New York, 1985. (48) Ames, W. F. Numerical Methods for Partial Differential Equations; Academic Press, Inc.: London, 1977. (49) Li, R.; Chen, Z.; Wu, W. Generalized Difference Methods for Differential Equations; Marcel Dekker: New York, 2000.

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Table 1. Calculated (Numerical) Electric Field at the Apex of a Hemispherical Dropa number of mesh points

numerical electric field at the apex

20 50 100 150 200 250

2.287 2.628 2.801 2.868 2.903 2.924

number of mesh points

numerical electric field at the apex

300 400 500 800 1000

2.939 2.958 2.969 2.986 2.992

a The first and third columns corresponds to the number of mesh points in the radial direction (i.e., along the capacitor plate). Electric field values are dimensionless (see Figure 4). The results converge to the true value of the electric field, which is equal to three (see Figure 5), as the number of mesh points increases.

etries. This analysis showed that in all cases the iterative scheme converges to the final solution without any stability problems. The following third-order difference formula47 was used to calculate the component of the electrostatic field along the radial coordinate (x direction) from the values of the electric potential at the nodal points

∇V Bx )

2Vi+4 - 9Vi+3 + 18Vi+2 - 11Vi+1 6h

(5)

where h is the distance between the nodes and i is an index referring to the nodal points in the radial direction. A similar formula was used for the calculation of the component of the electrostatic field along the axis of symmetry (i.e., z direction). The sensitivity of the numerical results to the number of mesh points was studied next. The value of the electric field at the drop apex was considered for this purpose, because (1) this value includes the additional error due to the curved drop boundary (i.e., the error of linear interpolation mentioned earlier), (2) the maximum gradient of the electric potential and, hence, the maximum numerical error occurs at the drop apex (see the following), and (3) the accuracy of the calculated electric field at the drop apex is crucial for the simulation of drop profiles (see next section). Table 1 shows the numerical values of the dimensionless electric field at the apex of a hemispherical drop for different numbers of mesh points. The results converge to the true electric field, which is equal to 3 (see the following), with increasing the number of mesh points. The accuracy of the finite difference results can be further improved by calculating the order of the discretization error.47 Let Ei be the calculated electric field at the drop apex for mesh length hi. Taking the leading term in the discretization error proportional to hp, the value of p can be estimated from47

2p )

E2 - E1 E3 - E2

(6)

where h1 ) 2h2 ) 4h3. Using the values of Table 1 for 250, 500, and 1000 mesh points as inputs results in a value of p ) 0.938 for the order of the discretization error. Then, the magnitude of the electric field at any node can be calculated with higher accuracy as47

E)

h2pE1 - h1pE2 h2p - h1p

(7)

For instance, by substituting the results of Table 1 for 100 and 150 mesh points in eq 7 the electric field at the apex

Figure 4. The distribution of the electrostatic field along with the equipotential lines calculated over the integration domain. The length of the arrows signifies the magnitude of the electric field. Dimensionless variables were used for the calculation, so that the electric field is 1 (E ) 1) far from the drop, where the electric field is uniform. The maximum electric field (the highest density of equipotential lines) occurs at the apex of the drop. The electric field then decreases continually from the apex to the contact point.

is calculated as E ) 3.012, significantly more accurate that the original values shown in Table 1. Similar analysis was conducted for different numbers of mesh points. It was found out that by solving the problem using 200 and 400 mesh points and taking advantage of eq 7, the electric field can be calculated efficiently with respect to computation time and a maximum error of less than 1%. Figure 4 illustrates the calculated distribution of the electrostatic field (arrows) and the equipotential lines, when a conducting sessile drop is formed in a parallel plate capacitor. Dimensionless variables were defined so that the magnitude of the electric field is unity far from the drop, where the distribution of the electric field is uniform. The figure shows that the electric field is maximum at the apex, and it is almost 0 at the contact point of the drop. This was anticipated, because the drop is conducting and there is no charge density at the contact point. To validate the formulation and the implementation of the numerical scheme, the results were compared with the analytical solutions, which exist for the simple case of hemispherical drops. For this case, the method of images50 can be used to calculate the distribution of the electric field at the drop surface as25

E ) 3 cos(θ), 0 e θ e π/2 along the drop surface (8) where θ is the polar angle measured from the drop apex. Figure 5 shows the numerical and analytical electric fields calculated along the drop surface (i.e., from the apex to the contact point). The figure shows that the numerical results agree well with the analytical solution, validating the numerical scheme. The figure also illustrates the distribution of the radial and the vertical components of the electric field, calculated numerically (see eq 5). Next, the effect of the drop geometry on the distribution of the electric field was studied. The electrostatic field distributions were calculated along the surface of hypothetical drops, represented by elliptical shapes with different aspect ratios (see Figure 6). It was found that the intensity of the electric field increases at the drop apex with increasing aspect ratio of the drop, while in all cases the electric field is 0 at the contact point. The calculated electrostatic field distribution is provided as (50) Jeans, J. Mathematical Theory of Electricity and Magnetism; Cambridge University Press: Cambridge, U.K., 1960; p 185.

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Figure 7. Coordinate system used for the integration of the drop shapes in the electric field, that is, the drop-shape module.

respectively. When the drop is a conducting liquid there is no electric field inside the drop and the electric field is normal to the drop surface. Thus, the governing equation, that is, eq 1, for conducting drops can be simplified as Figure 5. Comparison of the analytical solution (solid line) with the numerically calculated electric field at several points along the drop surface (circles). The good agreement validates the numerical scheme. The illustrated radial and the vertical components of the electric field were used in the calculation of the numerical electric field. The polar angle was measured from the apex to the contact point.

(

γ

)

1 1 1 + ) ∆P0 + (∆F)gz + (a)‚E(a)2 n R1 R2 2

(10)

The two-dimensional differential form of eq 10 can be obtained when the principal radii of curvature (i.e., R1 and R2) are replaced with the corresponding differential terms for the axisymmetric shape43

dφ 1 ) R1 ds

(11-1)

1 sin φ ) R2 x

(11-2)

where φ is the angle of inclination of the interface to the horizontal and s is the arc length from the apex (see Figure 7). As a result of the symmetrical nature of the problem, the curvature at the apex is constant in all directions,43 that is, the two principal radii of curvature are equal.

1 1 1 ) ) )b R1 R2 R0 Figure 6. Effect of the drop geometry on the distribution of the electric field. Synthetic/hypothetical elliptical drops are used for demonstration.

input to the drop-shape module, which calculates numerical drop profiles in the electric field (see next section). Drop-Shape Module The drop-shape module numerically integrates the Young-Laplace equation, that is, eq 1, to simulate the shape of conducting drops when both gravity and electric field are present. The values of the surface tension, drop apex curvature, and electric field distribution are needed as inputs to this module, which are calculated by the optimization scheme and the electrostatic-field module of ADSA-EF (see Figure 2). The calculated numerical drop profiles will then be fitted to the experimental ones, through an optimization process (see Figure 2). The electrical pressure on the right-hand side of eq 1, ∆Pe, depends on the magnitude and direction of the electric field, as well as the permittivity of the fluids25,26,28

1 ∆Pe ) [(a)E(a)2 - (b)E(b)2 + ((b) - (a))Et2] n n 2

(9)

where En and Et are the normal and the tangential components of the electric field at the drop surface,  is the permittivity of the fluid, and superscripts a and b refer to the surrounding fluid and the drop liquid,

at the apex (s ) 0) (12)

where R0 is the radius of curvature and b is the curvature, both at the apex. Moreover, by defining the apex as the origin (i.e., z ) 0 at the apex), the gravitational term vanishes, and eq 10 for this point reduces to

1 2bγ ) ∆P0 + (a)E(a)2 n 2

at the apex (s ) 0)

(13)

The second term on the right-hand side of the eq 13 is a known constant, and its value can be calculated by the electrostatic-field module. Therefore, the pressure difference at the reference (apex of the drop) can be expressed as

(

∆P0 ) γ 2b -

K γ

)

at the apex (s ) 0)

(14)

where K is the known electric-field term in eq 13. Substituting eqs 11 and 14 into eq 10 yields

K 1 dφ sin φ ) 2b - + (a)E(a)2 n ds γ 2γ x

(15)

Equation 15 together with the geometric relations43

dx ) cos φ ds

(16)

dz ) sin φ ds

(17)

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Figure 8. Prediction of the drop shapes using the ALFI-EF scheme. The scheme calls the electrostatic-field and the drop-shape modules iteratively to converge to the final shape of the drop. An initial hemispherical drop converges to the experimental image in three iterations.

form a set of first-order differential equations in terms of φ, x, and z as functions of the arc length, s. The boundary conditions for the drop-shape problem can be defined as (see Figure 7)

φ(0) ) x(0) ) z(0) ) 0

at the apex (s ) 0)

(18)

dφ )b ds

at the apex (s ) 0)

(19-1)

dx )1 ds

at the apex (s ) 0)

(19-2)

dz )0 ds

at the apex (s ) 0)

(19-3)

A fourth-order Runge-Kutta51 scheme with an adaptive step-size control was implemented to solve this system of differential equations (i.e., eq 15-19). The stability of the program has been tested by generating numerical drop profiles for a wide range of input parameters (within the stability limit of the drop), that is, surface tension, apex curvature, and magnitude of the electric field. No limitation was observed with respect to the computational effort or stability of the program. Axisymmetric Liquid Fluid InterfacesElectric Fields Axisymmetric liquid fluid interfaceselectric fields (ALFI-EF) is a side product of ADSA-EF, which can be used as a predictive tool to simulate and study drop shapes in the electric field. The required values for ALFI-EF are (51) Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes The Art of Scientific Computing; SpringerVerlag: New York, 1986; p 710.

the drop properties (i.e., surface tension and the curvature at the apex), the capacitor geometry, and the magnitude of the applied electric potential. Then the scheme generates numerical drop profiles by employing the electrostaticfield and the drop-shape modules. As mentioned previously, the shape of a drop in the electric field is affected by the electric pressure at the drop surface (see Figure 1). Conversely, the magnitude and the distribution of the electric field within the capacitor are affected by the drop shape (see Figure 6). Therefore, to predict a drop shape in the electric field, both the Young-Laplace equation and the Laplace equation should be solved simultaneously, forming a free boundary problem. Using the successive approximation approach, the free boundary problem is uncoupled into two simpler and independent problems, which are solved successively as follows: 1. The procedure starts from an initial shape of the drop (e.g., a hemispherical shape). 2. The given drop shape is assumed to be fixed and independent of the electric field. 3. The electrostatic field distribution is calculated along the drop surface, by employing the electrostatic-field module. 4. The shape of the drop is updated by employing the drop-shape module, given the calculated electrostatic field distribution from step 3. 5. The cycle (steps 2-4) is repeated until convergence is achieved. It turns out that drop shapes can be predicted quite well using this procedure, even at high voltages. As an illustration, Figure 8 shows numerically generated drop profiles for a sessile drop of water on a poly(methyl methacrylate) (PMMA)-coated surface, when an electric potential of 7.5 kV is applied (see the following for the

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Figure 9. Calculated electric field (dimensionless) at the apex of the drop, at each iteration of ALFI-EF shown in Figure 8. The electric field is assumed 0 at the initial iteration (iteration 0).

experimental setup). The figure shows that the scheme converges to the experimental drop image in three iterations. Figure 9 illustrates the calculated electric field at the apex of the drop at each iteration. Experiments and Results A series of experiments were conducted to illustrate the ADSA-EF methodology. For this purpose, the change in the surface tension of water as a result of an applied electric field was investigated, using a sessile drop configuration. The experiments were conducted by forming drops of distilled water on a PMMA-coated silicon wafer, inside a capacitor. Small sessile drops were used in the electric field because small drops are stable over a wider range of applied electric field. However, in the absence of an electric field, such small drops result in very spherical shapes that provide little information about the liquid surface tension. Therefore, a larger drop, that is, more deformed due to gravity, was used for zero electric field. The drop was growing slowly (about 1 mm/min) during the experiment to cause an advancing three-phase line. Due to contact angle hysteresis, an advancing contact line is desirable to ensure the applicability of the YoungLaplace equation.52 The validity of the assumption of perfect conductivity of distilled water was examined before by Notz and Basaran,30 by comparing the electrical relaxation time to the drop formation time.53 It was shown that this assumption is valid when the liquid flow rate is small. The capacitor is made of two parallel disks with the radius of 30 mm and a separation distance of 6 mm (see Figure 3). The plates are horizontal with respect to the direction of gravity. The upper disk of the capacitor was connected to a high-voltage power supply, and the lower disk was grounded. The drop and the capacitor plates share the same axis of symmetry, which is along the direction of gravity. The body of the electric field cell was manufactured from Delrin, a polymer which is chemically inert and has good machining characteristics. The body allows the upper plate to be placed at variable distances from the lower disk. The plates of the capacitor were made of copper because it is both easily machined and has good electrical conductivity. The volume of the drop was controlled by a motor-driven syringe and a stepper motor controller (model 18705, Oriel Instruments, CT, U.S.A.). The drop was illuminated by a white light source (model V-WLP1000, Newport Corp., Fountain Valley, CA, U.S.A.). Images were acquired using a system consisting of a microscope (52) Kwok, D. Y.; Neumann, A. W. Adv. Colloid Interface Sci. 1999, 81, 167. (53) Melcher, J. R.; Taylor, G. I. Annu. Rev. Fluid Mech. 1969, 1, 111.

Figure 10. Comparison of several pixels from the experimental drop profile (extracted by edge detection) with the simulated drop shape, for sessile drops of water on a PMMA-coated wafers at 7.5 kV.

(Apozoom, Leitz Wetzlar, Germany), a charge-coupled device camera (model 4815-5000, Cohu Co., U.S.A.), and a digital video processor (Parallax Graphics, CA, U.S.A.). The latter performs the frame grabbing and digitization of the image to 640 by 480 resolution with 256 gray levels. The images were stored in a SUN workstation (Sparc Station-10, Sun Microsystems, Inc., U.S.A.) for further analysis. The Canny edge detection technique54 was used to extract the experimental profile of the drop. Given the drop profiles as input, the electric field distribution was calculated by the electrostatic-field module. The apex curvature of the drop was calculated by fitting a portion of an ellipse to the pixels of the experimental drop profile close to the apex. Then the curvature was calculated from

b)

ap an2

(20)

where ap and an are the axes of the ellipse parallel and normal to the axis of symmetry, z. Next, the numerical drop profiles were generated using the drop-shape module. Figure 10 shows the comparison of the experimental drop profile (extracted by edge detection) with the simulated drop shape, for a sessile drop of water on a PMMA-coated wafer when an electric potential of 7.5 kV is applied. The two curves completely coincide; therefore, for better graphical representation only several arbitrary pixels of the experimental profile are shown. The figure indicates good agreement between the experimental and numerical profiles in the electric field. This agreement is quantified by calculation of the correlation coefficient between the experimental drop profile and the numerically generated one. Then by means of a manual optimization, the surface tension was adjusted to maximize the correlation coefficient. Figure 11 shows the calculated correlation coefficients for several values of surface tension. The figure shows that the optimum surface tension (corresponding to the maximum correlation coefficient) is shifted by 1.4 mJ/m2 when an electric potential of 7.5 kV is applied. That is, the liquid surface tension increases with increasing the magnitude of the electric field. (54) Parker, J. Algorithms for Image Processing and Computer Vision; John Wiley & Sons: New York, 1997.

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Figure 11. Correlation coefficient, between the experimental and the numerical drop profiles versus the surface tension. The correlation coefficient was calculated for different values of the surface tension (i.e., 72.6, 72.8, ..., 74.6) for drops of water on the PMMA surface at zero and 7.5 kV. The graph indicates that the optimum surface tension (corresponding to the maximum correlation coefficient) is shifted by 1.4 (mJ/m2) when an electric potential of 7.5 kV is applied.

Figure 12. Optimum values of the surface tension (corresponding to the maximum correlation coefficient) versus the magnitude of the applied electrostatic potential. A regression line is fitted to the observed points (dashed line) suggesting that the surface tension of water increases with increasing electric field.

A similar procedure was performed for drop images obtained at different electric potentials. Figure 12 shows the calculated surface tension (i.e., the surface tension at which the maximum correlation coefficient was obtained) versus the applied electric potential. A linear regression line was fitted to the observed points. Figure 12 shows the slope and the intercept of the fitted line. The significance of the observed positive trend in the surface tension was examined using a statistical test of hypothesis.55 That is, the null hypothesis of H0:SLOPE ) 0 was tested versus H1:SLOPE * 0. The variation of the experimental error around the regression line (i.e., the model variance) was estimated by n

2

S )

∑ i)1

2 (γobs - γest i i )

n-2

(21)

(55) Walpole, R. E.; Myers, R. H.; Myers, S. L.; Ye, K. Probability and Statistics for Engineers and Scientists; Prentice Hall, Inc.: Upper Saddle River, NJ, 2002; p 364.

where γobs is the observed surface tension value, γest is the estimated value by the linear model, and n is the number of observations. Then the statistic t0 was calculated as

t0 )

x

SLOPEobs S

n

(Vi - V h )2 ∑ i)1

where SLOPEobs is the calculated slope of the regression line and Vi is the applied electric potential (i.e., 0, 2, 4, 6, 7.5 kV). The resulting value of the statistic (i.e., t0 ) 5.79) was compared with the value of the t distribution with n - 2 degrees of freedom and R ) 0.05 significance level (i.e., t0.05;3 ) 3.18). The null hypothesis was rejected because the calculated statistic is larger than the value of the t distribution (i.e., t0 > t0.05;3). That is, the increase in the value of the surface tension (i.e., the positive slope) as a result of applied electric potential is significant at the confidence level of 95%. It is believed that the observed change in the water surface tension might be the result of the rearrangement

Methodology To Study Drop Shape and Surface Tension

of the molecules at the drop surface after applying the electric field, that is, the electric field enhances the average cluster size and the cluster integrity of water molecular structure,56 hence, increasing the surface tension. Such information about the effect of the electric fields on surface tension was not available heretofore. Summary Development of a new methodology called ADSA-EF was presented. The axisymmetric shape of the drop was the only assumption made in the development of ADSAEF, which is not a restrictive assumption in most practical cases. The new methodology is applicable when both gravity and electrostatic forces are present and can be employed to determine the effect of the electric field on the surface tension of drops, which is of great importance from the fundamental point of view as well as a wide range of industrial applications. ADSA-EF can also be used to (56) Chaplin, M. F. Biophys. Chem. 1999, 83, 211.

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measure liquid surface tensions in microgravity conditions, when the drops are not deformed by the gravity and the current drop-shape techniques are not applicable. As a side product of ADSA-EF, an iterative scheme, called ALFI-EF, was developed. ALFI-EF can predict and simulate the shape of conducting drops (and bubbles) in the electric field, when both gravity and electric field forces are present. As an illustration of the ADSA-EF methodology, the surface tension of water in the electric field was investigated. The results suggest that the liquid surface tension increases by about 2% with increasing magnitude of the electric potential up to 106 V/m. Acknowledgment. This investigation was financially supported by the Canadian Space Agency (Contract No. 9F007-006051/001/ST) and a University of Toronto Open Fellowship. LA0494167