Determination of Surface Tension and Contact Angle from the Shapes

Surface properties of materials, such as surface tensions, contact angles, and .... determining the shape of the interface (capillary number and geome...
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Langmuir 2006, 22, 10053-10060

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Determination of Surface Tension and Contact Angle from the Shapes of Axisymmetric Fluid Interfaces without Use of Apex Coordinates M. Guadalupe Cabezas,† Arash Bateni,‡ Jose´ M. Montanero,† and A. Wilhelm Neumann*,‡ Department of Electronic and Electro-mechanical Engineering, UniVersity of Extremadura, AVda. de ElVas s/n, 06071 Badajoz, Spain, and Department of Mechanical and Industrial Engineering, UniVersity of Toronto, 5 King’s College Road, Toronto, Ontario M5S 3G8, Canada ReceiVed July 4, 2006. In Final Form: September 7, 2006 Drop shape techniques, such as axisymmetric drop shape analysis, are widely used to measure surface properties, as they are accurate and reliable. Nevertheless, they are not applicable in experimental studies dealing with fluid configurations that do not present an apex. A new methodology is presented for measuring interfacial properties of liquids, such as surface tension and contact angles, by analyzing the shape of an axisymmetric liquid-fluid interface without use of apex coordinates. The theoretical shape of the interface is generated numerically as a function of surface tension and some geometrical parameters at the starting point of the interface, e.g., contact angle and radius of the interface. Then, the numerical shape is fitted to the experimental profile, taking the interfacial properties as adjustable parameters. The best fit identifies the true values of surface tension and contact angle. Comparison between the experimental and the theoretical profiles is performed using the theoretical image fitting analysis (TIFA) strategy. The new method, TIFA-axisymmetric interfaces (TIFA-AI), is applicable to any axisymmetric experimental configuration (with or without apex). The versatility and accuracy of TIFA-AI is shown by considering various configurations: liquid bridges, sessile and pendant drops, and liquid lenses.

1. Introduction Surface properties of materials, such as surface tensions, contact angles, and hydrophobicity of solids, are of interest from both fundamental and practical points of view. These quantities have attracted growing attention recently due to their crucial role in advanced technologies such as micro-electromechanical systems (MEMS) and labs-on-chips, where surface forces are dominant. The shape of a liquid-fluid interface depends on the surface properties and is given by the Laplace equation. Therefore, surface properties can be measured by analyzing the experimental shape of a liquid-fluid interface and comparing it with the solution of the Laplace equation. On the basis of this principle, drop shape methods, such as axisymmetric drop shape analysis (ADSA), have been developed.1-6 These methods are widely used, as they are very accurate and reliable. They calculate the theoretical shape of an axisymmetric interface with apex (i.e., a drop or a bubble) by numerically solving the Laplace equation using the initial conditions at the apex (see, for instance, ref 7). Then, the theoretical shape of the configuration can be calculated depending on only two variables: the capillary number and the curvature at the apex. An image of the fluid configuration is acquired in an experiment, and the experimental profile extracted by an edge detection procedure. An error function is defined that measures the distance between the theoretical and the experimental * To whom correspondence should be addressed. Tel: (416) 978-1270. Fax: (416) 978-7753. E-mail: [email protected]. † University of Extremadura. ‡ University of Toronto. (1) Rotenberg, Y.; Boruvka, L.; Neumann, A. W. J. Colloid Interface Sci. 1983, 93, 169-183. (2) Anastasiadis, S. H.; Chen, J. K.; Koberstein, J. T.; Siegel, A. F.; Sohn, J. E.; Emerson, J. A. J. Colloid Interface Sci. 1987, 119, 55-66. (3) Pallas, N. R.; Harrison, Y. Colloids Surf. 1990, 43, 169-194. (4) Hansen, F. K. J. Colloid Interface Sci. 1993, 160, 209-217. (5) Demarquette, N. R.; Kamal, M. R. Polym. Eng. Sci. 1994, 34, 1823-1833. (6) Loglio, G.; Tesei, U.; Pandolfini, P.; Cini, R. Colloids Surf., A 1996, 114, 23-30. (7) 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; Vol. 1, Chapter 10.

profiles. The error depends on the variables determining the shape of the interface (mentioned above) and its orientation and position in the image. The coordinates of the apex are used to define the interface position in the image. Finally, the values of these five variables are adjusted to find the minimum error, i.e., the theoretical profile that best fits the experimental one. The values for the surface properties are those corresponding to the minimum error. The accuracy of ADSA-type techniques depends crucially on the quality of the experimental profile extracted by the edge detection procedure. For that reason, edge detectors in ADSA are powerful gradient-based edge detection techniques, such as those of Sobel or Canny.8,9 A new approach that eliminates the need for an independent edge detector was developed recently.10,11 This new method, termed theoretical image fitting analysis (TIFA), combines the image processing tasks with the comparison with the theoretical shape. TIFA, as edge detectors in current drop shape techniques, is gradient based. However, it differs from them in one crucial respect: TIFA takes into account that the extracted edge should correspond to a Laplacian drop profile. While edge detectors select edge pixels individually (by considering if the gradient at each pixel is a maximum), TIFA searches for the Laplacian profile whose corresponding set of pixels in the image have the highest accumulated gradient. TIFA results for pendant drops and captive bubble experiments were compared with those of ADSA, showing good agreement.10 It was demonstrated that TIFA is robust and noise-resistant. Both approaches, ADSA and TIFA, have one important common feature: they require the liquid-fluid interface to present an apex. This is a major drawback for certain applications. Many experimental studies deal with axisymmetric fluid configurations (8) Hoorfar, M.; Neumann, A. W. J. Adhes. 2004, 80, 727-743. (9) Zuo, Y. Y.; Ding, M.; Bateni, A.; Hoorfar, M.; Neumann, A. W. Colloids Surf., A 2004, 250, 233-246. (10) Cabezas, M. G.; Bateni, A.; Montanero, J. M.; Neumann, A. W. Colloid Surf., A 2005, 255, 193-200. (11) Cabezas, M. G.; Bateni, A.; Montanero, J. M.; Neumann, A. W. Appl. Surf. Sci. 2004, 238, 480-484.

10.1021/la061928t CCC: $33.50 © 2006 American Chemical Society Published on Web 10/18/2006

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without apex. In those cases, neither the standard drop shape techniques nor TIFA are applicable. For instance, experiments under microgravity conditions frequently focus on an axisymmetric fluid configuration known as liquid bridge (see, for example, refs 12-15). The equilibrium shapes, stability limits, and both linear and nonlinear dynamics of liquid bridges have been extensively studied over the past years both theoretically and experimentally. In the experiments, an accurate knowledge of the surface tension is crucial to obtain reliable results. The simple procedures developed for that purpose are based on the same principles as drop shape techniques. The experimental profile is extracted from liquid bridge images manually12,13 or by simple edge detection procedures.14,15 Then, the theoretical shape (a second-order analytical approximation15 or the numerical solution of the Laplace equation13,14) is fitted to the experimental profile, taking the surface tension as an adjustable parameter. However, these procedures are mainly manual, and the measured surface tension values are not accurate. To the best of our knowledge, no procedure has been developed to measure the surface tension from the shape of a liquid bridge automatically and accurately. In line tension research, experiments deal with configurations that present a three-phase line. Liquid-liquid-fluid systems are of special interest, as there is no solid involved, which avoids the issue of contact angle hysteresis. The most popular liquidliquid-fluid system is that formed by a liquid lens floating atop a second liquid (see ref 16 and references within). In these experiments, the dependence of contact angles on the size of the lens is analyzed. The value of the line tension is indirectly calculated from the three interfacial tensions involved (previously measured with techniques such as ADSA) and from the size of the lens and the contact angles at the three phase line (measured from a liquid-lens image). Due to an unavoidable instability of the experimental setup, the liquid lens has to remain attached to a capillary.17 This also allows us to control the volume of the lens and to perform dynamic experiments. However, “hanging” liquid lenses do not present an apex. Hence, existing drop shape techniques are not applicable, and special procedures have been designed for situations where contact angles cannot be obtained from ADSA- or TIFA-type methods, either because the apex of the drop is not available or because the Laplace equation is not applicable.17,18 In these approaches, the experimental profile of the fluid system is extracted using edge detection procedures Then, polynomials of different degrees are fitted to the experimental profile. The required geometrical quantities, particularly contact angles, are measured from those polynomials. To the best of our knowledge, no procedure has been developed that fits the solution of Laplace equation to the experimental liquid-lens profile in the absence of an apex. A generalized methodology for measuring surface properties from any axisymmetric interface (AI), with or without apex, is presented in this paper. As in other drop shape techniques, the Laplace equation is solved to calculate the theoretical shape of (12) Ramos, A.; Gonza´lez, H.; Castellanos, A. Phys. Fluids 1994, 6, 32063208. (13) Meseguer, J.; Slobozhanin, L. A.; Perales, J. M. AdV. Space Res. 1995, 16, 5-14. (14) Ahrens, S.; Falk, F.; Grossbach, R.; Langbein, D. MicrograVity Sci. Technol. 1994, 7, 2-5. (15) Meseguer, J.; Espino, J. L.; Perales, J. M.; Lavero´n-Simavilla A. Eur. J. Mech. B - Fluid 2003, 22, 355-368. (16) Amirfazli, A.; Neumann, A. W. AdV. Colloid Interface Sci. 2004, 110, 121-141. (17) Chen, P.; Susnar, S. S.; Amirfazli, A.; Mak, C.; Neumann, A. W. Langmuir 1997, 13, 3035-3042. (18) Bateni, A.; Susnar, S. S.; Amirfazli, A.; Neumann, A. W. Colloids Surf. A-Physicochem. Eng. Asp. 2003, 219, 215-231.

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the interface. Then, the surface properties, among other parameters, are adjusted to fit the theoretical shape to the experimental one. However, the theoretical profile is now calculated depending on a different set of adjustable parameters. While drop shape techniques make use of the apex geometry as initial conditions to solve the Laplace equation,7 the AI methodology uses the geometry of the interface at a reference level (different from that of the apex), i.e., the curvature, the inclination, and the radius of the interface at the point where the interface intersects the horizontal reference plane.19 For sessile drops, liquid bridges, or liquid lenses, the reference plane is usually the plane containing the three-phase line. For pendant drops, any plane above the apex is chosen. Both approaches (ADSA and TIFA) used in drop shape techniques can be implemented to compare the experimental and the theoretical profiles. In any case, the error function depends on the variables determining the shape of the interface (capillary number and geometry at the reference level) and its position and orientation in the image. Now, the position of the interface is given by the coordinates of the point where the symmetry axis intersects the reference plane, instead of those of the apex in other drop shape techniques. Thus, the optimization parameters will be different from those used in ADSA or TIFA, and very different mathematical optimization procedures will have to be used. Both approaches (ADSA and TIFA) for comparing the theoretical and experimental profiles may be used in combination with the AI methodology. However, TIFA has been selected in this paper, as it performs the edge detection and the comparison of the profiles in only one step. Both the optimization and the (crucial) estimation procedures in TIFA will be re-developed. Then, the range of validity of TIFA-AI will be established from a comparison with other available techniques. To this end, liquid bridges, pendant drops, sessile drops, and liquid lenses will be considered. The organization of the paper is as follows. In Section 2, the TIFA-AI method is presented in detail. The calculation of the theoretical shape is described in Section 2.1, the TIFA strategy is briefly reviewed in Section 2.2, and the new optimization procedure is described in Section 2.3. For the sake of simplicity, the figures and examples in the Methodology section are only given for the liquid bridge configuration. Experiments and results are presented in Section 3. Finally, a summary is presented in Section 4.

2. Methodology 2.1. Theoretical Shape. Generation of numerical (theoretical) shapes of axisymmetric liquid-fluid interfaces is the core of the AI methodology. As mentioned in the Introduction, this methodology is applicable to any configuration dealing with an axisymmetric liquid-fluid interface. The shape of an axisymmetric liquid-fluid interface, as shown in Figure 1, is governed by the Laplace equation:

C(zT) ) C0 -

∆Fg T z γ

(1a)

where C(zT) is twice the local mean curvature at a point of the interface

C(zT) )

[

1 + (xTz )2 xT

]

- xTzz [1 + (xTz )2]-3/2

(1b)

(19) Cabezas M. G. Ph.D. Thesis, Universidad de Extremadura, 2005.

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Figure 1. Axisymmetric liquid-fluid interface without apex (a) and with apex (b) and the coordinate system (xT, zT) used in the numerical integration of the Laplace equation.

The subscript z indicates a derivative with respect to zT. Equation 1 shows that the shape of the interface depends on the following set of parameters {g, ∆F, γ, C0}, where g is the gravitation acceleration, ∆F the density difference across the interface, γ the interfacial tension, and C0 twice the local mean curvature of the interface at the reference level zT ) 0. Equation 1 can be solved as a two-point boundary value problem (BVP)20 or as an initial value problem (IVP).20 The BVP approach has been traditionally preferred by liquid bridge researchers, using the radii at the two ends of the interface as boundary conditions (see, for instance, refs 12 and 13). However, a BVP-based numerical scheme would suffer from stability problems because the range of values of γ and C0sas yet unknownsfor which the BVP yields a solution is quite limited. As an alternative, in the AI methodology, the shape of a theoretical interface is calculated using the IVP approach.19 In this approach, all the initial conditions are defined at the reference level zT ) 0. For interfaces that do not present an apex (liquid bridges and certain liquid lenses) and sessile drops, the horizontal reference plane is located at the three-phase line (see Figure 1a). For pendant drops, it is located above the apex, so a small section containing the apex is ignored (see Figure 1b). The initial conditions are

Figure 2. (a) Image of a liquid bridge of water surrounded by air held between a capillary and a lower support with sharp edges. (b) Sketch of AI methodology cominbed with ADSA strategy for profile comparison: experimental profile obtained by edge detection (circles) and theoretical profile (line). (c) Gradient image obtained after applying the 3 × 3 Sobel operator to image (a). (d) Theoretical gradient image corresponding to a calculated theoretical profile.

where R0 is the radius of the interface, and θ0 its inclination at the reference level. Note that when the reference contains the three-phase line, the value of θ0 corresponds to the contact angle. The vertical length, L, of the interface is only needed to define the upper cutoff level and does not play a role in the integration of eq 1. The AI methodology calculates the theoretical profile of an axisymmetric interface depending on the set of parameters {g, ∆F, γ, C0, R0, θ0}. 2.2. Comparison of Theoretical and Experimental Profiles. The surface properties are measured by comparing the calculated theoretical shape of the axisymmetric interface with the experimental shape contained in a digital image (see Figure 2a). In drop shape techniques, two different strategies have been used for making this comparison, both being applicable to the study of axisymmetric interfaces without apex. The ADSA strategy first extracts the experimental profile from the image by using advanced edge detection procedures, and then fits the Laplacian profile to the set of points of the experimental profile (see Figure 2b). On the other hand, TIFA constructs a black and

white theoretical (gradient) image containing the Laplacian profile and fits it to the experimental (gradient) image (see Figure 2c and d). In both cases, the fitting involves the minimization of an error function that evaluates the difference between the experimental and the theoretical profiles. The error function used by ADSA measures the distance of the Laplacian profile to the set of points of the experimental profile. In TIFA, the error function measures the difference between the theoretical and the experimental (gradient) images. However, in both cases, the same parameters are considered as unknowns in the multivariate minimization process. In addition to the quantities determining the shape (those used to solve the Laplace equation), the parameters defining the position and orientation of the interface are also considered. In the AI methodology, the position of the interface is defined by the coordinates, not of the apex, but of a reference point, given by the intersection of the symmetry axis and the reference plane. In this paper, the TIFA-AI method has been developed by combining the TIFA strategy with the AI methodology. The principles of the TIFA strategy are briefly described below. The experimental shape of the interface is contained in a digital image acquired in the experiment. Mathematically, this image is characterized by its gray intensity function I(i, j) that takes a value between 0 (black) and 255 (white) for each pixel (i, j) (see Figure 2a). TIFA calculates the gradient image G(i, j) corresponding to the experimental image I(i, j) by using the 3 × 3 Sobel operator, as it is believed to be less sensitive to noise than other operators due to its smoothing performance.21 Figure 2b shows the gradient image for the experimental image shown in Figure 2a. The value of the gradient function G(i, j) is high (light pixels) at the contour of the interface (where the gray value changes rapidly) and low (dark pixels) on both sides of the interface. The theoretical counterpart of G(i, j) is a binary image generated numerically by integrating the Laplace equation (as explained above). The value of the theoretical gradient image, GT(i, j), is 255 (white pixel) along the theoretical interface and 0 (black

(20) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Fannery, B. P. Numerical Recipes in C: The Art of Scientific Computing, 2nd ed.; Cambridge University Press: New York, 1992; Chapters 16-17.

(21) Seul, M.; O’Gorman, L.; Sammon, M. J. In Practical Algorithms for Image Analysis. Description, Examples, and Code; Cambridge University Press: New York, 2000; Chapter 3.

xT(zT ) 0) ) R0 xTz (zT ) 0) ) tan θ0

(2)

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pixel) for the rest of the image (see Figure 2c). Therefore, the theoretical gradient image is defined as

GT(i,j) )

{

255 ∀ (i, j) ∈ {(ip, jp)} 0 ∀ (i, j) ∉ {(ip, jp)}

(3)

where {(ip, jp)} is the set of pixels corresponding to the theoretical profile. The error function

)

∑[G(i, j) - GT(i, j)]2

Figure 3. Calculation of the radius and center point of the lower disk from an image.

(4)

(i,j)

evaluates the pixel-by-pixel difference between the two images. Substitution of eq 3 into eq 4 yields

 ) K - 255

∑ [2‚G(i, j) - 255]

(5)

(ip,jp)

where K is a constant for each experimental image which does not play a role in the minimization of the error function. Note that the summation in eq 5 is limited to the set of pixels {(ip, jp)} corresponding to the theoretical interface. The procedure employed to calculate the error (eq 5) corresponding to an axisymmetric Laplacian profile without apex is the same as that used for profiles with apex (details can be found elsewhere10). Briefly, this procedure consists of three steps: 1. The axisymmetric Laplacian profile is situated in the image considering the position of the reference point, i.e., the intersection of the symmetry axis and the reference plane, given by its coordinates (xI0, zI0 ) in the image, and the orientation, R, of its axis (i.e., the orientation of gravity). 2. The corresponding continuous profile in the image is obtained applying the same optical effects caused by the image acquisition system. These have been previously calibrated and modeled using a calibration grid.10,22 3. The continuous profile is discretized by selecting only one point of the interface at each row of the image. Interpolation is used to calculate the gradient value at that point. The TIFA-AI error function depends on the parameters determining the shape of the interface {g, ∆F, γ, C0, R0, θ0}, and those determining its position and orientation in the image {R, xI0, zI0}. Among them, the values of g and ∆F are usually considered as known. It should be noted that the quantity R0 may be known in advance for particular experimental configurations (e.g., the radius of the lower disk of a liquid bridge). In some configurations (e.g., the pendant drop), the value of zI0 only affects the limit of integration and can also be assumed as known. However, for the sake of generality, all these quantities are assumed as unknowns in this formulation. The values of all these unknown parameters are adjusted to determine the set of pixels {(ip, jp)} that minimizes the error function (eq 5). Therefore, the optimization module needs to calculate seven unknown quantities {γ, C0, R0, θ0, R, xI0, zI0 }. 2.3. Optimization Procedure. The error function (γ, C0, R0, θ0, R, xI0, zI0) measures the difference between the experimental and the theoretical gradient images. TIFA-AI employs the Nelder-Mead simplex technique as the minimization method. As for all multivariate optimization techniques, the convergence of this method essentially relies on the use of good estimates of the optimization parameters as initial values. It was found that sufficiently accurate initial values can only be obtained when the (22) Cheng, P.; Li, D.; Boruvka, L.; Rotenberg, Y.; Neumann, A. W. Colloids Surf. 1990, 43, 151-167.

Figure 4. Calculated liquid bridge profiles for the same set of shape parameters but different surface tensions: continuous line, γ ) 72.00 mJ/m2; dashed line, 20% over; and dotte line, 20% lower.

parameters are estimated in a particular order, i.e., step 1: the orientation of the interface R, the coordinates (xI0, zI0) of the reference point in the image, and the radius R0 of the interface at the reference level; step 2: the inclination, θ0, and the curvature, C0, at the reference level; and step 3: the surface tension γ. Simple image processing and geometrical principles are used to estimate the parameters at the first step. The angle R is estimated by analyzing a plumb-line image acquired before the experiment. The parameters {R0, xI0, zI0} can generally be estimated by analyzing the experimental image of the interface. In the particular case of liquid bridges, the image of the lower disk (i.e., the lower end of the interface) is preferred for this purpose. The coordinates of the right and left edges of the disk are determined by analyzing the gray level of the pixels, from which the radius R0 and the coordinates of the center point (xI0, zI0) are estimated (see Figure 3). The estimates of the parameters θ0 and C0 are calculated in the second step. Figure 4 shows three profiles of a liquid bridge calculated using different values of the surface tension and the same values of these parameters (θ0 ) 0.4695 and C0 ) 12.80 cm-1). As expected from eq 1, the curves overlap each other near the lower disk, i.e., over a vertical distance of about L/5 from the lower disk (limited by the horizontal dashed line). In this region, the shape of the liquid bridge depends essentially on the parameters θ0 and C0 and is basically independent of the value of the surface tension, γ. Therefore, the inclination and the curvature can be estimated by matching the theoretical and experimental shapes of the interface over the region L/5, using an arbitrary value of the surface tension. In particular, the error function (eq 5) is used such that the summation is applied only to the pixels (ip, jp) that belong to that region. The resulting error function, 1/5, depends essentially on C0 and θ0 and is practically

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Table 1. Estimates and the Final Results Obtained by TIFA-AI for the Liquid Bridge of Water Shown in Figure 2aa trial γ0 ) 30 mJ/m2 γ0 ) 60 mJ/m2 γ0 ) 72 mJ/m2 a

estimates result estimates result estimates result

γ (mJ/m2)

C0 (cm-1)

xΙ0

R0 (cm)

xI0

R × 102

44.52 72.00 66.67 72.00 69.27 72.00

13.53 12.80 12.99 12.80 12.90 12.80

0.5259 0.5073 0.5172 0.5073 0.5158 0.5073

0.1049 0.1051 0.1049 0.1051 0.1049 0.1051

316.5 316.1 316.5 316.1 316.5 316.1

1.159 1.025 1.159 1.025 1.159 1.025

Each row corresponds to a different initial value γ0 for the surface tension.

Figure 6. Image of (a) a pendant drop and (b) a liquid bridge of hexadecane obtained in an experiment, along with the fitted theoretical profiles calculated by TIFA-AI.

Figure 5. Shapes of the theoretical interface corresponding to the three steps of the estimation process: step 1 (dotted line), step 2 (dashed line), and step 3 (continuous line).

respect to the (arbitrary) starting value of the surface tension for the liquid bridge in Figure 2a. The three trials converge to the same optimum set of parameters {γ ) 72.00 mJ/m2, C0 ) 12.80 cm-1, R0 ) 0.1051 cm, θ0 ) 0.4695, R ) 1.025 × 10-2, xI0 ) 316.1, zI0 ) 412.2}, regardless of the starting value of γ.

independent of γ. Again, the Nelder-Mead simplex method is used to calculate the values of C0 and θ0 corresponding to the minimum of 1/5. An arbitrary value of γ along with good initial values for C0 and θ0 is used in this calculation. The initial values of the inclination and the curvature are calculated by geometrical analysis. A set of 15 points of the profile close to the lower end of the interface are calculated by searching for the points of maximum gradient,10,11 and a parabola is fitted to them. The inclination and curvature of the parabola are used as the corresponding initial values. Finally, the surface tension, γ, is estimated at the last step by analyzing the shape of the whole interface. In particular, the minimum of the error function (eq 5) is found from a onevariable minimization process, using the estimated values of {R, xI0, zI0, R0, C0, θ0} as constants. Figure 5 shows the comparison of the theoretical and experimental interfaces of a liquid bridge at the three steps of the estimating process. The dotted curve is generated using the estimated quantities R ) 1.159 × 10-2, R0 ) 0.1049 cm, xI0 ) 316.5, zI0 ) 412.2 and arbitrary values of γ ) 60 mJ/m2, C0 ) 10 cm-1 and θ0 ) 0.2450. The curve starts at the lower disk (the lower end of the experimental profile) but does not match its shape. The dashed curve is generated after the parameters C0 ) 12.99 cm-1 and θ0 ) 0.4773 are also estimated. The curve matches the shape of the experimental interface over a region close to the lower disk. The continuous line is the theoretical profile after all parameters are estimated (γ ) 66.67 mJ/m2). The shape of the curve agrees quite well with the experimental interface. After good initial values (estimates) of the parameters are calculated, the error function (γ, C0, R0, θ0, R, xI0, zI0) is minimized using the multivariate Nelder-Mead simplex optimization. Table 1 examines the robustness of the scheme with

Various experiments were conducted using different liquid-fluid configurations to examine the accuracy and the range of applicability of the new methodology. TIFA-AI was compared with the best available technique for each configuration. For drops, TIFA-AI was compared with ADSA that is well-known to produce accurate results.8,9 For liquid lenses, TIFA-AI was compared with a custommade polynomial fitting technique developed recently.18 It should be noted that the purpose of this section is to compare and validate the measuring techniques and not to report literature values for the interfacial properties of liquids. The latter would require more complex experimental procedures. 3.1. Liquid Bridges and Pendant Drops. In the first round of experiments, images of pendant drops and liquid bridges were acquired in similar experimental conditions. TIFA-AI was applied to measure the surface tension from both fluid configurations. The results obtained were compared with each other and with those obtained by processing the pendant drops images with TIFA10 and ADSA.7 The procedure used for these experiments is as follows. The liquid bridge is formed between a capillary and the bottom disk inside a test chamber. The capillary is connected to a syringe and can be displaced along its axis. The bottom disk is fixed to the opposite side of the test chamber such that both the capillary and the disk remain in coaxial alignment. The edge of the disk is sufficiently sharp to anchor the liquid and to prevent the motion of the three-phase contact line. In an experiment, a pendant drop was first formed by injecting liquid through the capillary. It is well-known that the closer an interface is to its stability limit the higher the accuracy of the surface tension measurement; hence, fairly large (well deformed) pendant drops were used in this experiment.8 A series of 10 images of the drop were acquired and stored on a computer for processing (see Figure 6a). Next, the needle-drop system was axially moved until the drop touched the bottom disk, forming a liquid bridge between

3. Experiments and Results

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Table 2. Surface Tension Obtained from Liquid Bridge Images Processed by TIFA-AI and from Pendant Drop Images Processed by TIFA-AI, TIFA, and ADSAa surface tension (mJ/m2) configuration liquid hexadecane water

experiment A B C A B C

liquid bridge

pendant drop

TIFA-AI

TIFA-AI

TIFA

ADSA

27.07 ( 0.02 27.10 ( 0.03 27.13 ( 0.01 71.95 ( 0.11 71.84 ( 0.04 71.93 ( 0.03

26.93 ( 0.02 27.05 ( 0.02 27.02 ( 0.02 71.36 ( 0.05 71.85 ( 0.09 71.94 ( 0.07

26.92 ( 0.01 27.03 ( 0.01 27.04 ( 0.01 71.39 ( 0.03 71.93 ( 0.08 72.12 ( 0.02

26.89 ( 0.01 27.02 ( 0.01 27.00 ( 0.01 71.37 ( 0.03 72.12 ( 0.09 72.21 ( 0.04

a A series of 10 images of each configuration is obtained in each experiment. The result presented here is the average of the measurements from the 10 images.

the needle and the lower disk. The volume of the liquid bridge was reduced until the interface was close to its stability limit, i.e., a situation of high sensitivity of the interface shape.23 Finally, a series of 10 images was acquired from the liquid bridge (see Figure 6b). The whole experimental process was performed in a short period of time of ∼2 min, for which the experimental conditions were assumed to be constant. Distilled water and hexadecane were studied by conducting several experiments, according to the above procedure. Table 2 shows the average of the 10 measurements obtained from the images of each experiment. The surface tension for the liquid bridges was calculated by the new method TIFA-AI. The pendant drop images were processed by TIFA-AI, TIFA,10 and ADSA.7 Good agreement (differences of the order of 1%) between the three techniques was found. Figure 6 shows the theoretical profiles obtained from TIFAAI for the pendant drop and the liquid bridge images. Note that, to process the pendant drop image with TIFA-AI, the apex of the drop is ignored; hence, a reference level (zT ) 0) above the apex is chosen. Figure 6a shows that the section of the interface considered does not include the apex. 3.2. Sessile Drops. Contact angle measurement in sessile drop experiments using ADSA-P is commonly performed to characterize solid surfaces (see, for instance, refs 24 and 25) and to evaluate line tension.26 To this end, TIFA-AI can also be employed. The reference plane is located at the solid surface, containing the three-phase line, so the contact angle θ0 is calculated as an optimization parameter. The Laplacian curve is fitted to the drop image from the contact point to a certain height of the drop, i.e., the area near the drop apex is neglected (see Figure 7). Table 3 shows the results for a sequence of 16 images corresponding to a typical dynamic sessile drop experiment. In this experiment, a sessile drop of o-xylene was formed on a Teflon-coated silicon wafer (Teflon AF 1600 from Dupont Co., London, ON). The drop was growing during the experiment to allow measurement of advancing contact angles. Experimental images were acquired approximately every 1.5 s and were processed using both ADSA and TIFA-AI. The results show good agreement between the two methods. However, the scatter of the measurements (i.e., 95% calculated confidence interval) is slightly higher for the TIFAAI method, possibly because TIFA-AI uses less experimental information, as it neglects a section of the drop that includes the apex. 3.3. Liquid-Lens Systems. A further application of the new methodology is the analysis of liquid-lens images (see Figure 8). This liquid-liquid-fluid system is of particular interest in line tension research, as a three-phase line is formed without the presence of a solid body.16 In these experiments, the line tension is indirectly calculated from the interfacial tensions, the radii of the lens, and the contact angles at the three-phase line. The interfacial tensions associated with the three interfaces are obtained previously by other (23) Cabezas, M. G.; Montanero, J. M. J. Comput. Methods Sci. Eng. 2004, 4, 75-85. (24) Kwok, D. Y.; Gietzelt, T.; Grundke, K.; Jacobasch, H. J.; Neumann, A. W. Langmuir 1997, 13, 2880-2894. (25) Long, J.; Chen, P. Langmuir 2001, 17, 2965-2972. (26) Amirfazli, A.; Ha¨nig, S.; Mu¨ller, A.; Neumann, A. W. Langmuir 2000, 16, 2024-2031.

Figure 7. Image of a sessile drop of o-xylene obtained in an experiment, along with the fitted theoretical profile calculated by TIFA-AI. Table 3. Contact Angle of Sessile Drop of o-Xylene on Teflon Calculated Using ADSA and TIFA-AI for a Sequence of Images of a Dynamic Experimenta contact angles (deg)

a

time (s)

ADSA

TIFA-AI

124.29 125.71 127.14 128.57 130.00 131.43 132.86 134.29 135.71 137.15 138.57 140.00 141.43 142.86 144.29 145.71 average

64.93 64.78 64.80 65.08 64.71 64.81 64.74 64.81 65.10 64.60 64.63 64.75 65.00 65.11 65.29 64.86 64.88 ( 0.19

64.80 65.03 64.60 64.84 64.97 65.04 65.06 65.01 64.91 64.74 64.64 64.37 65.60 65.75 65.13 65.08 64.97 ( 0.34

Diameter of the drop is about 1 cm.

techniques, such as ADSA. The lens radii and the contact angles at the three-phase line are obtained from the experimental image of a liquid lens. Therefore, measurement of line tension for such systems relies on high-accuracy measurement of contact angles of the two liquid-fluid interfaces.17 The liquid lenses used in this experiment were generated as follows. A pendant drop of the upper liquid was formed at the tip of a capillary.

Determination of Surface Tension and Contact Angle

Langmuir, Vol. 22, No. 24, 2006 10059 a Laplacian shape, while APF approximates a given section of the interface by a polynomial shape).

4. Summary

Figure 8. Image of a liquid-lens system formed by dodecane floating on water, along with the fitted theoretical profiles calculated by TIFA-AI. Table 4. Contact Angle of the Upper and Lower Interfaces for a Sequence of Images of a Liquid-Lens System of Dodecane Floating on Water Taken during an Experiment, Calculated Using TIFA-AI and the Polynomial Fitting Methoda TIFA-AI

polynomial

image

θ1 (deg)

θ2 (deg)

θ1 (deg)

Θ2 (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 average

29.95 30.14 30.06 30.19 30.03 30.12 29.96 29.94 30.13 30.14 29.96 30.00 30.00 30.14 30.07 30.13 30.06 ( 0.08

3.51 3.33 2.14 3.06 2.98 3.81 3.14 3.05 3.14 3.34 4.18 3.17 3.13 3.47 3.22 3.10 3.24 (0.43

30.47 30.31 30.47 30.07 30.34 30.75 30.42 30.26 30.30 30.27 30.01 30.09 30.48 30.69 30.63 29.77 30.33 ( 0.26

3.31 3.37 3.39 3.60 3.46 3.45 3.46 3.28 3.47 3.45 3.37 3.44 3.37 3.51 3.29 3.42 3.41 (0.08

a

Diameter of the three-phase contact line is 0.46 cm.

The lower liquid was contained in a glass cylinder to ensure that the liquid lens formed is circular. To observe the lower liquid-air interfaces from a side view of the fluid configuration, the level of liquid in the glass cylinder has to be above the rim of the cylinder, i.e., the cylinder was overfilled with liquid. The upper liquid drop was lowered so that it touched the lower liquid-air interface, and a liquid lens was formed floating on the lower liquid. The resulting contact line represents the intersection of the three interfaces among the two immiscible liquids and the ambient gaseous phase. In this experimental configuration, it is not possible to remove the upper needle, as the system is mechanically unstable, i.e., a free drop will tend to move away. Current drop-shape techniques (e.g., ADSA) are not applicable to this configuration since the shape of the interface does not include an apex; however, as explained above, TIFA-AI can deal with any axisymmetric interface, with or without apex. The results provided by TIFA-AI are compared with those given by an automated polynomial fitting (APF) technique.18 This technique extracts the profile of the liquid-fluid interface using edge detection, fits thirdorder polynomials to a part of the extracted profile, and measures the contact angle as the slope of the fitted polynomial at the contact point. Sophisticated analysis is required to determine the optimum order of the polynomial and the length of the profile used for fitting. Details of the polynomial fitting technique can be found elsewhere.18,27 Table 4 shows the contact angle of upper and lower interfaces measured by the above two methods for a sequence of 16 images of a static liquid lens of dodecane floating on water/air. The results indicate very good agreement to within (0.2°. This agreement is remarkable since completely different approaches are used for image processing (TIFA strategy vs edge detection in APF), and different theoretical shapes are fitted to the experimental one (TIFA-AI fits (27) Tavassoli, S. Masters Thesis, University of Toronto, 2006.

A generalized methodology, AI, was presented to measure interfacial properties from the shape of any axisymmetric liquidfluid interface (with or without apex). This methodology is applicable when any section of an axisymmetric interface is available/visible in an experiment. Typical applications are the surface tension measurement using liquid bridges, pendant/sessile drops, and contact angle measurement in sessile drop and liquidlens systems. This methodology can be implemented using two strategies for comparing the experimental and the theoretical profiles: the strategy used by traditional drop shape methods, such as ADSA or a recently developed image processing approach, TIFA.10,11 The latter fits a theoretical (Laplacian) image of an interface to the experimental one without the need for an independent edge detector. It is believed to be robust and noiseresistant; hence, it is very promising when acquisition of highquality experimental images is not possible. For that reason, the TIFA strategy was combined with the new AI methodology to develop TIFA-AI. Generation of numerical (Laplacian) shapes of an axisymmetric liquid-fluid interface without apex is the core of this methodology. The module for generating numerical profiles in the original TIFA was not applicable for fluid configurations without apex, so a new module was developed for this purpose. Both boundary value and initial value strategies were implemented and examined. It was shown that an initial value formulation results in a more stable numerical scheme. The shape and boundary parameters for generating numerical shapes of the interface were defined accordingly. TIFA-AI use substantially different parameters in the optimization procedure. A multiparameter optimization scheme was developed to calculate the unknown parameters (e.g., surface tension and contact angle) by fitting the theoretical shape of the interface to the experimental one. It was shown that accurate initial values (estimates) of these parameters are needed for convergence to the global minimum. A precise step-by-step procedure was presented to estimate these parameters. Finally, experiments with liquid bridges, pendant drops, sessile drops, and liquid lenses were conducted to examine the accuracy, versatility, and range of applicability of the new methodology. For each experiment, TIFA-AI was compared with the best available technique for that particular configuration. Good agreement was obtained. In brief, the advantages of the new method can be summarized as follows. (i) It is applicable to a wide range of experimental configurations, where any section of an axisymmetric liquid-fluid interface is available/visible. Examples of the standard experimental configurations are liquid bridges, pendant/sessile drops/bubbles, and liquid lenses systems. (ii) TIFA-AI can be employed to measure both surface tension and contact angles. Furthermore, the contact angle results are of particular interest, since, unlike for existing drop-shape methods, they are calculated as an optimization parameter. (iii) TIFA-AI can be employed to measure surface tension from the shape of a liquid bridge. Compared with pendant/sessile drop configurations, a liquid bridge with less liquid provides more accurate surface tension results.23 This is a key advantage when test liquids are scarce or expensive. The only apparent drawback of TIFA-AI is the rather large computer power and computation time required. The algorithm

10060 Langmuir, Vol. 22, No. 24, 2006

gives the results in around 5 min using a Pentium V 600 MHz processor. No attempt was made, as yet, to reduce the computation time. Acknowledgment. We thank H. Tavana for providing the images of the sessile drop experiments, S. Tavassoli for the images of the liquid-lens experiments, and Dr. S. S. Susnar for his collaboration in liquid bridge and pendant drop experiments.

Cabezas et al.

This research was financially supported by the Ministerio de Ciencia y Tecnologı´a (Spain) through Grant No. ESP2003-02859, the Consejerı´a de Educacio´n y Ciencia de la Junta de Extremadura, the Fondo Social Europeo (M.G.C. and J.M.M.), and by Natural Sciences and Engineering Research Council (NSERC) of Canada under Grant No. 8278, and a NSERC of Canada Postdoctoral Fellowship (A.B.). LA061928T