pubs.acs.org/Langmuir © 2009 American Chemical Society
Methodology for High Accuracy Contact Angle Measurement† A. Kalantarian, R. David, and A. W. Neumann* Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Canada Received June 4, 2009. Revised Manuscript Received July 13, 2009 A new version of axisymmetric drop shape analysis (ADSA) called ADSA-NA (ADSA-no apex) was developed for measuring interfacial properties for drop configurations without an apex. ADSA-NA facilitates contact angle measurements on drops with a capillary protruding into the drop. Thus a much simpler experimental setup, not involving formation of a complete drop from below through a hole in the test surface, may be used. The contact angles of long-chained alkanes on a commercial fluoropolymer, Teflon AF 1600, were measured using the new method. A new numerical scheme was incorporated into the image processing to improve the location of the contact points of the liquid meniscus with the solid substrate to subpixel resolution. The images acquired in the experiments were also analyzed by a different drop shape technique called theoretical image fitting analysis-axisymmetric interfaces (TIFA-AI). The results were compared with literature values obtained by means of the standard ADSA for sessile drops with the apex. Comparison of the results from ADSA-NA with those from TIFA-AI and ADSA reveals that, with different numerical strategies and experimental setups, contact angles can be measured with an accuracy of less than 0.2°. Contact angles and surface tensions measured from drops with no apex, i.e., by means of ADSA-NA and TIFA-AI, were considerably less scattered than those from complete drops with apex. ADSA-NA was also used to explore sources of improvement in contact angle resolution. It was found that using an accurate value of surface tension as an input enhances the accuracy of contact angle measurements.
Introduction Improvements in computer software and optical devices in the past three decades have led to considerable development of drop shape techniques in the area of surface thermodynamics.1,2 In essence, the shape of a drop depends on the combined effects of gravitational and interfacial forces. When the gravitational and interfacial forces are comparable, the interfacial properties can be determined from the analysis of the shape of a drop.2 The advantages of drop shape techniques are numerous. They require only small amounts of the liquid and are applicable to various situations, including extreme temperature and pressure.3 Since the profile of the drop is recorded, drop shape methods can be used for studying, e.g., changes in surface tension with time due to various causes,4 as long as the changes are slow enough so that the Laplace equation of capillarity remains applicable. Axisymmetric drop shape analysis (ADSA) is a long-standing and widely used drop-shape technique for measuring interfacial properties.4 Typically, such methods as ADSA rely on matching theoretical profiles calculated from the Laplace equation to experimental drop profiles. The key point is to have the algorithm search for that specific surface tension that produces the best fit with the experimental profile. Rotenberg et al.5 developed the first generation of ADSA (written in Fortran). An objective function was defined as the sum of the squares of the normal distances † Part of the “Langmuir 25th Year: Wetting and superhydrophobicity” special issue. *Corresponding author. E-mail:
[email protected].
(1) Hartland, S. Surface and Interfacial Tension: Measurement, Theory, and Applications; Marcel Dekker: New York, 2004. (2) Adamson, A. W.; Gast, A. P. Physical Chemistry of Surfaces, 6th ed.; Wiley: New York, 1997. (3) Holgado-Terriza, J. A.; Gomez-Lopera, J. F.; Luque-Escamilla, P. L.; AtaeAllah, C.; Cabrerizo-Vilchez, M. A. Colloids Surf., A: Physicochem. Eng. Aspects 1999, 156, 579–586. (4) Hoorfar, M.; Neumann, A. W. Adv. Colloid Interface Sci. 2006, 121, 25–49. (5) Rotenberg, Y.; Boruvka, L.; Neumann, A. W. Colloid Interface Sci. 1983, 93, 169–183.
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between the experimental profile points with corresponding points on the theoretical profile. The Newton-Raphson method in conjunction with incremental loading was used to minimize the objective function to produce the best match between the experimental and theoretical profiles. In the first generation of ADSA, extracting the drop interface coordinates was performed manually. Cheng et al.6 improved the first generation of ADSA by implementing computer-based edge detection to extract the drop interface automatically. An edge operator, Sobel,7 in conjunction with subpixel resolution, and optical distortion correction techniques were incorporated into the ADSA program to achieve higher accuracy for detecting the drop interface. The first generation of ADSA was found to give accurate results except for very large and flat sessile drops, where the program failed. As the sessile drops become very large and flat, the radius of curvature at the apex becomes very large, so that computer algorithms may become unstable. To overcome the deficiencies of the numerical schemes of that first generation, a second generation of ADSA was developed by del Rio and Neumann8 using more efficient algorithms (written in C). In this version, the curvature at the apex, which approaches zero for very flat sessile drops, instead of the radius of curvature at the apex was used as an optimization parameter to resolve the convergence problem. While the broadest application of ADSA has been in the determination of surface tension, it has also had a significant impact on contact angle research. Because of the necessity of having the apex of a drop as part of the drop image, the most convenient constellation for a contact angle measurement, (6) Cheng, P.; Li, D.; Boruvka, L.; Rotenberg, Y.; Neumann, A. W. Colloids Surf. 1990, 43, 151–167. (7) Duda, R. O.; Hart, P. E. Pattern Classification and Scene Analysis; Wiley: New York, 1973. (8) del Rio, O. I.; Neumann, A. W. J. Colloid Interface Sci. 1997, 196, 136–147.
Published on Web 08/14/2009
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i.e., a drop in contact with a vertical capillary, could not be used. Instead, drops had to be formed from below through a hole in the solid surface.9 But apart from this complication, the strategy for the contact angle measurement was straightforward: The shape of the drop was optimized for the liquid surface tension, and the contact angle was determined as a geometrical parameter from the intersection of the drop profile with the solid. The vertical location of the solid surface was determined from the digitized drop image to pixel resolution. This methodology proved to be very successful: The reproducibility of contact angle measurement was improved by essentially an order of magnitude, to 0.2-0.3 degrees on well-prepared solid surfaces,9,10 compared to direct tangent methods. The ADSA strategy is based on the comparison of a theoretical profile line and the experimental points representing the drop interface extracted through edge detection. In a new drop shape technique called theoretical image fitting analysis (TIFA), a new approach completely different from the approach of ADSA was introduced.11,12 The difference between the strategy of ADSA and TIFA is that, in TIFA, the interfacial properties are calculated by fitting the whole theoretical image, i.e., not the profile line, to the experimental image of the drop. The theoretical image is a black and white image of the drop generated by numerically solving the Laplace equation. An error function measures the pixel-by-pixel difference between the whole theoretical and experimental images. The interfacial properties are found by fitting the theoretical image to the experimental one by minimizing the error function. The remarkable feature of TIFA is that it operates without using edge detection algorithms. In fact, image analysis is tied to the optimization process in TIFA, and it is not a separate module as in ADSA. The restriction of ADSA and TIFA to drops or bubbles with an apex limits the application of the methodologies. Liquid bridges13,14 do not have an apex, and the study of contact angles of floating lenses15 is best accomplished by having a capillary protrude into the liquid lens, to provide mechanical stability, since, for optical reasons, the cuvette containing the bulk liquid has to be filled more than level-a constellation that does not allow mechanical stability of a free floating liquid lens. All these concerns motivated the development of a different version of TIFA, called TIFA-AI (theoretical image fitting analysis for axisymmetric interfaces).16,17 In this method, the geometry of the interface at a reference level different from the apex is used to solve the Laplace equation. Therefore, two new additional optimization parameters compared to TIFA were defined: the radius of the profile, and its inclination at the reference level. Recently, TIFA-AI was used to measure the size dependence of contact angles in sessile drop experiments.18 In that study, a syringe needle was mounted vertically with its tip less (9) Kwok, D.; Lin, R.; Mui, M.; Neumann, A. W. Colloids Surf., A: Physicochem. Eng. Aspects 1996, 116, 63–77. (10) Tavana, H.; Neumann, A. W. Adv. Colloid Interface Sci. 2007, 132, 1–32. (11) Cabezas, M. G.; Bateni, A.; Montanero, J. M.; Neumann, A. W. Appl. Surf. Sci. 2004, 238, 480–484. (12) Cabezas, M. G.; Bateni, A.; Montanero, J. M.; Neumann, A. W. Colloids Surf., A: Physicochem. Eng. Aspects 2005, 255, 193–200. (13) Meseguer, J.; Espino, J. L.; Perales, J. M.; Laveron-Simavilla, A. Eur. J. Mech., B: Fluids 2003, 22, 355–368. (14) Ferrera, C.; Montanero, J.; Cabezas, M. G. Meas. Sci. Technol. 2007, 18, 3713–3723. (15) David, R.; Dobson, S. M.; Tavassoli, Z.; Cabezas, M. G.; Neumann, A. W. Colloids Surf., A: Physicochem. Eng. Aspects 2009, 333, 12–18. (16) Cabezas, M. G.; Bateni, A.; Montanero, J. M.; Neumann, A. W. Langmuir 2006, 22, 10053–10060. (17) Cabezas, M. G.; Montanero, J. M.; Ferrera, C. Meas. Sci. Technol. 2007, 18, 1637–1650. (18) David, R.; Park, M. K.; Kalantarian, A.; Neumann, A. W. Colloid Polym. Sci. DOI: 10.1007/s00396-009-2077-1.
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Figure 1. Image of a dodecane drop on a Teflon AF 1600-coated surface. The drop was formed by injecting the liquid through a vertical stainless steel needle.
than 1 mm above the solid surface (Figure 1) to ensure contact of the tip with the liquid drop for any drop size. The solid level (the contact points at the solid-liquid interface) was determined manually with an accuracy of (1 pixel and used as an input in TIFA-AI. Typical scatter of 0.05° within a single run and typical run-to-run scatter of 0.2° was reached in that study. The main purpose of the present study is to develop a new version of ADSA to be called ADSA-NA (axisymmetric drop shape analysis-no apex) that does not require information about the apex region, to facilitate contact angle measurements on drops with a capillary protruding into the drop, i.e., an alternative to TIFA-AI. Since the accuracy of contact angle measurement has now moved into the order of magnitude of (0.1°, it is a legitimate and indeed important question to ask whether these high accuracies are real and do not somehow reflect peculiarities of the image and the image processing. A comparison of results obtained from the same drop by means of ADSA-NA and TIFA-AI can answer this question since they handle the same image quite differently: ADSA finds an optimum profile line and matches it to theoretical profiles, whereas TIFA avoids the use of edge detection altogether and matches the whole drop image to a theoretical two-dimensional projection of the drop, not just the profile line. Contact angle measurements were performed here on thin films of a commercial fluoropolymer, Teflon AF 1600, for four longchained alkanes: decane, dodecane, tetradecane, and hexadecane. The images acquired were also used to investigate several other remaining issues: a refinement in the determination of the coordinates of the contact point between the three interfaces, and the question whether, for the purpose of contact angle measurement, the surface tension value should be simultaneously optimized or preferably used as input. Finally, the experimental setup to be used is the same as that used in ref 18, i.e., a syringe with a thin needle forming and protruding into the drop, a setup that is much easier to handle than the setup used in reference 10.
Methodology ADSA-NA has the same structure as ADSA and consists of three main modules. The first module is the image processing, which extracts the experimental profiles of drops. The second module generates the theoretical profiles by numerical integration of the Laplace equation. The last module is the optimization procedure to find the best fit of the theoretical Laplacian curves to the experimental profile. The following provides a detailed description of the method. Image Processing. In this module, the experimental profile, consisting of an array of coordinate points, is extracted from the gray scale image of the drop acquired in the experiment (Figure 1). DOI: 10.1021/la902016j
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Figure 2. The binary image corresponding to Figure 1 after applying the Canny edge detector. The edges detected by Canny are shown in black. The circled area shows the region of the drop close to the solid surface and its reflection in the solid substrate.
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Figure 3. A typical example of a gray level profile perpendicular to a drop interface (open circles). Point A is the pixel that was detected by Canny as the drop profile coordinate. The solid line shows a fitted natural spline to the profile. Point B is the new location of the drop interface with subpixel resolution.
The gray level for every pixel in the image ranges from 0 to 255, representing black and white, respectively. Extracting the drop profile is performed in two stages just as in ADSA: pixel resolution and subpixel resolution. In ADSA, originally the Sobel edge detector, written in Fortran language, was implemented to detect the drop profile to pixel resolution. Recently, a more robust edge detector called Canny19 was implemented in a new version of ADSA called ADSA-CB (ADSA for Captive Bubble).20 It was shown that the Canny edge detector is more resistant to noise in the image than Sobel.20 Here, the Canny edge detector, available as a built-in function in the MATLAB Image Processing Toolbox (The Mathworks), is used for the first stage of image processing, i.e., determination of the experimental profile to pixel resolution. After applying Canny to the gray scale image of the drop, a binary image of the same size as the original image is constructed as the output of Canny, in which any pixel that belongs to an edge is assigned the value of 1, and all other pixels are assigned a value of 0. Figure 2 shows a binary image corresponding to Figure 1 after applying the Canny edge detector. The binary image includes the pixels representing the drop interface, the needle, and the reflection of the drop at the solid surface. Since the solid substrate is smooth and shiny, the reflection of the drop interface close to the solid surface appears in the image and can be detected by Canny (circled area in Figure 2). A specific procedure was implemented in ADSA-NA in order to extract just the pixels that belong to the drop profile. The binary image is searched from the top along the rows, and the two edge pixels with the value of one in each row are selected as the needle profile. The distance between these two pixels is the needle diameter at each row. Once the difference between the calculated needle diameter for a row and the calculated diameter for the previous row reaches more than four pixels, the search is stopped, and the two pixels in that row are chosen as the upper boundaries of the drop profile. The search continues from the upper boundary pixels of the drop. For each row, the edge pixels with the maximum and minimum x coordinate are selected as the drop profile. In the second stage, the drop profile coordinates are improved to subpixel resolution using the original gray scale image of the drop.6,21 Two different methods were implemented. In the first method, for every pixel of the drop profile, a natural cubic spline is fitted to a gray level profile perpendicular to the drop interface.6
where x is the direction normal to the drop interface, g1 and g2 correspond to the plateau gray levels at the two sides of the edge (Figure 3), W is the edge width, and x0 is the point whose gray level is the midpoint of the high and the low plateaus. The parameters g1, g2, x0, and W are the fitting parameters. The parameter x0 is considered as the new location of the drop interface. In general, optical distortion in the experimental image, produced by microscope lenses and/or digitizing board, can affect the numerical results. Therefore, a calibration grid is used to correct the optical distortion as well as to calculate the scale of the image.6,22 The grid image is acquired in the experiment, and it is used to find a function that maps any pixel of the experimental image to the corresponding coordinate point on the actual grid. Detailed information can be found elsewhere.6 The location of the contact points at the solid-liquid interface is crucial because ADSA-NA eventually measures the contact angle at this location. Detecting the contact points using the drop image is not trivial. The region around the corner at the solidliquid interface is usually fuzzy, and it is hard to detect the corner points by direct inspection. Even a robust nongradient corner detector, e.g. SUSAN,23 does not give satisfactory results, and approaches such as the B-spline snake-based method24 cannot be applied because the apex region does not appear. Typical edge detectors, such as Canny or Sobel, do not detect these contact points and their reflections as a sharp corner. Instead, the physically sharp corner appears as several pixels aligned vertically (circled area in Figure 2). In the standard ADSA, the vertical level of the solid surface was determined prior to the implementation of subpixel resolution. The upper pixel was selected as the contact point in the case of two vertically aligned pixels and the middle
(19) Canny, J. F. IEEE Trans. Pattern Anal. Mach. Intell. 1986, 8, 679–698. (20) Zuo, Y. Y.; Ding, M. B.; Bateni, A.; Hoorfar, M.; Neumann, A. W. Colloids Surf., A: Physicochem. Eng. Aspects 2004, 250, 233–246. (21) Vega, E. J.; Montanero, J. M.; Fernandez, J. Exp. Fluids DOI 10.1007/ s00348-009-657-y.
(22) Green, W. B.; Jepsen, P. L.; Kreznar, J. E.; Ruiz, R. M.; Schwartz, A. A.; Seidman, J. B. Appl. Opt. 1975, 14, 105–114. (23) Smith, S.; Brady, J. Int. J. Comput. Vision 1997, 23, 45–78. (24) Stalder, A.; Kulik, G.; Sage, D.; Barbieri, L.; Hoffmann, P. Colloids Surf., A: Physicochem. Eng. Aspects 2006, 286, 92–103.
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The point whose gray level is the midpoint of the high and the low plateaus of the fitted curve is selected as the new location of the drop interface (Figure 3). In the second method, for every pixel of the drop profile, the sigmoid function (eq 1) is fitted to a gray level profile perpendicular to the drop interface.21 gðxÞ ¼
g1 -g2 þ g2 1 þ expððx -x0 Þ=WÞ
ð1Þ
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Figure 4. Strategies for determining the contact points at the solid-liquid interface. The first method (a) detected the contact point (point A) as (877.1,669.5), and the second method (b) detected the contact point (point B) as (877.6,670.1).
one in the case of three such pixels. This strategy has an accuracy of (1 pixel, but it can introduce significant errors for the cases with more than three vertically aligned pixels, which occurs for sessile drops whose contact angles are close to 90°. Therefore, a new module was written to detect the contact points more accurately. Two different methods were implemented for determining the contact points at the interface with the solid substrate. In the first method, the solid level, i.e. the vertical coordinate of the contact points, is calculated as the average of the vertical coordinates of the pixels aligned vertically at the contact point region. Then 10 points of the drop profile in the neighborhood of the corner are fitted to a straight line. The horizontal coordinate of the contact point is found as the intersection of the fitted line and the calculated solid level (Figure 4a). In the second method, the first 10 points of the drop profile in the neighborhood of the corner are fitted to a straight line. Then another straight line is fitted to the reflection of the drop profile in the neighborhood of the corner. The intersection point of these two straight lines is chosen as the contact point at the interface with the solid substrate (Figure 4b). The accuracy of this method depends on the quality of the reflection of the drop profile, which is sometimes fuzzy. The vertical coordinates of the points at the solid-liquid interface (or solid level) are physically the same for a series of images captured in a run. Therefore, the calculated values of the solid level are averaged over all images in a run, and the average is then used as an input to ADSA-NA for each image in the run. It should be noted that in the situation when there is no reflection in the image, the contact points at the solid-liquid interface should be determined by other means. Generating Theoretical Laplacian Curve. The Laplace equation governs the balance between the surface tension and the external forces, such as gravity. The Laplace equation for an axisymmetric liquid-fluid interface (Figure 5) can be written as dφ sin φ ¼ 2b þ cz ds x c ¼
ðΔFÞg γ
ð2Þ
ð3Þ
where c is the capillary constant, γ is the liquid-fluid interfacial tension, g is the gravitational acceleration, ΔF is the density Langmuir 2009, 25(24), 14146–14154
Figure 5. Coordinate system used in the numerical solution of the Laplace equation for axisymmetric liquid-fluid interfaces without the apex. The drop is attached to the needle at the top and to the solid surface at the bottom.
difference between the two bulk phases, φ is the angle of inclination of the interface to the horizontal (x direction), s is the arc length along the interface, and b is the mean curvature of the interface at the reference level (z = 0). Analyzing the drop shape without the apex (Figure 5) introduces two new nonzero boundary conditions at the reference level for generating the Laplacian curves: at s ¼ 0 :
φ ¼ θ 0 , x ¼ R0 , z ¼ 0
ð4Þ
where R0 is the radius of the interface at the reference level, and θ0 is its inclination. Note that, when the reference level contains the three-phase line, as in the case of a sessile drop, the value of θ0 is the contact angle.16 In this case, the z direction increases upward, and g becomes negative. Therefore the value of the capillary constant, c, should be negative to account for the pressure change due to the effect of gravity inside the drop and along the z direction. Equations 2-4 show that the shape of a drop interface without the apex depends on parameters c, b, R0, and θ0. Taking these parameters as inputs, the theoretical profile can be calculated by numerical integration of eq 2. The numerical integration stops DOI: 10.1021/la902016j
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Figure 6. Sketch of ADSA-NA for profile comparison: schematic of the experimental profile obtained by edge detection (solid dots in X-Z coordinate system) and theoretical profile (solid lines in x-z coordinate system); (x0,z0) is the offset between the coordinate systems, and R is the rotation angle.
when the vertical coordinate, z, reaches the maximum vertical coordinate of the experimental profile. Optimization Procedure. Similar to the standard ADSA, the objective function is defined as the sum of the squares of the normal distances between all the experimental points of the drop profile and the theoretical profile (Figure 6). E ¼
N X
ei
ð5Þ
1
1 ei ¼ ½ðxi -Xi Þ2 þ ðzi -Zi Þ2 2
ð6Þ
Here, (Xi,Zi) are the measured drop coordinates, and (xi,zi) are the closest Laplacian coordinates to (Xi,Zi). However, in general, the two coordinate systems do not coincide, therefore their offset and rotation should be considered. Then, eq 6 can be written as 1 ei ¼ ½ðex Þ2 þ ðez Þ2 2
ð7Þ
ex ¼ xi -x0 -Xi cos R þ Zi sin R
ð8Þ
ez ¼ zi -z0 -Xi sin R -Zi cos R
ð9Þ
where (x0,z0) is the offset between the coordinate systems (experimental vs theoretical) and R is the rotation angle. The value of the objective function, E, is a function of a set of optimization parameters: the parameters determining the shape of a Laplacian curve {b,c,R0,θ0}, and the parameters determining the position of a Laplacian curve with respect to the experimental profile {x0,z0,R}. These seven optimization parameters in ADSANA are the same as those of TIFA-AI except for the substitution of c for γ. In the case of sessile drops, the parameters θ0, R0, and z0 are not independent. In fact, the same Laplacian curves can be generated using different starting points, i.e. different θ0, R0, and z0. In addition, the location of the contact points or the solid level (represented by z0) is physically fixed. Therefore, the value of z0 is constant and should not be optimized. The parameter z0 is determined as the vertical coordinate of the contact points at the solid-liquid interface discussed in the Image Processing 14150 DOI: 10.1021/la902016j
section. Fixing the value of z0 leads to a total of six optimization parameters in ADSA-NA and TIFA-AI. The goal of the optimization is to calculate the values of the optimization parameters that minimize E, thus giving the best fit between the experimental profile and a Laplacian curve. Minimizing the objective function, E, is a multidimensional nonlinear least-squares problem that requires an iterative optimization procedure. Here the Nelder-Mead simplex is used;25 it is a nongradient optimization method. In the original ADSA, Levenberg-Marquardt and Newton-Raphson8 were used as the gradient optimization methods, which require the first and second derivatives of the objective function for the Newton-Raphson and the first derivative of the objective function for LevenbergMarquardt. Compared to the gradient methods, Nelder-Mead is computationally easier to implement since it does not require a derivative of the objective function, but it consumes more computer time. Like all optimization techniques, the convergence of the Nelder-Mead simplex method depends crucially on the initial estimate of the optimization parameters. In the first step, the initial values are estimated by geometrical analysis of the extracted experimental profile. In the particular case of a sessile drop, the parameters R0 and x0 are estimated by simple calculations using the coordinates of the contact points at the solidliquid interface detected in the image processing. The parameters b and θ0 are estimated by fitting a third-order polynomial to the region close to the solid surface.26 The curvature and the inclination of the polynomial at the contact points are considered as the initial values of b and θ0, respectively. The parameter R is estimated by analyzing the image of a plumb line taken in the experiment. It was shown that the shapes of Laplacian curves close to the solid surface depend only on the values of {b,R0,θ0} and virtually not on the value of surface tension.16 Therefore an improved estimate of parameters {b,R0,θ0} can be obtained by matching the experimental profile and the theoretical profile over the region close to the solid surface. For this calculation, the lower half of the experimental profile was used, and the optimization process was carried out just for the parameters {b,R0,θ0} for an arbitrary value of surface tension, i.e., c. At the last step, the capillary constant, c, is estimated by matching the whole experimental profile to the theoretical profile. In this case, the optimization process is just performed for the capillary constant, and the other parameters are considered constant. It should be noted that the procedure used to estimate the initial values of optimization parameters is similar to the procedure in TIFA-AI.16 After estimating good initial values of the optimization parameters, the optimization process is carried out for all the optimization parameters, and it converges if the change in the value of every optimization parameter and the change in the value of the objective function reaches a given accuracy (tolerance). The minimum of the objective function determines the Laplacian curve that best fits the given experimental profile, from which interfacial properties can be readily found.
Experimental Procedure Materials. The experimental liquids are listed in Table 1. The surface tension of experimental liquids was measured using pendant drop experiments analyzed by ADSA.4 The measured (25) Nelder, J.; Mead, R. Comput. J. 1965, 7, 308–313. (26) Bateni, A.; Susnar, S. S.; Amirfazli, A.; Neumann, A. W. Colloids Surf., A: Physicochem. Eng. Aspects 1986, 219, 215–231.
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Article Table 1. Experimental Liquids and Their Surface Tension at 24°Ca
liquid
brand
purity
γ (mJ/m2) (this work)
γ (mJ/m2) (Perry’s handbook27)
n-decane Sigma-Aldrich 99+% 23.49 ( 0.01 23.48 n-dodecane Sigma-Aldrich 99+% 25.07 ( 0.04 25.02 n-tetradecane Sigma-Aldrich 99+% 26.26 ( 0.01 26.24 n-hexadecane Aldrich 99+% 27.17 ( 0.03 27.24 a The surface tensions for this work were measured using pendant drop experiments analyzed by ADSA. The errors given are the 95% confidence limits.
values of surface tension show good agreement with literature values (Table 1). The solid surfaces were Si wafers dip-coated with Teflon AF 1600 (DuPont), as described elsewhere.10 Briefly, a wafer of silicon was cut to smaller surfaces about 1 cm in diameter and soaked in chromic acid for 24 h. Then, the surfaces were washed with distilled water and dried under a heating lamp. After coating with Teflon AF 1600, the solid surfaces were annealed at 165 °C. Tubing and glassware used in the experiments were cleaned by soaking in chromic acid for 24 h. Syringes and needles were cleaned by sonication in ethanol (3 10 min) and distilled water (1 10 min). Contact Angle Measurements. The experimental liquid was loaded into a syringe that was mounted to a stepper motor. The stepper motor was used to control the rate of advancing and receding of the drop front. A drop of the experimental liquid was formed on the solid surface by allowing the motor to push liquid through Teflon tubing and then down through a vertical stainless steel needle. The needle remained attached to the drop to allow the drop volume to be changed. The size of the needle was 0.3 mm i.d. and 0.6 mm o.d. The solid surface was leveled using a bubble level. The drop was illuminated from behind by a white light projector through frosted glass. A charge-coupled device (CCD) camera (Sony XCD-SX900) was used to capture the images with a horizontal microscope (Wild Heerbrugg 400076) at 5.8 magnification. The microscope and camera were mounted together on an axial translation stage for focusing, and the camera was leveled using the bubble level. The tilting angle of the camera was zeroed using a plumb line. Images were captured while the drop size was increased (advancing contact angle) and then decreased (receding contact angle). The images were 1024 768 pixels in size, and the physical pixel size was about 6 μm. The typical range of drop diameters was 3 - 6 mm, and a typical rate of motion of the drop front was 0.1 mm/min. The sample and the optical apparatus were on a vibration isolation table.
Results and Discussion ADSA-NA Measurements. Contact angle measurements were performed for each liquid on one solid surface with two or three different runs. Each run consisted of enlarging and shrinking of a drop formed on a different spot of the solid surface. The results of ADSA-NA for contact angle and contact radius are shown in Figure 7 for a typical run with dodecane on Teflon AF 1600. Each run typically contained 100 images overall. The volume and surface area of the drop are also outputs of ADSANA. Figure 8 illustrates the Laplacian curve (white line) superimposed on the drop interface calculated for a dodecane drop by ADSA-NA. It should be noted that since the solid surfaces used were homogeneous and flat, the drops were fairly axisymmetric, i.e., few pixels difference between the two sides of the drop. However, drops with high asymmetry were excluded from the analysis. One of the input parameters in ADSA-NA is the tolerance value in the optimization process. The iterative optimization process stops if the relative change in every optimization parameter and the objective function reaches this tolerance. The default value in the Nelder-Mead function in MATLAB is 10-4, but there is no guarantee that this is adequate. To determine the Langmuir 2009, 25(24), 14146–14154
Figure 7. Results of ADSA-NA for a typical run with dodecane on Teflon AF 1600. The advancing angle was measured as 63.87 ( 0.02 for this run. The literature value from ref 10 is 63.78° ( 0.28.
Figure 8. A dodecane drop on a Teflon AF 1600 coated surface. The white line is the Laplacian fit by ADSA-NA. DOI: 10.1021/la902016j
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Table 2. Advancing Contact Angles (Degrees) of Sample Liquids on Teflon AF 1600 Analyzed by ADSA-NA (Using Two Different Sub-Pixel Resolution Methods) and TIFA-AI and the Comparison with Literature Valuesa ADSA-NA spline
sigmoid
TIFA-AI
ref 10
decane
RunA RunB
59.37 ( 0.04 59.43 ( 0.02
59.30 ( 0.04 59.43 ( 0.06
59.37 ( 0.05 59.43 ( 0.02
59.29 ( 0.18
dodecane
RunA RunB
63.87 ( 0.02 63.86 ( 0.02
63.80 ( 0.03 63.81 ( 0.02
63.85 ( 0.02 63.85 ( 0.03
63.78 ( 0.32
RunA RunB RunC
67.16 ( 0.03 67.20 ( 0.02 67.15 ( 0.02
67.13 ( 0.03 67.18 ( 0.03 67.11 ( 0.02
67.01 ( 0.02 67.07 ( 0.03 66.99 ( 0.01
tetradecane
67.15 ( 0.14
RunA 69.44 ( 0.02 69.40 ( 0.01 69.32 ( 0.02 RunB 69.43 ( 0.02 69.40 ( 0.02 69.27 ( 0.02 69.48 ( 0.16 RunC 69.44 ( 0.02 69.41 ( 0.03 69.23 ( 0.02 a The errors given for ADSA-NA and TIFA-AI are the standard deviations, and the errors given for ref 10 are the 95% confidence limits. The values for ref 10 were averaged over five runs.
hexadecane
required tolerance for ADSA-NA and TIFA-AI, the advancing contact angles of dodecane and tetradecane on Teflon AF 1600 were measured for one sample run for each liquid with tolerance values ranging from 10-4 to 10-8. On the basis of this analysis, the required tolerances for ADSA-NA and TIFA-AI are found to be 10-6 and 10-4, respectively. Comparison with ADSA. Table 2 shows the advancing contact angles of all sample liquids on Teflon AF 1600 analyzed by ADSA-NA and TIFA-AI. The results of ADSA-NA were calculated using the two different subpixel resolution methods: spline function and sigmoid function. Advancing contact angles were calculated for the largest drops (around 40 images), for which drop size dependence is insignificant. The last column of Table 2 shows the previously reported values10 obtained by ADSA using regular sessile drops, i.e., sessile drops with the apex. The contact angle for each liquid in ref 10 was obtained by averaging the contact angles over five different runs on five solid sample surfaces, each containing about 100 images for advancing contact angles. The errors given in Table 2 represent standard deviations for ADSA-NA and TIFA-AI, but 95% confidence limits for the data in reference.10 In the present experiments with both ADSA-NA and TIFA-AI, since the liquid surface tensions were known, the surface tension was held fixed during the optimization, i.e., it was input (discussed further in the Optimization Parameters section below). In contrast, surface tension was optimized in ref 10. The contact angles obtained by means of ADSA-NA with the two subpixel resolution methods agree to within 0.1° and show similar scatter, i.e., image-to-image standard deviation, for most sample runs. This similarity shows that the different strategies of extracting the edge all yield essentially the same answer. Here, the spline method is preferred because it requires less computation time. The good agreement between ADSA-NA and TIFA-AI demonstrates the independence of the contact angle measurements from both the numerical method and the image processing. The good agreement between ADSA-NA and previous ADSA results demonstrates the independence of the measurements from experimental technique. There is a slightly closer agreement between ADSA-NA and ADSA than between TIFA-AI and ADSA. Overall, with different experimenters, experimental setup, and numerical method, average advancing contact angles are reproducible to no worse than ∼0.2°. (27) Green, D. W.; Perry, R. H. Perry’s Chemical Engineers’ Handbook, 8th ed.; McGraw-Hill: New York, 2008.
14152 DOI: 10.1021/la902016j
The typical image-to-image scatter (standard deviation) for ADSA-NA and TIFA-AI in advancing contact angles is about 0.02°, considerably lower than the scatter of about 0.5° obtained with ADSA in reference.10 Sources for this improvement may include the nonoptimization of c and the greater number of pixels per image. Run-to-run scatter for ADSA-NA and TIFA-AI (∼ 0.05°) is also lower than the previous results in reference.10 Comparing the results in this study with those in reference 18 shows slight improvement in the image-to-image and run-to-run scatter. It is also intriguing that much of the image-to-image scatter is not random and is well reproduced from ADSA-NA to TIFA-AI. This suggests that the causes for the scatter reported in Table 2 for these two methods are physical. Here, the range of contact angle for sample liquids is about 10°. However, ADSA-NA strategy is indeed applicable to a wider range of contact angle since it is based on the original ADSA strategy for measuring contact angles, and ADSA has been tested extensively for a wide range of contact angle (10-170°).29,30 Moreover, TIFA-AI was tested recently18 for an identical setup for a wider range of contact angle (40-85°). To compare the quality of the solid surfaces used in this study and the previous studies,10,28 contact angle hysteresis, i.e. the difference between advancing contact angle and receding contact angle, was calculated for each run. The receding contact angle was the linearly extrapolated estimate at the time of the last measured advancing angle. Table 3 illustrates the contact angle hysteresis for each run obtained by ADSA-NA and TIFA-AI and the comparison with previous results obtained by standard ADSA. Apparently there is no significant difference between the surfaces used in ref 10 and those used here. Solid Surface Location. As mentioned in the Image Analysis section, two different methods were implemented in ADSA-NA to detect the contact points at the solid-liquid interface automatically (Figure 4). In TIFA-AI, these contact points are selected manually by eye, with an estimated accuracy of ( 1 pixel. To examine the influence of different methods of locating the solid surface, the advancing contact angles of dodecane and tetradecane on Teflon AF 1600 were measured with two different methods. The results are shown in Table 4. The contact angles obtained with the two methods agree to within 0.1°. The standard (28) Tavana, H.; Jehnichen, D.; Grundke, K.; Hair, M. L.; Neumann, A. W. Adv. Colloid Interface Sci. 2007, 134-135, 236–248. (29) Tavana, H.; Amirfazli, A.; Neumann, A. W. Langmuir 2006, 22, 5556–5559. (30) Alvarez, J.; Amirfazli, A.; Neumann, A. W. Colloids Surf., A: Physicochem. Eng. Aspects 1999, 156, 163–176.
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Table 3. Contact Angle Hysteresis (Degrees) of Sample Liquids on Teflon AF 1600 Analyzed by ADSA-NA and TIFA-AI and the Comparison with Literature Values ADSA-NA
TIFA-AI
ref 28
decane
RunA RunB
5.6° 5.6°
5.6° 5.6°
5.7°
dodecane
RunA RunB
5.2° 5.2°
5.2° 5.2°
5.4°
tetradecane
RunA RunB RunC
5.4° 5.3° 5.4°
5.4° 5.4° 5.5°
5.6°
hexadecane
RunA RunB RunC
5.4° 5.3° 5.3°
5.4° 5.3° 5.4°
5.3°
Table 4. The Effect of Different Methods of Solid Detection on the Advancing Contact Angles θadv (Degrees) of Dodecane and Tetradecane on Teflon AF 1600a liquid
run
method
θadv
method 1 63.87 ( 0.02 method 2 63.80 ( 0.02 RunB method 1 63.86 ( 0.02 method 2 63.78 ( 0.04 tetradecane RunA method 1 67.16 ( 0.03 method 2 67.08 ( 0.04 RunB method 1 67.20 ( 0.02 method 2 67.13 ( 0.04 RunC method 1 67.15 ( 0.02 method 2 67.07 ( 0.02 a Method 1 and method 2 are illustrated in Figure 4a,b, respectively. The errors given are the standard deviations. dodecane
RunA
deviations are substantially smaller than the differences between the contact angles calculated with the two methods. Hence it is worthwhile to pursue the underlying question. The average solid level calculated by method 1 is lower than that calculated by method 2 by about 2.2 μm (0.38 pixel) for dodecane and 2.3 μm (0.40 pixel) for tetradecane. Method 1 is preferred here because it shows slightly lower scatter, i.e., smaller image-to-image standard deviation. The difference between the two methods shows that solid surface positioning is a significant source of uncertainty in contact angle measurement. To ensure that it is indeed ADSA-NA and TIFA-AI that are compared, the solid level detected by ADSA-NA for each run was also used as an input for TIFA-AI. Optimization Parameters. In ADSA-NA, the tilt angle of the camera, R, is optimized to correct the vertical alignment of the camera. To examine the effect of fixing or optimizing R on the results, the advancing contact angles of dodecane, tetradecane, and hexadecane on Teflon AF 1600 for one sample run were measured with fixed R, which was calculated from analyzing the image of the plumb line. No significant change was observed for the average advancing angles and their scatter compared to the results with optimized R. The reason is that R was small (less than 0.1°) in the experiments. Therefore, if the experiment is performed carefully enough, R does not need to be optimized, which makes the optimization faster. It should be noted that the horizontal alignment of the camera was corrected using a bubble level in the experiment. It is known31 that, for an inclination angle of about 0.1° and a contact angle of about 50°, the error for the measured contact (31) Sobolev, V. D.; Starov, V. M.; Velarde, M. G. Colloid J. 2003, 65, 611–614.
Langmuir 2009, 25(24), 14146–14154
angle is about 10-4 degree, which is an insignificant error in the measurements. All the results shown in Table 2 and Table 3 for ADSA-NA and TIFA-AI were obtained for known surface tension, i.e., the measurements from Table 1 were used as input. However, in previous results10 analyzed by ADSA, surface tension, i.e. capillary constant, was optimized. To investigate the effect of optimizing surface tension in ADSA-NA and TIFA-AI on the results, the advancing contact angle of each liquid was analyzed with surface tension as an optimization parameter (Table 5). It is found that the necessary tolerances for ADSA-NA and TIFA-AI in this case are 10-9 and 10-6, respectively. Comparing the results for each run in Table 2 with the corresponding run in Table 5 shows that there is at most 0.1° change in the average advancing contact angle for each run with variable surface tension compared to the case of fixed surface tension. In addition, the scatter of the measured contact angle for every liquid is higher when surface tension is variable in the optimization, for both ADSA-NA and TIFA-AI. The typical image-to-image scatter for ADSA-NA and TIFA-AI for advancing contact angles in Table 5 is about 0.05°, which is higher than the scatter of 0.02° for the strategy of using surface tension as input. However, the imageto-image scatter is still considerably lower than the previous results in ref 10. The underlying reason for higher scatter of contact angles in Table 5 compared to Table 2 presumably is that the optimization takes place over an expanded parameter space, allowing the surface tension to take on unphysical values with spurious compensating adjustments in the contact angle. Comparing the surface tensions in Table 5 with those in Table 1 reveals that there is an average of 0.2 mJ/m2 error when compared with Table 1. There is also an average discrepancy of 0.2 mJ/m2 between the results of ADSA-NA and TIFA-AI, which is likely due to their different numerical strategies. Nevertheless, the accuracy of the surface tension reported here far surpasses that obtained from sessile drop and the standard ADSA procedure. On the basis of the above discussion for the case of adjustable surface tension, the contact angles in Table 2 (fixed surface tension) are preferred to those in Table 5 (variable surface tension). However, if surface tension is unknown in contact angle measurements, it can be optimized in the analysis, which leads to contact angle results with slightly higher scatter and higher processing time compared to the case of using a known value of surface tension measured accurately before, e.g., with pendant drop experiments. In this case, the measured value of contact angle differs up to 0.1° from the case of a known value of surface tension. It should be noted that using an incorrect fixed value of surface tension can affect the measured contact angle and leads to systematic errors. To investigate this effect, the advancing contact angle of dodecane was measured for different fixed values of surface tension. It is found that changing the surface tension by (1 mJ/m2 from the true value changes the contact angle by (0.3°. Therefore, if the value of surface tension changes in the experiment, e.g., due to impurities, it should be determined simultaneously with contact angle.
Conclusions The new contact angle methods ADSA-NA and TIFA-AI, when applied to the same drop image, produce results that agree to within 0.1°. The agreement with literature values DOI: 10.1021/la902016j
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Kalantarian et al. Table 5. Advancing Contact Angles (Degrees) of Sample Liquids on Teflon AF 1600 with Optimized Surface Tensiona ADSA-NA
TIFA-AI
run
θadv
γ
θadv
γ
decane
RunA RunB
59.29 ( 0.05 59.36 ( 0.04
23.87 ( 0.10 23.70 ( 0.10
59.36 ( 0.09 59.43 ( 0.10
23.58 ( 0.32 23.51 ( 0.33
dodecane
RunA RunB
63.91 ( 0.04 63.95 ( 0.04
24.95 ( 0.14 24.91 ( 0.10
63.94 ( 0.04 63.95 ( 0.05
24.76 ( 0.12 24.71 ( 0.17
tetradecane
RunA RunB RunC
67.21 ( 0.05 67.28 ( 0.05 67.20 ( 0.04
26.08 ( 0.11 26.15 ( 0.12 26.25 ( 0.08
67.08 ( 0.05 67.15 ( 0.04 67.06 ( 0.05
26.00 ( 0.11 25.97 ( 0.12 26.06 ( 0.12
RunA 69.45 ( 0.04 27.27 ( 0.10 69.38 ( 0.03 RunB 69.45 ( 0.05 27.23 ( 0.11 69.34 ( 0.04 RunC 69.46 ( 0.05 27.28 ( 0.09 69.37 ( 0.03 a θadv is the average advancing contact angle, and γ (mJ/m2) is the surface tension. The errors given are the standard deviations.
hexadecane
obtained from the studies of sessile drops with an apex by means of ADSA is approximately (0.2-0.3 degrees. The error limits for ADSA-NA and TIFA-AI are tighter at least by a factor of 5 than those obtained with the conventional ADSA. These results suggest that the accuracies reported are not artifacts of details of experimentation or of mathematical or image analysis procedure. Contact angle measurements even at
14154 DOI: 10.1021/la902016j
27.00 ( 0.08 27.01 ( 0.19 26.90 ( 0.07
that high level of accuracy reflect physicochemical properties of the solid surface. Acknowledgment. This work was supported by Natural Sciences and Engineering Research Council (NSERC) of Canada Grant 8278 and an open fellowship from University of Toronto to A.K.
Langmuir 2009, 25(24), 14146–14154
