DEVELOPMENT OF AXISYMMETRIC DROP SHAPE ANALYSIS - NO APEX (ADSA-NA) Ali Kalantarian

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1 DEVELOPMENT OF AXISYMMETRIC DROP SHAPE ANALYSIS - NO APEX (ADSA-NA) by Ali Kalantarian A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mechanical and Industrial Engineering University of Toronto Copyright c 2011 by Ali Kalantarian

2 Abstract DEVELOPMENT OF AXISYMMETRIC DROP SHAPE ANALYSIS - NO APEX (ADSA-NA) Ali Kalantarian Doctor of Philosophy Graduate Department of Mechanical and Industrial Engineering University of Toronto 2011 The main purpose of this thesis is the development of a new methodology of contact angle measurement called ADSA-NA (Axisymmetric Drop Shape Analysis - No Apex) that analyzes drop shape configurations with no apex. Thus ADSA-NA facilitates contact angle measurements on drops with a capillary protruding into the drop. This development is desirable because the original ADSA has some limitations for contact angle measurement, and there is a need for the improvement of the accuracy of contact angle measurement. To develop ADSA-NA, a new reference point other than the apex and a new set of optimization parameters different from those of ADSA had to be defined. The three main modules of ADSA had also to be modified; the image analysis, the numerical integration of the Laplace equation for generating theoretical curves, and the optimization procedure. It was shown that ADSA-NA significantly enhances the precision of contact angle and surface tension measurements (by at least a factor of 5) compared to those obtained from sessile drops using ADSA. Computational as well as design aspects of ADSA-NA were scrutinized in depth, well beyond the level of scrutiny in the original ADSA. On the computational side, the results obtained from one and the same drop image were compared using different gradient ii

3 and non-gradient edge detection strategies and different gradient and non-gradient optimization methods. It was found that the difference between the results of different edge detection strategies is minimal. It was also found that all the optimization methods yield the same answer with eight significant figures for one and the same image. The determination of the location of the solid surface in the drop image was also further refined. On the design side, the effect of controllable experimental factors such as drop height and drop volume on the accuracy of surface tension measurement was studied. It was shown that drop height is the dominant experimental factor, and larger drop heights yield lower surface tension errors. iii

4 Dedication To my parents, Mitra and Massood To my wife, Mandana iv

5 Acknowledgements I would like to thank God for giving me the opportunity and strength to gain knowledge and helping me to complete my studies. I would like to take this opportunity to express my sincere gratitude to my supervisor, Prof A. W. Neumann. He provided me with advice and support throughout my Ph.D. studies. I have had a pleasure of learning a lot of things from him. I would also like to thank my co-supervisor, Prof. E. J. Acosta, for his help and advice. I would like to thank all my colleagues, Dr. S. Saad, Dr. R. David, and Ms. Z. Policova, at the Laboratory of Applied Surface Thermodynamics for their help and friendship. life. I would like to thank my parents for their continued love and support throughout my And finally, I am indebted to my wife for her support and encouragement. v

6 Contents 1 Introduction Overview Purpose and Scope of the Thesis Overview of Drop Shape Techniques 12 3 Axisymmetric Drop Shape Analysis-No Apex (ADSA-NA) Introduction Image Analysis Drop Profile Detection with Pixel Resolution Drop Profile Detection with Sub-Pixel Resolution Solid Surface Location Generating Theoretical Laplacian Curves Optimization Procedure ADSA-NA Measurements Introduction Experimental Procedure Contact Angle Experiments Materials and Solid Surfaces Preparation Contact Angle Measurements Comparison of ADSA-NA and TIFA-AI with ADSA Effect of Solid Surface Location Effects of Optimization Parameters Examination of Image Analysis and Optimization Strategies Introduction Image Analysis Edge Detector Solid Surface Location Optimization Procedure vi

7 6 Optimization of Experimental Geometry of ADSA-NA Introduction Effect of Volume at Constant Drop Height Effects of Drop Height, Drop Volume, Pedestal Width, and Drop Asymmetry Objective Function Curvature at Its Minimum and the Accuracy of Surface Tension Measurement Summary and Future Work Summary Future Work Appendices 153 A System of Equations for Levenberg-Marquardt and Newton-Raphson 154 A.1 Laplace Equation of Capillarity A.2 Optimization Procedure A.2.1 Error Estimate A.2.2 Objective Function A.2.3 Optimization Methods A.2.4 Evaluation of Partial Derivatives A.2.5 System of Differential Equations B Drop Size Dependence of Contact Angles 172 B.1 Introduction B.2 Drop Size Dependence B.3 Effect of Edge Detection B.4 Effect of Drop Asymmetry Bibliography 182 vii

8 List of Tables 4.1 Experimental liquids and their surface tension at 24 C. The surface tensions for this thesis were measured using pendant drop experiments analyzed by ADSA. The errors given are the 95% confidence limits The effect of tolerance value in the optimization procedure on the advancing contact angles θ adv (degrees) of dodecane and tetradecane on Teflon AF 1600 analyzed by ADSA-NA and TIFA-AI. The errors given are the standard deviations. T is the average computation time per image in seconds 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 values. The errors given for ADSA-NA and TIFA-AI are the standard deviations, and the errors given for Ref. [21] are the 95% confidence limits. The values for Ref. [21] had been averaged over five runs Contact angle hysteresis (degrees) of sample liquids on Teflon AF 1600 analyzed by ADSA-NA and TIFA-AI and the comparison with literature values The effect of two different strategies of solid detection on the advancing contact angles θ adv (degrees) of dodecane and tetradecane on Teflon AF The profile and the reflection strategies are illustrated in figures 3.5(a) and 3.5(b), respectively. The errors given are the standard deviations Advancing contact angles (degrees) of sample liquids on Teflon AF 1600 with optimized surface tension. θ adv is the average advancing contact angle, and γ (mj/m 2 ) is the surface tension. The errors given are the standard deviations The effect of tolerance value in the optimization procedure on the advancing contact angles θ adv (degrees) of dodecane and tetradecane on Teflon AF 1600 with optimized surface tension analyzed by ADSA-NA and TIFA-AI. θ adv is the average advancing contact angle, and γ (mj/m 2 ) is the surface tension. The errors given are the standard deviations. T is the average computation time per image in seconds Advancing contact angle θ adv (degrees) of dodecane on Teflon AF 1600 analyzed by ADSA-NA using different fixed values of surface tension (γ (mj/m 2 )). The errors given are the standard deviations viii

9 5.1 Advancing contact angles (degrees) of four sample alkanes on Teflon AF 1600 with fixed surface tension analyzed by ADSA-NA using different edge detections methods. The errors given are the standard deviations. The surface tension values measured by ADSA using pendant drop experiments (Table 4.1) were used as inputs Advancing contact angles (degrees) of four sample alkanes on Teflon AF 1600 with optimized surface tension analyzed by ADSA-NA using different edge detection methods. The errors given are the standard deviations Advancing contact angles (degrees) of sample liquids on Teflon AF 1600 with optimized surface tension analyzed by ADSA-NA using the two different sub-pixel resolution methods. θ adv is the average advancing contact angle, and γ (mj/m 2 ) is the surface tension. The errors given are the standard deviations. Canny was used as the edge detector The effect of different strategies of solid detection (using the Canny edge detection) on the advancing contact angles θ adv (degrees) of dodecane and tetradecane on Teflon AF The errors given are the standard deviations Difference between the calculated solid surface locations by Canny on the one hand and LoG, SUSAN, and Sobel on the other hand. The difference is in pixels. The minus sign means the calculated solid location is lower than the one calculated by Canny. The modified reflection strategy was used for detecting the solid surface location Advancing contact angles (degrees) of four sample alkanes on Teflon AF 1600 with fixed surface tension analyzed by ADSA-NA using different optimization methods. The errors given are the standard deviations. The surface tension values measured by ADSA using pendant drop experiments (Table 4.1) were used as inputs Advancing contact angles (degrees) of four sample alkanes on Teflon AF 1600 with optimized surface tension analyzed by ADSA-NA using different optimization methods. The errors given are the standard deviations Experimental liquids used in the experiments with a pedestal and their surface tension at 24 C. The surface tensions measured using pendant drop experiments were analyzed by ADSA. The errors given are the 95% confidence limits Surface tensions (mj/m 2 ) of sample liquids obtained from constant height experiments with a pedestal. Each number corresponds to one run and each pair corresponds to one surface. The errors given are the standard deviations Coefficient of volume in the regression function, m, for surface tension error fitted to volume (Equation 6.1) for all runs of each sample liquid obtained in constant height experiments ix

10 6.4 Coefficient of volume in the regression function, m, for surface tension error fitted to volume (Equation 6.1) obtained in constant height experiments. Each number is the average of the calculated coefficient m for all runs of each sample liquid for each pedestal width Sample results of constant volume experiments obtained for dodecane with three different pedestal widths Coefficients of the regression function for surface tension error fitted to volume (a), height (b), asymmetry (c), and pedestal width (d) for all drops of the sample liquids obtained in constant volume experiments. The errors given are the 95% confidence intervals, and R 2 is the square of the correlation coefficient Results of constant volume experiments obtained for water using a pedestal width of 6.0 mm B.1 Sigma (J/m) of all runs of DMCPS, dodecane, cyclododecatriene, and water obtained from constant height experiments with a pedestal. Each number corresponds to one run and each pair corresponds to one surface. 179 B.2 Advancing contact angles (degrees) of all runs of DMCPS, dodecane, cyclododecatriene, and water obtained from constant height experiments with a pedestal. Each number corresponds to one run and each pair corresponds to one surface. The errors given are the standard deviations B.3 Advancing contact angles (degrees) and sigma (J/m) of four long-chained alkanes on Teflon AF 1600 using a capillary. The errors given are the standard deviations x

11 List of Figures 1.1 Schematic of the contact angle (θ). γ lv, γ sv, and γ sl are the liquid-vapour interfacial tension, the solid-vapour interfacial tension, and the solid-liquid interfacial tension, respectively 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 General procedure of Axisymmetric Drop Shape Analysis (ADSA) The binary image corresponding to Figure 1.2 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 Criteria for the identification of contact points at the solid-liquid interface for the cases of contact angle smaller and greater than A typical example of a gray level profile perpendicular to a drop interface (open circles). Point A is the pixel which 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 sub-pixel resolution An image of the calibration grid. The size of the squares is mm Strategies for determining the contact points at the solid-liquid interface. The profile strategy (a) detected the contact point (point A) as (877.1,669.5), and the reflection strategy (b) detected the contact point (point B) as (877.6,670.1) 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 Sketch of ADSA-NA for profile comparison: schematic of the experimental profile obtained by edge detection (solid dots in X-Z coordinate system) and the Laplacian profile (solid lines in x-z coordinate system); (x 0, z 0 ) is the offset between the coordinate systems, and α is the rotation angle Schematic diagram of the experimental setup of ADSA-NA Repeat unit of: poly[4,5-difluoro-2,2-bis(trifluoromethyl)-1,3-dioxole-cotetrafluoroethyene], 65 mol% dioxole (Teflon AF 1600) xi

12 4.3 Results of ADSA-NA for a typical run with dodecane on Teflon AF The advancing angle was measured as ± 0.02 for this run. The literature value from reference [21] is ± A dodecane drop on a Teflon AF 1600 coated surface. The white line is the Laplacian fit by ADSA-NA Results of ADSA-NA and TIFA-AI for advancing contact angle for a typical run with decane on Teflon AF The advancing angle for the largest drops of this run was measured by ADSA-NA and TIFA-AI as ± 0.04 and ± 0.05, respectively Advancing and receding contact angles of dodecane on Teflon AF The receding angle is time dependent Modified reflection strategy for determining the contact points at the solidliquid interface of a dodecane drop on Teflon AF The original reflection strategy detected the contact point (point B in Figure 3.5(b)) as (877.6,670.1), and the modified reflection strategy detected the contact point (point C) as (877.0,670.2), for this specific drop image Flowchart of identifying outliers using Pierce s criterion Image of a dodecane drop on a Teflon AF 1600 coated surface formed by injecting the liquid through an inverted vertical stainless steel pedestal Images of different drops of DMCPS on Teflon AF 1600 coated surfaces obtained from constant drop height experiments. Drop (a) was formed using a pedestal width of 1.6 mm with a volume of 5 µl. Drop (b) was formed using a pedestal width of 2.8 mm with a volume of 20 µl. Drop (c) was formed using a pedestal width of 3.8 mm with a volume of 35 µl Images of different drops of dodecane on Teflon AF 1600 coated surfaces obtained from constant drop height experiments. Drop (a) was formed using a pedestal width of 1.6 mm with a volume of 8 µl. Drop (b) was formed using a pedestal width of 2.8 mm with a volume of 29 µl. Drop (c) was formed using a pedestal width of 3.8 mm with a volume of 50 µl Images of different drops of cyclododecatriene on Teflon AF 1600 coated surfaces obtained from constant drop height experiments. Drop (a) was formed using a pedestal width of 1.6 mm with a volume of 12 µl. Drop (b) was formed using a pedestal width of 2.8 mm with a volume of 35 µl. Drop (c) was formed using a pedestal width of 3.8 mm with a volume of 58 µl Images of different drops of water on Teflon AF 1600 coated surfaces obtained from constant drop height experiments. Drop (a) was formed using a pedestal width of 1.6 mm with a volume of 11 µl. Drop (b) was formed using a pedestal width of 2.8 mm with a volume of 45 µl. Drop (c) was formed using a pedestal width of 3.8 mm with a volume of 78 µl xii

13 6.6 Images of a drop of dodecane on a Teflon AF 1600 coated surface captured at different vertical positions of the pedestal obtained from constant volume experiments. The drop was formed using a pedestal width of 3.8 mm with a volume of 36 µl. The surface tension of dodecane using the drop images at pedestal positions (a), (b), (c), and (d) was measured as mj/m 2, mj/m 2, mj/m 2, and mj/m 2, respectively Surface tension versus drop height for DMCPS on Teflon AF 1600 obtained from constant volume experiments. The measured value of γ using pendant drop experiments analyzed by ADSA is mj/m 2 (broken straight line in the figure) Surface tension versus drop height for dodecane on Teflon AF 1600 obtained from constant volume experiments. The measured value of γ using pendant drop experiments analyzed by ADSA is mj/m 2 (broken straight lines in the figure) Surface tension versus drop height for cyclododecatriene on Teflon AF 1600 obtained from constant volume experiments. The measured value of γ using pendant drop experiments analyzed by ADSA is mj/m 2 (broken straight line in the figure) Surface tension versus drop height for water on Teflon AF 1600 obtained from constant volume experiments. The measured value of γ using pendant drop experiments analyzed by ADSA is mj/m 2 (broken straight line in the figure) Theoretical Laplacian profiles calculated for sessile drops. Open symbols, 10% higher surface tension than solid symbols (with the same curvature at the top of the profile and the same contact angle) Images of different drops of water on a Teflon AF 1600 coated surface formed using a pedestal width of 6.0 mm. The volumes of drops (a), (b), (c), and (d) are 157 µl, 240 µl, 305 µl, and 334 µl, respectively. The surface tension of water using the drop images (a), (b), (c), and (d) was measured as mj/m 2, mj/m 2, mj/m 2, and mj/m 2, respectively Objective function value versus hypothetical surface tension for different drops of dodecane on Teflon AF 1600 with pedestal width of 1.6 mm. The output values of surface tension (arrows in the figure) are (mj/m 2 ) and (mj/m 2 ) for drop 1 and drop 2, respectively. The measured value of γ using pendant drop experiments analyzed by ADSA is mj/m 2 (broken vertical line in the figure) Output value of surface tension versus the curvature of the objective function value at its minimum for all drops of DMCPS obtained from constant volume experiments. The measured value of γ using pendant drop experiments analyzed by ADSA is mj/m 2 (broken horizontal line in the figure) xiii

14 6.15 Output value of surface tension versus the curvature of the objective function value at its minimum for all drops of dodecane obtained from constant volume experiments. The measured value of γ using pendant drop experiments analyzed by ADSA is mj/m 2 (broken horizontal line in the figure) Output value of surface tension versus the curvature of the objective function value at its minimum for all drops of cyclododecatriene obtained from constant volume experiments. The measured value of γ using pendant drop experiments analyzed by ADSA is mj/m 2 (broken horizontal line in the figure) Output value of surface tension versus the curvature of the objective function value at its minimum for all drops of water obtained from constant volume experiments. The measured value of γ using pendant drop experiments analyzed by ADSA is mj/m 2 (broken horizontal line in the figure) Curvature of the objective function value at its minimum versus nondimensional drop height for all drops of DMCPS, dodecane, cyclododecatriene, and water obtained from constant volume experiments Parabolas fitted to the data shown in Figure The curve corresponding to each liquid was obtained by fitting a parabola to the data shown for that liquid in Figure B.1 (a) Advancing contact angles versus contact radius (r) for a sample run of dodecane on Teflon AF 1600 using a pedestal width of 1.6 mm. (b) cos θ adv versus 1/r for the same run. The solid line is the linear fit to the data (c) Asymmetry calculated for the same run B.2 Advancing contact angles and the calculated sigma for a sample run of dodecane on Teflon AF 1600 obtained with different edge detectors and sub-pixel resolution methods B.3 Sigma versus the slope of the asymmetry for all runs of DMCPS, dodecane, cyclododecatriene, and water obtained from constant height experiments. The solid line is the linear fit to the data xiv

15 Chapter 1 Introduction 1.1 Overview When a liquid is in contact with a solid surface, the contact angle is defined as the angle made between the solid and a tangent aligned with the liquid at the point of contact with the solid (Figure 1.1). The study of wetting, adhesion, and flotation requires knowledge of the contact angle. Wetting is the ability of a liquid to maintain contact with a solid surface. The contact angle provides an inverse measure of wettability. A contact angle very close to zero indicates wetting of the surface, and the fluid will tend to spread over a large area of the surface. There are various situations where good contact is desired between a liquid and an oily or waxy surface. For example, insecticides should fully wet the surface of waxy leaves to protect the leaves from insects and disease, and animal dips should properly coat the greasy hair of animals to repel vermin and lice [1]. In studies of biological adhesion, the free energy of adhesion is determined using 1

16 2 the contact angle. This free energy establishes whether cells and proteins will adhere to foreign surfaces [2, 3]. In the mining industry, flotation plays an important role in separating the minerals from the non-minerals in crushed ore. The mineral particles float to the surface by attaching themselves to air bubbles. If the contact angle at the interface is finite, then the particle may be stably located at the interface [4 7]. Impurities and adulterants present in or added to a liquid may alter the contact angle considerably [1]. Wetting agents or detergents change the contact angle from a large value, greater than 90, to a value often much smaller than 90. Conversely, waterproofing agents applied to cloth cause the contact angle of water in contact with the cloth to be larger than 90. Different methods have been developed for measuring contact angles. The most common techniques are direct measurement from sessile drops using a goniometer [1, 8 10], the Wilhelmy balance method [8, 11, 12], the capillary rise method [8, 13], and drop shape methods [14 16]. A widely used conventional method of contact angle measurement is the direct measurement of contact angle from sessile drops using a telescope equipped with a goniometer eyepiece [1, 8]. In this technique, the contact angle is measured by aligning a tangent with the drop profile at the contact point of the solid-liquid interface. A precision of ±2 is claimed at best for the direct technique. Although the direct method is easy to implement, it is subjective and dependent on the experience of the operator [8]. For a liquid with known surface tension, the Wilhelmy balance method or the capillary rise method can be used to measure contact angles. In the Wilhelmy method, a smooth

17 3 vertical plate is brought into contact with the liquid, and the downward force exerted by the liquid on the plate is measured using an electrobalance [1, 8]. The contact angle is calculated using the following relation: f = pγ lv cos θv ρg (1.1) where f is the measured force, p is the perimeter of the plate, γ lv is the liquid surface tension, θ is the contact angle, V is displaced liquid volume, ρ is the liquid-fluid density difference, and g is the gravitational acceleration. An alternative method for measuring the contact angle of a liquid with known surface tension is the capillary rise at a vertical plate. In this method, the solid surface is also aligned vertically and brought into contact with the liquid. Instead of the capillary force as in the Wilhelmy method, the capillary rise h is measured in this method. The contact angle is calculated using the following relation: sin θ = 1 ρgh2 2γ lv (1.2) Although the precision of contact angles obtained with the capillary rise method and the Wilhelmy method is far better by an order of magnitude than the direct method, these techniques have several drawbacks. They are not easy to handle experimentally, and they require relatively large amounts of liquid [8]. Furthermore, the plate used in the Wilhelmy method must have the same composition and morphology on both sides [8].

18 4 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, 9]. It is noted that the earliest work in drop shape techniques goes back to the late nineteenth century by Bashforth and Adams [17]. They generated tables containing the solution of the Laplace equation for sessile drop profiles for different surface tension and radius of curvatures at the apex of the drop. 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 [1]. The advantages of drop shape techniques are numerous. They require only small amounts of the liquid [14] and are applicable to various situations, including extreme temperature and pressure [18]. If the profile of the drop is recorded, drop shape methods can be used in dynamic systems, e.g. involving changes in surface tension with time due to various causes such as adsorption of surface-active substances [14], 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 dropshape technique for measuring interfacial properties [14]. ADSA acquires images of a drop (or bubble) and extracts the experimental profile of the drop using edge detection techniques [14]. Then it fits a theoretical profile of the drop to the extracted experimental profile, taking surface tension and geometrical information at the apex as adjustable parameters, i.e. as optimization parameters. An error function is defined to measure the distance between the theoretical curve and the experimental profile. An optimization

19 5 procedure finds the best match between experimental and theoretical profiles by minimizing the error function. The best match provides the value of the surface tension, contact angle, surface area, and drop volume. 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 drop possessing an apex in ADSA, i.e. the geometry of the drop interface at the apex is used as the reference level to solve the Laplace equation, the most convenient constellation for a contact angle measurement, 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 [19]. It is noted that the reason to form sessile drops from below the surface, rather than depositing from the top, was to ensure that measured contact angles were proper advancing contact angles. Depositing a drop from the top causes vibrations of the drop which may lead to contact angles intermediate between advancing and receding contact angles [20]. Apart from the complication of supplying the liquid, the strategy for contact angle measurement using ADSA is straightforward: The shape of the drop is optimized for the liquid surface tension, and the contact angle is determined as a geometrical parameter from the intersection of the drop profile with the solid. An accuracy of ± , i.e. the scatter on a single, given surface, for contact angle measurements was reached by ADSA on well-prepared solid surfaces [19, 21]. Despite the success of ADSA in contact angle research, the accuracy of contact angle measurement by ADSA, i.e. ± , is still not at a level to perform studies that require higher accuracy. As an example, the study of drop size dependent contact angles requires

20 6 at least an accuracy of ±0.1 [22]. The current accuracy of contact angle measurement by ADSA also does not allow to investigate whether the reproducibility of contact angle measurements reached by ADSA implies the real physical reproducibility of solid surfaces. This limitation prevents investigation of very small changes in contact angles due to possible differences between the structure of different solid samples or the microstructure of solid surfaces. In spite of advantages compared to the experimental setups of the Wilhelmy method and the capillary rise method, the experimental setup of ADSA is not convenient, because it involves drilling a hole into the solid through which the liquid can be injected, and perfect sealing of the hole to prevent leakage [21]. The alternative and preferable constellation for contact angle measurement is forming sessile drops with a capillary protruding into the drop (Figure 1.2). This experimental setup is much easier to handle than the setup used in ADSA. However, ADSA cannot analyze such drop shape configurations with no apex. An example of a situation where a capillary in contact with the liquid is necessary is the study of contact angles of floating lenses [23]. Such a study 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.

21 7 1.2 Purpose and Scope of the Thesis The main purpose of this thesis is to establish a new methodology of contact angle measurement by developing 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 of the drop. Thus, ADSA-NA facilitates contact angle measurements on drops with a capillary protruding into the drop (Figure 1.2). This experimental setup of ADSA-NA is much simpler than the setup used in the original ADSA, i.e. formation of a complete drop from below through a hole in the test surface. ADSA-NA extends the applicability of the ADSA methodology to a wider range of liquid-fluid interfaces, for instance, liquid bridges [24, 25], as they occur in oil recovery [26, 27]. Since there is no geometrical information available from the apex region, a new reference point other than the apex has to be defined in ADSA-NA. For the case of sessile drops, the reference level can be positioned at the solid surface. Therefore, a new set of optimization parameters different from those of ADSA has to be formulated. The main difference between the proposed new optimization scheme and ADSA is two new additional optimization parameters, the horizontal coordinate of the profile and its inclination at the reference level; thus for the case of sessile drops, contact angle and contact radius become the new optimization parameters. Consequently, a new set of partial differential equations of the objective function with respect to the new optimization parameters has to be derived. Development of ADSA-NA also requires modifying the three main building blocks of ADSA; the image analysis part, the numerical integration of the Laplace

22 8 equation for generating theoretical curves, and the optimization procedure. It is noted that MATLAB was chosen as the programming environment for ADSA-NA. In this thesis, ADSA-NA was tested extensively for measuring contact angles of different liquids on smooth and homogeneous thin films of a commercial fluoropolymer, Teflon AF To investigate the effects of different numerical strategies and experimental setups on the accuracy of contact angle measurement, the images acquired in the experiments were also analyzed by a different drop shape technique called Theoretical Image Fitting Analysis-Axisymmetric Interfaces (TIFA-AI) [28], and the results were compared with literature values obtained by means of the standard ADSA for sessile drops with the apex. Early work in this thesis showed that the accuracy of the surface tension, i.e. the deviation from the true value, obtained by ADSA-NA was improved significantly compared to that obtained from sessile drops using the standard ADSA (see Chapter 4), although the accuracy of pendant drops could not be reached. To further improve the accuracy of contact angle and surface tension measurement, two main aspects of ADSA-NA were scrutinized in depth, well beyond the level of scrutiny in the original ADSA. First, the computational aspects of ADSA-NA were explored for which the results obtained from one and the same drop image were compared using different edge detection strategies and different optimization methods. After optimizing the computational aspects, the experimental geometry of ADSA-NA was studied, i.e. the effect of controllable experimental factors such as drop height and drop volume on the accuracy of surface tension measurement.

23 9 The overall organization of this thesis is as follows: In chapter 2, the Laplace equation of capillarity will be discussed, and a review of drop shape techniques will be presented. A detailed description of the ADSA-NA methodology will be given in Chapter 3. This chapter provides extensive information on the three main building blocks of ADSA-NA: the image analysis and edge detection, the numerical integration of the Laplacian equation for generating theoretical curves, and the optimization procedure. It is noted that these three main components are common blocks of all versions of ADSA methodology. In Chapter 4, the experimental procedure of ADSA-NA and the results of contact angle measurements for four different alkanes will be illustrated. The effect of refining solid surface location and the effects of optimization parameters on the accuracy of contact angle measurement will also be discussed in this chapter. The effect of different edge detectors and different optimization methods on the accuracy of contact angle measurement will be investigated in Chapter 5. In Chapter 6, the effect of different experimental parameters such as drop height and drop volume on the accuracy of surface tension measurement will be discussed. The summary of this thesis and subjects proposed for future work will be presented in Chapter 7.

24 10 Vapour lv sv sl Liquid Solid Figure 1.1: Schematic of the contact angle (θ). γ lv, γ sv, and γ sl are the liquid-vapour interfacial tension, the solid-vapour interfacial tension, and the solid-liquid interfacial tension, respectively.

25 Figure 1.2: 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. 11

26 Chapter 2 Overview of Drop Shape Techniques Mathematically, the balance between surface tension and external forces, such as gravity, is governed by the Laplace equation [29, 30]. This equation relates the pressure difference across an liquid-fluid interface to the curvature of the interface and surface tension: γ( 1 R R 2 ) = P (2.1) where γ is the liquid-fluid interfacial tension, R 1 and R 2 are the two principal radii of curvature, and P is the pressure difference across the interface. In the absence of external forces other than gravity, the pressure difference is a linear function of the elevation: P = P 0 + ( ρ)gz (2.2) where P 0 is the pressure difference at a reference plane, ρ is the density difference 12

27 13 between the two bulk phases, g is the gravitational acceleration, and z is the vertical height of the drop measured from the reference level. Thus, for a given value of γ, the shape of a drop may be determined for known values of physical properties, i.e. density difference and gravity, and known geometrical quantities, i.e. R 1 and R 2. The inverse, i.e. determination of the interfacial tension γ, from the shape, is also possible in principle; however this is a much more difficult task. Mathematically, the integration of the Laplace equation (Equation 2.1) is straightforward only for cylindrical menisci, i.e. menisci for which one of the principal radii of curvatures is zero. For a general irregular meniscus, mathematical integration would be very difficult. For the specific case of axisymmetric drops, e.g. sessile drops and pendant drops, numerical procedures have been devised [14]. Fortunately, obtaining axial symmetry is not difficult for most sessile drop and pendant drop systems. Drop shape techniques have been developed to measure the interfacial tension of liquids from the shape of pendant drops or sessile drops or bubbles [31, 32]. Typically, such methods rely on matching theoretical profiles calculated from the numerical integration of the Laplace equation (Equation 3.3) to the measured shape of the drop. For sessile drops, once the surface tension and the principal radii of curvature of the drop have been calculated using these methods, the Laplace equation can be integrated to compute the contact angle. As mentioned in Chapter 1, the earliest work in the analysis of axisymmetric drops was that of Bashforth and Adams [17] who generated sessile drop profiles for different surface tension and radius of curvatures at the apex of the drop. The task of determining the

28 14 surface tension and contact angle from the actual profile became a matter of interpolation from their tables. Blaisdell [33] and Tawde and Parvatikar [34] extended Bashforth and Adams tables, and Fordham [35] and Mills [36] generated equivalent tables for pendant drops. Hartland and Hartley [37] presented numerous solutions in tabulated form for determining the interfacial tensions of different axisymmetric drop shapes. To integrate the appropriate form of the Laplace equation, a FORTRAN computer program was used. The major drawback in their method is the data acquisition. Only a few preselected points are measured to describe the drop interface. The location of these points is critical and must be determined with high accuracy since these points correspond to special features, such as inflection points on the interface. The other disadvantage of the generated tables is that they are applicable only to drops of a certain size and shape range. Malcolm and Paynter [38] proposed an analytic method for the determination of contact angle and surface tension from sessile drop configurations. However, the data points are specific points on the drop interface, i.e. drop height and drop equatorial diameter, and the method is also limited to sessile drops with the contact angle greater than 90. Maze and Burnet [39, 40] developed a more accurate method for the measurement of interfacial tensions from the shape of sessile drops. Their method was reviewed by Hoorfar and Neumann [14]. In essence, the method consists of a non-linear regression procedure in which a calculated Laplacian curve is fitted to a number of arbitrarily selected coordinate points on the drop profile, and the best fit is obtained by optimizing

29 15 two parameters. The advantage of this method is that the measured drop profile is described by a set of coordinate points with no particular significance assigned to any of the points. To start the fitting procedure, reasonable estimates of drop shape and size are required; otherwise the calculated Laplacian curve will not converge to the measured drop profile. The initial estimates are obtained, indirectly, using values from the tables of Bashforth and Adams [17]. Despite improvement in strategy, there are several drawbacks in this method. The error function, i.e. the difference between the theoretical profile and the experimental profile, is defined as the sum of the squares of the horizontal distance between the theoretical curve and the experimental drop profile. This definition is not particularly appropriate for sessile drops whose shapes are strongly influenced by gravity. In fact, large drops of low surface tension tend to flatten near the apex, and any data point near the apex region may cause a large error, even though such points may be located close to the best-fitting theoretical curve. The other problem in the method is that the predetermination of the apex of the drop is crucial since it acts as the origin of the calculated curves. Maze and Burnet [40] realized that locating the origin is an important task and modified their original program, but only the vertical displacement of the origin was adjusted in this work and not the horizontal displacement. Huh and Reed [41] developed a similar method to the work of Maze and Burnet [40]. However, the objective function was defined by a weak approximation of the normal distance, and the apex point must still be predetermined by the user. Furthermore, their method is applicable only to sessile drops with contact angles greater than 90. Major credit for introducing image processing techniques to drop shape methods has

30 16 to go to Girault et al. [42] who used a video image profile digitizer for the drop profile acquisition. However, their method is limited to pendant drops, and it requires the determination of the inflexion plane and the volume of the drop. Anastasiadis et al. [43] developed a method that combines digital image processing with robust shape comparison routines [44, 45]. Hoorfar and Neumann [14] reviewed the method. In essence, the robust shape comparison routine used in this method compares vectors or line segments on the experimental profile with the corresponding vectors on the theoretical profiles, instead of comparing individual points on the experimental profile with the corresponding points on the theoretical curve, as it is compared in the least square methods. In general, it is recognized that outliers, i.e. erroneous data points of an experimental drop profile, can have a large impact on the results of the least squares methods. In fact, the outliers pull the least square fits toward them too much. Accordingly, robust methods have been created to modify least squares methods so that the outliers have less influence on the final estimates. However, if the fitting residuals (the distances between the experimental points with the corresponding points on the theoretical curve) have a Gaussian shape, which is the case in many experiments [14, 43], robust methods are less appropriate than the least squares methods [46]. It is noted that using appropriate hardware and lighting conditions in the experiments to obtain high quality images in conjunction with powerful edge detection techniques dramatically decreases the chance of introducing outliers [14]. Anastasiadis et. al. claimed that their method requires less computing power, and it is resistant to the presence of outliers. The major drawback in their technique is that it requires the determination of a reference

31 17 point on the profile to which the position of all other points on the profile is related. The reference point can be either the drop apex or the center of the drop, defined as the intersection of the vertical axis of the symmetry and the horizontal maximum diameter. Thus, the accuracy of the results depends not only on the accuracy of the drop profile coordinates but also on the accuracy of the determination of the reference point. The other problem of the method is the high sensitivity to the initial value of the optimization parameter because of using an exhaustive, i.e. direct, search for finding the best fit which requires a very accurate initial value of the optimization parameter. Rotenberg at al. [47] developed the first generation of Axisymmetric Drop Shape Analysis (ADSA), which fits the experimental drop profile to a Laplacian curve using a nonlinear regression procedure. ADSA predates the work of Anastasiadis et al. To measure the discrepancy between experimental drop profile and the theoretical Laplacian curve, an objective function is defined as the sum of the squares of the normal distances between the measured points on the drop profile and the theoretical curve, giving the same weight to every measured point. The remarkable feature in ADSA which distinguishes this method from the previous drop shape methods is that the location of the apex of the drop is assumed to be unknown, and the coordinates of the apex are regarded as independent variables of the objective function, i.e. optimization parameters. Thus, the drop shape can be measured from any convenient reference frame, and any measured point on the drop profile is equally important. The first generation of ADSA uses the incrementally loaded Newton-Raphson method to minimize the objective function. No table is required for ADSA, and no particular initial values are required for any of the

32 18 optimization parameters, i.e. the surface tension, the radius of curvature at the apex, and the coordinates of the apex. This first generation of ADSA was written in FORTRAN and it can analyze both sessile and pendant drops. Jennings and Pallas [48] developed another techniques which was built upon ADSA. The technique uses a rotational discrimination method instead of the incrementally loaded Newton-Raphson method to minimize the objective function. There are some problems with their technique. The objective function employed is continuous but not the first derivative, violating the truncation convergence requirement as discussed in their technique. The other problem is the input requirement of their program, which must be supplied with initial guesses for the parameters of angle of rotation, aspect ratio of the digitizing board, and the surface tension value. These initial guesses may not be easily obtained in some cases. Skinner et al. [49], Moy et al. [50] and Alvarez et al. [51] developed a modified version of ADSA called ADSA-Diameter (ADSA-D) in which the contact angle of a sessile drop is calculated from its contact diameter or maximum diameter by viewing the drop from above. The ADSA-D algorithm was specifically designed for the measurement of low contact angles on non-ideal and/or hydrophilic surfaces (e.g. biological surfaces). The required input data for ADSA-D are the liquid surface tension, density difference across the liquid-fluid interface, drop volume, the contact diameter or the equatorial diameter of the drop, and the gravitational acceleration. The simplicity and accuracy of the first generation of ADSA was improved by Cheng et al. [52] who implemented computer-based image processing techniques to extract the

33 19 drop interface automatically. An automated edge detection technique in conjunction with sub-pixel resolution and optical distortion correction techniques were incorporated into the ADSA program that improved considerably the accuracy of results and the efficiency of the first generation of ADSA. Cheng et al. [52 54] evaluated the performance of the first generation of ADSA for both pendant and sessile drops using synthetic drops. The randomness of the data as input to ADSA was evaluated. The points at five different locations of the profile were individually perturbed to test the influence of each location on the results. It was found that data points near the neck of a pendant drop or near the liquid-solid interface for a sessile drop have more impact on the results than points from other locations. The first generation of ADSA was found to give very accurate results except for very large and flat sessile drops, where the program failed [53]. 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. Cheng [54] also pointed out that it is difficult to achieve perfect alignment of the camera with a plumb line; there are errors associated with the coordinates of the plumb line defined manually on the screen of the computer using a mouse. To overcome the deficiencies of the numerical schemes of that first generation, a second generation of ADSA was developed by del Rio and Neumann [55, 56] 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 problem of very large and flat sessile drops. In this version, the tilt angle of the camera was also

34 20 considered as an optimization parameter to correct the vertical alignment of the camera. Yu et al. [57] introduced ADSA-Constrained Sessile Drop (ADSA-CSD), which is suitable for very low surface tension measurement. Constrained sessile drop is a novel drop configuration consisting of a sessile drop on a pedestal with a sharp knife-edge to prevent spreading or, in the case of insoluble monolayers in lung surfactant studies, film leakage. The drop is formed by pushing the liquid through a hole inside the pedestal, just as in ADSA contact angle measurements. ADSA-CSD has been extensively used in lung surfactant studies [58, 59]. Zuo et al. [60, 61] developed a modified version of ADSA called ADSA-Captive Bubble (ADSA-CB) to measure the surface tension of a captive bubble configuration. ADSA- CB facilitates pulmonary surfactant related studies. Since in captive bubble experiments for lung surfactant studies the images are usually fuzzy or noisy due to opacity of the surfactant suspension, a sophisticated image analysis technique in conjunction with the Canny edge detection [62] was implemented in ADSA-CB to remove noise from the image. The standard version of ADSA assumes that gravity is the only external force acting on the drop. To extend the applicability of ADSA for measuring the interfacial properties of liquids in an electric field, a new version of ADSA called ADSA-Electric Fields (ADSA- EF) was developed by Bateni et al. [63, 64]. ADSA-EF employs the sessile drop and constrained sessile drop configurations. Despite the same structure as ADSA, ADSA-EF is more complex, due to the presence of an electric field. A new module was incorporated in ADSA-EF to calculate the electric field distribution along the drop surface. The module for generating the theoretical profiles also had to be changed significantly in

35 21 ADSA-EF to account for the effect of the electric field. Drop shape methods including ADSA cannot measure accurately the surface tension of a nearly spherical drop. To investigate the source of this limitation, Hoorfar et. al [14] systematically scrutinized the entire ADSA technique including hardware and software. Since the original ADSA was a firm fixed package, it was modularized by Hoorfar et al. to facilitate implementing alternative numerical techniques for different experimental situations, e.g. poor images. A criterion called shape parameter was also introduced by Hoorfar et. al to provide a priori knowledge of the accuracy of surface tension measurement by ADSA without using the ADSA calculations. The shape parameter was defined as a dimensionless parameter that quantifies the difference in shape between a given experimental drop profile and an inscribed circle with a radius equal to the radius of curvature at the apex of the drop. A critical shape parameter, i.e. a value of a shape parameter below which ADSA will give the value of surface tension less accurate than an expected tolerance, was determined experimentally. In summary, what we call ADSA has two roots with respect to the structure and the strategy. With respect to the structure, ADSA consists of three main modules. The first module is the image analysis in which the drop profile coordinates (i.e. the experimental profile) are obtained from the image of the drop. The second module generates the theoretical profiles by numerical integration of the Laplace equation for known values of surface tension, density, gravity, and the curvature of the apex of the drop. The last module is the optimization procedure to find the best fit of the theoretical Laplacian curves to the experimental profile. The best fit identifies liquid-fluid interfacial tension,

36 22 contact angle (in the case of sessile drops), drop volume, surface area, radius of curvature at the apex, and the radius of the contact circle between the liquid and solid (in the case of sessile drops). The flowchart presented in Figure 2.1 shows the general procedure of ADSA for the determination of the interfacial properties from the shape of a drop or a bubble. With respect to the strategy, ADSA requires only those inputs that are physically necessary, i.e. the drop image, density difference, and gravity, with arbitrary selection of the experimental drop profile points instead of relying on specific profile points. Several different drop shape methods [65 73] have been developed after ADSA in most of which the same structure as ADSA has been implemented. Recently, a new drop shape technique called Theoretical Image Fitting Analysis (TIFA) [74, 75] was introduced that has a different structure and approach than ADSA. As mentioned, 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. However, 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. More precisely, the gradient of the experimental image, rather than the raw image, is employed in TIFA to minimize the effect of contrast and lighting conditions. An error function measures the pixel-by-pixel difference between the gradient of the whole theoretical and experimental images. The interfacial properties are found by fitting the gradient of the theoretical image to the gradient of the experimental image by minimizing the error function. The remarkable feature of TIFA is that it operates without using edge detection

37 23 algorithms. In fact, image analysis is tied to the optimization process in TIFA, and it is not a separate module as in ADSA. Although it is conceptually attractive to have a uniform package, this makes TIFA less flexible than ADSA. In some circumstances, e.g. noisy images, the image analysis needs to be modified. An example is captive bubble experiments as used in lung surfactant research, where the images are noisy due to inhomogeneities of the surfactant suspension. Analyzing such images requires a sophisticated noise removal module using edge detection techniques [60]. Finally, to extend the applicability of TIFA to drop configurations with no apex, such as liquid bridges [24, 25], a different version of TIFA, called TIFA-AI (Theoretical Image Fitting Analysis - Axisymmetric Interfaces) was developed [28, 76]. 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. TIFA-AI was originally developed for surface tension measurement. Recently, TIFA-AI was used to measure the size dependence of contact angles in sessile drop experiments [22]. In that study, a syringe needle was mounted vertically with its tip less than 1 mm above the solid surface (Figure 1.2) 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 contact angle of 0.05 within a single run and typical run-to-run scatter of 0.2 was reached in that study.

38 24 Physical Properties (Density and Gravity) Image Acquisition Image Analysis Generating Theoretical Laplacian Curves Optimization Procedure Surface Tension, Contact Angle, Surface Area, Drop Volume Figure 2.1: General procedure of Axisymmetric Drop Shape Analysis (ADSA)

39 Chapter 3 Axisymmetric Drop Shape Analysis-No Apex (ADSA-NA) 3.1 Introduction It was indicated in the previous chapter that many drop shape techniques developed in the last two decades follow the structure of ADSA. ADSA-NA has also the same structure as ADSA (Figure 2.1). However, to analyze drop shapes with no apex, the three main modules of the original ADSA all have to be modified. Because of the presence of a capillary in the image of the drop (Figure 1.2), the image analysis procedure has to be modified to only extract the experimental profile of the drop, not the capillary. Since a new reference point other than the apex has to be defined, the boundary conditions and input parameters for numerical integration of the Laplace equation has to Part of this chapter was previously published [77] and reproduced with permission. 25

40 26 be changed. Introducing a new reference point other than the apex requires a new set of optimization parameters different from those of ADSA and hence modification of the whole optimization procedure. It is noted that all the components of ADSA-NA were written in MATLAB. In this chapter, a typical strategy of the ADSA-NA components is presented. A more complete account of the ADSA-NA components and further alternatives for each part will be discussed in Chapter Image Analysis 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 1.2). The gray level for every pixel in the image is ranging 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 sub-pixel resolution Drop Profile Detection with Pixel Resolution In ADSA, originally the Sobel edge detector was implemented to detect the drop profile to pixel resolution. Recently, a more robust edge detector called Canny [62] was implemented in a new version of ADSA called ADSA-CB (ADSA for Captive Bubble) [60]. Canny is a gradient operator, and it uses a filter based on the first derivative of a Gaussian function defined as:

41 27 G(X) = X 1 2 2σ e G 2, G(Z) = 2πσG Z 1 2 2σ e G 2 (3.1) 2πσG where G X and G Z are the Gaussian function values in the horizontal and vertical directions, and σ G is called the standard deviation of the Gaussian filter, which is a userspecified parameter. The Canny algorithm consists of three main stages. In the first stage called smoothing, the Gaussian filter is applied to the image for noise reduction. The result is a slightly blurred version of the original image with the degree of blurring being determined by the value of σ G. The next stage is the non-maximum suppression in which a search is carried out to determine whether a pixel with a finite, i.e. non zero, gradient magnitude is a local maximum, and the edge is thinned down to only one pixel width. If the gradient magnitude at a pixel is larger than the gradients of its neighbors in the gradient direction, the pixel is selected as an edge. Otherwise the pixel is marked as background. In the last stage called hysteresis thresholding, two thresholds, which are user-specified parameters for the Canny, are applied to obtain an accurate and continuous edge. In the hysteresis thresholding, any pixel with a gradient magnitude above the high threshold (T h ) is marked as a genuine edge pixel, and any pixel with a gradient magnitude less than the low threshold (T l ) is marked not to be an edge pixel. Any pixel with a gradient magnitude greater than (T l ) and adjacent to edge pixels is also selected as an edge pixel. It was shown [60] that the Canny edge detector is not sensitive to the selection of the user-specified parameters, i.e. σ G, T l, and T h. As suggested in reference [60], the value

42 28 of σ G is assigned as 1.0 for a clean image. It was found that T h equal to the value of 0.2 and T l equal to one half of T h are adequate for most images. It was shown that the Canny edge detector is more resistant to noise in the image than Sobel [60]. Noise is the random variation in gray levels in an image [78] caused by uncertainty due to electronic devices, e.g. salt-and-pepper noise in signal transmission, and ambiguity due to poor focus. Here, the Canny edge detector, available as a built-in function in the MATLAB Image Processing Toolbox (The Mathworks), was used for the first stage of image processing, i.e. determination of the experimental profile to pixel resolution. The effect of different edge detectors on the results will be discussed and compared in Chapter 5. 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 one, and all other pixels are assigned a value of zero. Binary images are illustrated as black and white images in which edge pixels are shown as black and all other pixels are shown as white. Figure 3.1 shows the binary image of a dodecane drop on a Teflon AF 1600 coated surface (Figure 1.2) after applying the Canny edge detector to the original gray scale image. 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 3.1). A specific procedure was implemented in ADSA-NA in order to extract just the pixels that belong to the drop profile as follows:

43 29 The binary image is searched from the top along the rows where it is obvious that edge pixels represent the capillary, and the two edge pixels with the maximum and minimum X coordinate in each row are selected as the capillary profile. The distance between these two pixels is the capillary diameter at each row. Once the difference between the calculated capillary diameter for a row and the calculated diameter for the first 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. The search stops at the contact points at the solid-liquid interface. To determine the location of the contact points in the binary image in this stage, a preliminary procedure is devised that only aims to set stop points for the search at the lower boundaries of the drop profile. It is noted that a module was implemented in ADSA-NA to detect the contact points at the solid-liquid interface more accurately which will be discussed in Section To identify the contact points at the solid-liquid interface automatically, the program requires the user to specify whether the contact angle is less than or greater than 90. Determination of the contact points relies on comparing the X coordinates of successive edge pixels along the drop profile. For the case of a contact angle less than 90, edge pixels on the right hand side of the drop whose X coordinates are either equal to or greater than that of the preceding edge pixel are identified as drop profile coordinates. If an edge pixel (i + 1) is reached whose X coordinate (X i+1 ) is less than that of the preceding edge pixel (X i ), the search stops and the edge pixel with the largest X coordinate is selected as the

44 30 right contact point of the sessile drop with the solid (Figure 3.2). A similar algorithm is applied to edge pixels on the left hand side of the drop to find the left contact point. In this case, the X coordinate of an edge pixel (X i+1 ) must not be greater than that of the preceding one (X i ); otherwise, the preceding edge pixel is taken as the left contact point. For the case where the contact angle is greater than 90, during the search on the right hand side of the drop, the X coordinate of the searched edge pixel first increases (X i+1 > X i ) until it reaches the edge pixel which is at the maximum width of the drop on the right side. Then, the X coordinate of the searched edge pixel decreases (X i+1 < X i ) until it reaches the contact point on the right side of the drop, where the X coordinate of the searched edge pixel increases (X i+1 > X i ) again due to the reflection of the drop in the solid. The second increase in the X coordinate of the searched edge pixel means that the search has reached the contact point on the right side of the drop at the solid-liquid interface (Figure 3.2). To find the contact point at the left hand side, a similar strategy is used. In this case, the X coordinate of the searched edge pixel first decreases until it reaches the edge pixel which is at the maximum width of the drop on the left side. Then, the X coordinate of the searched edge pixel increases until it reaches the contact point on the left side of the drop where the X coordinate of the searched edge pixel decreases again due to the reflection of the drop in the solid. The second decrease in the X coordinate of the searched edge pixel means that the search has reached the contact point on the left side of the drop at the solid-liquid interface. In the above strategy, it is assumed that only one pixel represents the contact point. However, it often happens that several pixels with the same X coordinate exist in the

45 31 contact point region of the drop profile (circled area in Figure 3.1). In fact, typical edge detectors, such as Canny, do not detect these contact points and their reflections as a sharp corner. Instead, the physically sharp corner appears as several pixels aligned vertically. The strategy to resolve this issue is to choose the upper pixel of the vertically aligned pixels as the preliminary contact point. In the case where the solid surface has no reflection of the drop, the program allows the user to specify the two contact points on the left and right sides of the drop. In this case, the search for the contact point stops when an edge pixel is reached along the extracted drop profile whose Z coordinate is the same to that of the manually determined contact point Drop Profile Detection with Sub-Pixel Resolution In the second stage, the drop profile coordinates are improved to sub-pixel resolution using the original gray-scale image of the drop [52, 79]. Two different methods were implemented. In the first method, which is the same method used in the original ADSA, for every pixel of the drop profile, a natural cubic spline is fitted to a gray level profile perpendicular to the drop interface [52]. Figure 3.3 shows a typical example of a gray level profile perpendicular to the drop interface fitted with a natural cubic spline curve. The end condition for the natural cubic spline fit is that the second derivative equals to zero at each end, since the gray level profile across the drop profile should approach a plateau at each end. The point whose gray level is the mid-point of the high and the low plateaus of the fitted curve is selected as the new location of the drop interface.

46 32 In the second method, for every pixel of the drop profile, a sigmoid function (equation 3.2) is fitted to a gray level profile perpendicular to the drop interface [79]. g(x) = g 1 g exp((x x 0 )/W ) + g 2 (3.2) where x is the direction normal to the drop interface, g 1 and g 2 correspond to the plateau gray levels at the two sides of the edge (Figure 3.3), W is the edge width, and x 0 is the point whose gray level is the mid-point of the high and the low plateaus. The parameters g 1, g 2, x 0, and W are the fitting parameters. The parameter x 0 is considered as the new location of the drop interface. It is noted that it is generally not possible to fit a cubic spline curve or sigmoid function exactly perpendicular to the drop interface using the gray level values of the digitized drop image. The fitting can only be performed in horizontal, vertical, and diagonal directions. Therefore, the direction closest to the perpendicular to the profile is chosen from the three main directions for each drop profile point, and the fitting for the gray levels is performed in that preferred direction. 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 so that the coordinates of the experimental profile are given in terms of length (e.g. centimeters) [52, 80]. The calibration grid is a grid pattern on a 5 5 cm 2 optical glass mounted on a metal frame. The grid image is acquired in the experiment. Figure 3.4 shows a typical image of a grid that consists of small cells. The size of the squares

47 33 of the grid is mm 2. The grid is used to find a function that maps any pixel of the experimental image to the corresponding coordinate point on the actual grid. This method is the same method used in the original ADSA. A significant advantage of this method is that the aspect ratio of the digitizing board does not have to be known [52]. Detailed information can be found elsewhere [52] Solid Surface Location 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 solid-liquid interface is usually fuzzy, and it is hard to detect the corner points by direct inspection. Even a robust non-gradient corner detector, i.e. SUSAN [81], does not give satisfactory results, i.e. it detects several pixels as the corner point, instead of one, sometimes with none of these pixels near the true corner, and approaches such as a B-spline snake based method [82] cannot be applied to drops with no apex because it assumes a complete sessile sessile drop with the apex for detecting the contact points. Typical edge detectors, such as Canny, 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 3.1). In the standard ADSA, the vertical level of the solid surface was determined prior to the implementation of sub-pixel resolution. The upper pixel was selected as the contact point in the case of two vertically aligned pixels and the middle one in the case of three such pixels. This strategy has an accuracy of ±1 pixel, but it can

48 34 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 in ADSA-NA to detect the contact points more accurately. Two different strategies were implemented. In the first strategy, called the profile strategy, 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 rationale behind the selection of 10 points is as follows. It was shown [83] that the optimal number of pixels for a linear fit to the drop profile close to the solid surface was 30. This is equivalent to 10 pixels in the present study, since the pixel size here is three times larger than the pixel size in Ref. [83], where high magnification images of the drop corner was used. Figure 3.5(a) also suggests that minor changes in the number of points considered will not affect the location of point A significantly. The horizontal coordinate of the contact point is found as the intersection of the fitted line and the calculated solid level (Figure 3.5(a)). In the second strategy, called the reflection strategy, 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 10 points of 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 3.5(b)). The accuracy of the reflection strategy depends on the quality of the reflection of the drop profile, which is sometimes fuzzy (Figure 1.2).

49 35 The vertical coordinates of the points at the solid-liquid interface (or solid level) is 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 taken as input, i.e. manually detected corner points may have to be used. 3.3 Generating Theoretical Laplacian Curves As described in Chapter 2, the Laplace equation describes the mechanical equilibrium conditions for two homogeneous fluids separated by an interface: γ( 1 R R 2 ) = P (3.3) where γ is the liquid-fluid interfacial tension, R 1 and R 2 are the two principal radii of curvature, and P is the pressure difference across the interface. In the absence of external forces other than gravity, the pressure difference is a linear function of the elevation: P = P 0 + ( ρ)gz (3.4) where P 0 is the pressure difference at a reference plane, ρ is the density difference

50 36 between the two bulk phases, g is the gravitational acceleration, and z is the vertical height of the drop measured from the reference plane. For an axisymmetric liquid-fluid interface (Figure 3.6), the principal radius of curvature R 1 is related to the arc length and the angle of inclination of the interface to the horizontal by [84]: 1 = dφ R 1 ds (3.5) where φ is the angle of inclination of the interface to the horizontal (x direction), and s is the arc length along the interface. The second radius of curvature is given by: 1 = sin φ R 2 x (3.6) In the case of a sessile drop, the reference plane can be positioned at the solid-liquid interface. Therefore, from Equation 3.3, the pressure difference at the reference plane (z = 0) can be expressed as: P 0 = γ( 1 R R 2 ) = 2bγ (3.7) where b is the mean curvature of the interface at the reference plane (z = 0). Substituting Equations into Equation 3.3 yields: dφ ds = 2b + cz sinφ x (3.8)

51 37 c = ( ρ)g γ (3.9) where c is the capillary constant. Analyzing the drop shape without the apex (Figure 3.6) introduces two new non-zero boundary conditions at the reference level for generating the Laplacian curves: at s = 0 : φ = θ 0, x = R 0, z = 0 (3.10) where R 0 is the radius of the interface at the reference level, and θ 0 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 θ 0 is the contact angle [28]. 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. Equation 3.8 together with the geometrical relations dx ds = cos φ (3.11) dz ds = sin φ (3.12) form a set of first order differential equations for x, z, and φ as functions of the arc length s.

52 38 Equations ( ) show that the shape of a drop interface without the apex depends on parameters c, b, R 0, and θ 0. Taking these parameters as inputs, the theoretical profile can be calculated by simultaneous numerical integration of the above set of equations. The numerical integration stops when the vertical coordinate, z, reaches the maximum vertical coordinate of the experimental profile. The method used for integration is a variable step size Runge-Kutta [85] which uses a 4 th and 5 th order pair for higher accuracy. It is noted in the original ADSA, a program called Axisymmetric Liquid-Fluid Interface (ALFI) was implemented to generate theoretical profiles for given values of capillary constant (c) and curvature at the apex [55, 56]. ALFI uses the Bulirsch-Stoer method [86, 87] with an automatic adaptive step size to integrate the Laplace equation. In this thesis, we call the program that generates theoretical profiles of drop configurations with no apex as ALFI-No Apex (ALFI-NA). 3.4 Optimization Procedure As mentioned, ADSA-NA fits a Laplacian curve to the extracted experimental profile of the drop. Therefore, it is required to determine the deviation of the drop profile from the shape dictated by the Laplace equation. The experimental profile points are compared with a calculated Laplacian curve by computing the normal distances between all the experimental points of the drop profile and the calculated Laplacian curve. The error for the ith point, e i, can be computed, assuming that the coordinate systems for the experimental profile and the calculated Laplacian profile coincide, as the square of the

53 39 minimum distance d i : e i = 1 2 d2 i = 1 2 [(x i X i ) 2 + (z i Z i ) 2 ] (3.13) Here (X i, Z i ) are the measured drop coordinates, and (x i, z i ) are the closest Laplacian coordinates to (X i, Z i ) (Figure 3.7). However, in general the two coordinate systems do not coincide, therefore their offset and rotation should be considered. Then, equation (3.13) can be written (eliminating the subscript i) as: e = 1 2 [(e x) 2 + (e z ) 2 ] (3.14) e x = x i x 0 X i cosα + Z i sinα (3.15) e z = z i z 0 X i sinα Z i cosα (3.16) where (x 0, z 0 ) is the offset between the coordinate systems (experimental vs. theoretical) and α is the rotation angle, i.e. the angle between the experimental drop profile and the theoretical Laplacian coordinate systems (Figure 3.7). However, to evaluate these equations for every experimental point, it is necessary to find the closest point on the theoretical Laplacian curve. The distance between any experimental point and the theoretical curve is a function of the arc length s, and its minimum or normal distance can be written as:

54 40 de ds = f(s) = 0 (3.17) where derivatives of Equations are taken for constant values of the parameters b, c, x 0, z 0, and α: de ds = e de x x ds + e de z z ds (3.18) de x ds = dx ds = cos φ (3.19) de z ds = dz ds = sin φ (3.20) Equation 3.17 must be solved numerically for s. The most efficient method to do so is the iterative Newton-Raphson method: s i+1 = s i f(si ) f (s i ) (3.21) where for f (s i ), the second derivatives of Equations with respect to s are required: f (s) = d2 e ds = e d 2 e x 2 x ds + (de x d 2 e z 2 ds )2 + e z ds 2 + (de z ds )2 (3.22)

55 41 d 2 e x ds 2 = d2 x = sin φdφ ds2 ds (3.23) d 2 e z ds 2 = d2 z = cos φdφ ds2 ds (3.24) where dφ ds can be obtained from Equation 3.8. Newton-Raphson must be initialized with a first guess for s to ensure convergence to a minimum. A value of s corresponding to a theoretical point (calculated from the integration of the Laplace equation) with the shortest distance to the experimental point is chosen as the initial guess for s. Then, the derivatives of Equations are evaluated, and a new value of s is obtained using Equation The Laplace equation is then integrated to the new value of s. Iteration is continued until convergence is achieved. The final outcome is the closest point on the theoretical curve to the experimental point. The above procedure is then repeated for all the experimental points. As in the standard ADSA [84], to measure the agreement between the experimental profile of the drop and a calculated Laplacian curve, the objective function is defined as the sum of the weighted individual errors: E = N w i e i (3.25) 1 where e i can be evaluated from Equation 3.14 and w i is a weighting factor to account for the influence of the location of the ith point on the fitted curve. Since there is no evidence that some points have more influence on the fitted curve, w i is chosen here to

56 42 be 1.0. 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, R 0, θ 0 }, and the parameters determining the position of a Laplacian curve with respect to the experimental profile {x 0, z 0, α}. However, the parameters θ 0, R 0, and z 0 are not independent. In fact, the same Laplacian curves can be generated using different starting points, i.e. different θ 0, R 0, and z 0. In addition, the location of the contact points or the solid level (represented by z 0 ) is physically fixed. Therefore the value of z 0 is constant and should not be optimized. The parameter z 0 is determined as the vertical coordinate of the contact points at the solid-liquid interface discussed in Section Fixing the value of z 0 leads to overall six optimization parameters in ADSA-NA. It is noted that the optimization parameters in ADSA-NA are the same as those of TIFA-AI [28, 76] except for the substitution of c for γ. 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. In the original ADSA, Levenberg-Marquardt and Newton-Raphson [55] 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 Levenberg- Marquardt. In ADSA-NA, the Nelder-Mead simplex method [88, 89] was used at the first attempt; it is a non-gradient optimization method. It is noted that the Nelder-

57 43 Mead simplex method was also used in TIFA and TIFA-AI. 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. It is noted that the two gradient optimization methods were also implemented in ADSA-NA which will be discussed and compared with the Nelder-Mead simplex method in Chapter 5. The Nelder-Mead algorithm uses the concept of a simplex which is a polytope, i.e. a geometric object with flat sides, of N + 1 vertices in N dimensions. Examples of a simplex are a line segment on a line, a triangle on a plane and a tetrahedron in three-dimensional space. The method finds the minimum of a function of N variables by generating a simplex of (N+1) vertices. The method is a pattern search that compares function values at the vertices of the simplex. The worst vertex, where the function value is the largest, is rejected and replaced with a new vertex and a new simplex is formed. The simplest step is to replace the worst vertex with a vertex reflected through the centroid, i.e. average, of the remaining N vertices. If this vertex is better than the best current vertex, i.e. the vertex with a minimum function value, then the method tries stretching exponentially out along the reflection line. On the other hand, if this new vertex is not better than its previous value, the method shrinks the simplex toward a better vertex. The process generates a sequence of simplices (which might have different shapes), for which the function values at the vertices get smaller and smaller. The size of the simplex is reduced and the coordinates of the minimum point are found. 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,

58 44 the initial values are estimated by geometrical analysis of the extracted experimental profile. In the particular case of a sessile drop, the parameters R 0 (Figure 3.6) and x 0 (Figure 3.7) are estimated by simple calculations using the coordinates of the contact points at the solid-liquid interface detected in the image processing. The parameters b and θ 0 (Figure 3.6) are estimated by fitting a third-order polynomial to the region close to the solid surface [83]. The curvature and the inclination of the polynomial at the contact points are considered as the initial values of b and θ 0, respectively. The parameter α (Figure 3.7) 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, R 0, θ 0 } and virtually not on the value of surface tension [28]. Therefore an improved estimate of parameters {b, R 0, θ 0 } 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, R 0, θ 0 } 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. 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

59 45 determines the Laplacian curve that best fits the given experimental profile, from which interfacial properties can be readily found.

60 Figure 3.1: The binary image corresponding to Figure 1.2 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. 46

61 47 Z Drop profile X i 1 X i Drop profile X i 1 X i X i 1 X i Reflection X i 1 X i Reflection X i 1 X i X Figure 3.2: Criteria for the identification of contact points at the solid-liquid interface for the cases of contact angle smaller and greater than 90

62 gray level A B pixel location Figure 3.3: A typical example of a gray level profile perpendicular to a drop interface (open circles). Point A is the pixel which 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 sub-pixel resolution. Student Version of MATLAB

63 Figure 3.4: An image of the calibration grid. The size of the squares is mm 2. 49

64 50 number of pixel points in vertical direction A number of pixel points in horizontal direction (a) number of pixel points in vertical direction B number of pixel points in horizontal direction Student Version of MATLAB (b) Figure 3.5: Strategies for determining the contact points at the solid-liquid interface. The profile strategy (a) detected the contact point (point A) as (877.1,669.5), and the reflection strategy (b) detected the contact point (point B) as (877.6,670.1). Student Version of MATLAB

65 51 z s θ 0 x R 0 Figure 3.6: 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

66 52 z Z z 0 x 0 x X Figure 3.7: Sketch of ADSA-NA for profile comparison: schematic of the experimental profile obtained by edge detection (solid dots in X-Z coordinate system) and the Laplacian profile (solid lines in x-z coordinate system); (x 0, z 0 ) is the offset between the coordinate systems, and α is the rotation angle.

67 Chapter 4 ADSA-NA Measurements 4.1 Introduction In this chapter, the experimental procedure for ADSA-NA is described and the results of contact angle and surface tension measurements by ADSA-NA are presented for four long-chained alkanes: decane, dodecane, tetradecane, and hexadecane. The measured contact angles are compared with literature values obtained by the original ADSA setup, i.e. forming drops from below the surface. Since the accuracy of contact angle measurement by TIFA-AI (see Chapter 2) 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 one and the same drop by means of ADSA-NA and TIFA-AI can answer this question Part of this chapter was previously published [77] and reproduced with permission. 53

68 54 since they handle one and 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. Therefore, the images acquired in the experiments were also analyzed by TIFA-AI, and the results are compared with the results of ADSA-NA. The images acquired are 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. 4.2 Experimental Procedure Contact Angle Experiments Figure 4.1 shows the schematic diagram of the experimental setup of ADSA-NA. The setup was mounted on a vibration isolation table (Model , Technical Manufacturing Corp.) which prevented relaxation of the contact angle to lower values [90]. The experimental liquid was loaded into a 1.0 ml syringe (Gastight, Hamilton Co.) that was mounted to a stepper motor (Model 18705, Oriel Corp., USA). The stepper motor was used to control the rate of advancing and receding of the drop front by controlling the rate of flow into and out of the drop. For a range in drop volume of 20 µl, the travel distance of the motor is 1000 steps. The available speed range of the motor is 0-500

69 55 steps per second. A drop of the experimental liquid was formed on the solid surface by allowing the motor to push liquid through a Teflon tube (with an outer diameter of 1.6 mm) and then down through a vertical stainless steel needle (Figure 1.2). The needle remained attached to the drop to allow the drop volume to be changed. The size of the needle was 0.3 mm ID and 0.6 mm OD. The solid surface was leveled using a bubble level. The drop was illuminated from behind by a white light projector through frosted glass. The heavily frosted diffuser provides a uniformly lit background and minimizes heat transfer to the drop during image acquisition. A CCD camera (Sony XCD-SX900) was used to capture the images with a horizontal microscope (Wild Heerbrugg ) at 5.8 magnification. The CCD camera provides an analog video signal of the drop that is digitized using a frame grabber installed in a host computer. The digitized image consists of a fixed number of pixels that determines the resolution of the image ( ). Each pixel specifies the intensity of light or gray level (in the black-and-white case) in a minute fractional area of the image. The gray level is registered using an 8-bit number. Therefore, it is defined in the interval where 0 and 255 correspond to black and white, respectively. Thus, a digitized image is mathematically represented by an array of real numbers from 0 to 255. The microscope and camera were mounted together on an axial translation stage for focusing, and the camera was leveled using a bubble level. The tilting angle of the camera was zeroed using a plumb line. Images were captured at a rate of one image every two seconds while the drop size was increased slowly (advancing contact angle) and then decreased (receding contact angle). The physical pixel size was

70 56 approximately 6.5 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. It is noted that tubing and glassware used in the experiments were cleaned by soaking in chromic acid for 24 hours. Syringes and needles were cleaned by sonication in ethanol (3 10 min) and distilled water (1 10 min) Materials and Solid Surfaces Preparation The experimental liquids are listed in Table 4.1. The surface tensions of the liquids were measured using pendant drop experiments analyzed by ADSA [14]. The errors given for the measured values of surface tension are 95% confidence limits. The last column in Table 4.1 shows the literature values of surface tension obtained from reference [91]. It is noted that no confidence limits for the values of surface tension are reported in reference [91]. The measured values of surface tension show good agreement with literature values (Table 4.1). It should be emphasized that it is not the aim of this thesis to produce literature values of surface tension since that would require highy purified liquids with accurate control of temperature during the experiment. In our experiments, temperature could change by ± 0.5 which corresponds to approximately ± 0.05 mj/m 2 change in the measured value of surface tension. The coating material was Poly[4,5-difluoro-2,2- bis(trifluoromethyl)-1,3-dioxoleco-tetrafluoroethyene], 65 mol% dioxole (Teflon AF 1600) that was purchased from DuPont Co. It is a fluoropolymer with a glass transition temperature of Tg=160 C [92]. Figure 4.2 shows the repeat unit of Teflon AF To prepare the polymeric solution for coating, Teflon AF 1600 which was originally

71 57 a 6% solution in Fluorinert FC-75 (3M) was further diluted with this solvent at a 1:1 volumetric ratio. Silicon Wafers <100> (Silicon Sense, Nashua, NH) were selected as the substrate because of their smoothness, rigidity, and high surface tension. The last property causes the polymeric solution to spread on the surface uniformly during the coating process. The surfaces were prepared by dip-coating. Briefly, a wafer of silicon was cut to smaller surfaces about 1 cm in diameter and soaked in chromic acid for 24 hours. Then, the surfaces were washed with distilled water and dried under a heating lamp. The cleaned substrates were immersed vertically into the coating solution at a slow speed of cm/s and withdrawn at the same speed so that the substrate was covered with a layer of the coating material. To further improve adhesion of the polymer to the substrate and to remove the solvent completely, the coated surfaces were annealed inside an oven above the glass transion temperature (Tg) of the polymer for 24 hours [21]. Heating to above Tg also relieves strains in the films and hence makes the sample more uniform [93]. It is noted that a polymer below the glass transition temperature is hard and brittle, like glass, and beyond that is elastic and flexible. The annealing tempearature was 165 C. The oven was then turned off, and the surfaces were cooled down gradually to ambient temperature. Using atomic force microscope (AFM), it was shown [21] that the polymer film is very smooth with root-mean-square (RMS) roughness of 0.4 nm and maximum peak-to-valley distances of 2.0 nm. It is known that roughness on this scale does not influence the contact angles significantly [94]. The thickness of the polymer film on dip-coated surfaces is 470 nm, determined from ellipsometry measurements [21].

72 Contact Angle 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 4.3 for a typical run with dodecane on Teflon AF Each run typically contained 300 images overall. The volume and surface area of the drop are also outputs of ADSA-NA. Figure 4.4 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. there was no more than a few pixels difference between the two sides of the drop. One of the input parameters in ADSA-NA and TIFA-AI is the tolerance value in the optimization process. The iterative optimization process stops when the change in every optimization parameter and the objective function between two consecutive iterations 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 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 The tolerance values larger than 10 4 were not tested because it is desired to have at least an accuracy of 10 4 cm for the contact radius to guarantee an error of less than the physical pixel size, i.e cm. Table 4.2 shows

73 59 the measured contact angles and the average computation time per image for different tolerance values. Based on the results shown in Table 4.2, the required tolerances for ADSA-NA and TIFA-AI are found to be 10 6 and 10 4, respectively. It can also be seen that the average computation time per image is significantly larger for TIFA-AI compared to ADSA-NA, and for both of them, the average computation time increases when decreasing the tolerance value Comparison of ADSA-NA and TIFA-AI with ADSA The results of ADSA-NA and TIFA-AI for advancing contact angle are shown in Figure 4.5 for a typical run with decane on Teflon AF Table 4.3 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 sub-pixel resolution methods discussed in Chapter 3: sigmoid function and spline function. Advancing contact angles were calculated for the largest drops (around 100 images), for which drop size dependence is insignificant. The last column of Table 4.3 shows the previously reported values [21] obtained by ADSA using regular sessile drops, i.e. sessile drops with the apex. The contact angle for each liquid in reference [21] 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 4.3 represent standard deviations for ADSA-NA and TIFA-AI, but 95% confidence limits for the data in reference [21]. It is noted that standard deviations are used throughout this thesis to illustrate the variability and deviation within each run, i.e. image-to-image scatter, and

74 60 95% confidence limits were used in reference [21] to show run-to-run scatter. Here, the accuracy of contact angle measurement is considered as the devation within each run. 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 Section 4.3.3). In contrast, surface tension was optimized in reference [21]. The contact angles obtained by means of ADSA-NA with the two sub-pixel resolution methods agree to better than 0.1 and show similar scatter, i.e. image-to-image standard deviation, for most sample runs, see Table 4.3. This similarity shows that 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. It should be emphasized that 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. However, in TIFA, the interfacial properties are calculated by fitting the gradient of the whole theoretical image, i.e. not the profile line, to the gradient of the experimental image of the drop. 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

75 61 to no worse than 0.2. 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 [21]. Sources for this improvement may include the non-optimization of capillary constant (c), higher accuracy for detecting the solid surface location, 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 [21]. 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 (see Figure 4.5). This suggests that the causes for the scatter reported in Table 4.3 for these two methods are physical, e.g. the micro structure of the solid substrate. To compare the quality of the solid surfaces used in this study and the previous studies [21, 95], 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 (Figure 4.6), i.e. the point where the motor had been reversed to shrink the drop and cause the drop front to retreat. Table 4.4 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 reference [21] and those used here.

76 Effect of Solid Surface Location As explained in Section 3.2.3, two different strategies (the profile and the reflection strategies) were implemented in ADSA-NA to detect the contact points at the solid-liquid interface automatically (Figure 3.5). In the original TIFA-AI [28], these contact points were selected manually by eye, with an estimated accuracy of ±1 pixel. To examine the influence of different strategies of locating the solid surface, the advancing contact angles of dodecane and tetradecane on Teflon AF 1600 were determined with the profile and the reflection strategies. The results are shown in Table 4.5. The contact angles obtained with the two strategies agree to within 0.1. The standard deviations, i.e. image-toimage scatter within each run, are substantially smaller than the differences between the contact angles calculated for each run with the two strategies, i.e. the differences between the results of the two strategies are statistically significant. Hence it is worthwhile to pursue the underlying reason for this discrepancy between the two strategies. In fact, the average solid level calculated by the profile strategy is lower than that calculated by the reflection strategy by about 2.2µm (0.38 pixel) for dodecane and 2.3µm (0.40 pixel) for tetradecane. The Profile strategy was preferred so far because it shows slightly lower scatter, i.e. smaller image-to-image standard deviation. However, the reflection strategy has the potential to be further refined which will be shown in Chapter 5. The difference between the two strategies 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 were compared above, the solid level detected by ADSA-NA for each run was also used as an input for TIFA-AI.

77 Effects of Optimization Parameters In ADSA-NA, the tilt angle of the camera, α, is optimized to correct a possible minor vertical misalignment of the camera. To examine the effect of fixing or optimizing α on the results, the advancing contact angles of dodecane, tetradecane, and hexadecane on Teflon AF 1600 for one sample run were measured with fixed α, 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 α. The reason is that α was small (less than 0.1 ) in the experiments. Therefore, if the experiment is performed carefully enough, α does not need to be optimized, which makes the optimization faster. It should be noted that the horizontal alignment of the camera was corrected carefully using a bubble level in the experiment. Based on the study performed in reference [96] concerning the effect of horizontal alignment of the camera on the accuracy of measured contact angles, for the inclination angle of about 0.1, i.e. the accuracy of the bubble level, and the contact angle of about 50, the error for the measured contact angle is about 10 4 degree, which is an insignificant error in the measurements. All the results shown in Tables 4.3 and 4.4 for ADSA-NA and TIFA-AI were obtained for known surface tension, i.e. the measurements from Table 4.1 were used as input. However in previous results [21] analyzed by ADSA, capillary constant and hence surface tension 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 4.6). To determine

78 64 the required tolerance in the optimization process for ADSA-NA and TIFA-AI in this case, the advancing contact angles and surface tensions of dodecane and tetradecane on Teflon AF 1600 were measured by ADSA-NA and TIFA-AI for one sample run for each liquid with tolerance values ranging from 10 4 to Table 4.7 shows the measured contact angles and surface tensions and the average computation time per image for different tolerance values. Based on the results shown in Table 4.7, the necessary tolerances in the optimization process for ADSA-NA and TIFA-AI in the case of optimized surface tension are 10 9 and 10 6, respectively. It can also be seen that the average computation time per image is larger for TIFA-AI compared to ADSA-NA, and for both of them, the average computation time increases by decreasing the tolerance value. Table 4.7 also shows that the surface tension values measured by ADSA-NA at the necessary tolerance, i.e are larger than the values measured by TIFA-AI. This is also true for decane and hexadecane (see Table 4.6). It appears, in general, the values measured by TIFA-AI are slightly lower than the values measured by ADSA-NA for reasons not known yet. Comparing the results for each run in Table 4.3 with the corresponding run in Table 4.6 shows that there is at most 0.1 change in the average advancing contact angle for each run if the fixed input of surface tension is replaced by optimization of surface tension. In addition, the scatter of the measured contact angle for every liquid is higher when surface tension is optimized, for both ADSA-NA and TIFA-AI. The typical image-toimage scatter for ADSA-NA and TIFA-AI for advancing contact angles in Table 4.6 is about 0.05, which is higher than the scatter of 0.02 for the strategy of using surface tension as input. However, the image-to-image scatter is still considerably lower than

79 65 the previous results in reference [21]. The underlying reason for higher scatter of contact angles in Table 4.6 compared to Table 4.3 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 4.6 with those in Table 4.1 reveals that there is an average of 0.2 mj/m 2 error. There is also an average discrepancy of 0.2 mj/m 2 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, i.e. the deviation from the true value, reported here far surpasses (by at least a factor of 5) that obtained from sessile drop and the standard ADSA procedure. Based on the above discussion for the case of optimizable surface tension, the contact angles in Table 4.3 (fixed surface tension) are preferred to those in Table 4.6 (optimizable 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 longer 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 where the surface tension is known and used as input. It should be noted that using an incorrect fixed value of surface tension can affect the contact angle value and lead to systematic errors. To investigate this effect, the advancing contact angle of dodecane was determined for different fixed values of surface tension (Table 4.8). It can be seen that changing the surface tension by ±1 mj/m 2 from

80 66 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 is advisable to determine it simultaneously with the contact angle.

81 67 Controller Syringe Motor Light Source Microscope CCD Camera Diffuser Host Computer Figure 4.1: Schematic diagram of the experimental setup of ADSA-NA

82 are very smooth with root mean square (RMS) roughness of 0.4 nm and maximum peak-to-valley distances of 2.0 nm. Similar measurements for EGC-1700, ETMF, and ODMF surfaces yielded RMS roughness of 1.4 nm, 0.4 nm, and co th el sp H. Tavana, A.W. Neumann / Advances in Colloid 68 Figure 4.2: Repeat unit of: poly[4,5-difluoro-2,2-bis(trifluoromethyl)-1,3-dioxole-cotetrafluoroethyene], 65 mol% dioxole (Teflon AF 1600) Fig. 4. Repeat unit of: (a) poly[4,5-difluoro-2,2-bis(trifluoromethyl)-1,3-dioxole-co-tet heptafluorobutyl methacrylate) (EGC-1700), (c) poly(ethene-alt-n-(4-(perfluorohep (perfluoroheptylcarbonyl)aminobutyl)maleimide) (ODMF).

83 contact angle ( ) time(s) (a) 0.26 contact radius (cm) Student Version of MATLAB time(s) (b) Figure 4.3: Results of ADSA-NA for a typical run with dodecane on Teflon AF The advancing angle was measured as ± 0.02 for this run. The literature value from reference [21] is ± Student Version of MATLAB

84 70 Figure 4.4: A dodecane drop on a Teflon AF 1600 coated surface. The white line is the Laplacian fit by ADSA-NA Student Version of MATLAB

85 ADSA-NA TIFA-AI contact angle ( ) time (s) Figure 4.5: Results of ADSA-NA and TIFA-AI for advancing contact angle for a typical run with decane on Teflon AF The advancing angle for the largest drops of this run was measured by ADSA-NA and TIFA-AI as ± 0.04 and ± 0.05, respectively. Student Version of MATLAB

86 72 contact angle ( ) θ r = time(s) Figure 4.6: Advancing and receding contact angles of dodecane on Teflon AF The receding angle is time dependent.

87 73 Table 4.1: Experimental liquids and their surface tension at 24 C. The surface tensions for this thesis were measured using pendant drop experiments analyzed by ADSA. The errors given are the 95% confidence limits. γ (mj/m 2 ) γ (mj/m 2 ) Liquid Supplier Purity (this thesis) Ref. [91] n-decane Sigma-Aldrich 99+% ± n-dodecane Sigma-Aldrich 99+% ± n-tetradecane Sigma-Aldrich 99+% ± n-hexadecane Aldrich 99+% ± :DIPPR Project 801-Full Version

88 74 Table 4.2: The effect of tolerance value in the optimization procedure on the advancing contact angles θ adv (degrees) of dodecane and tetradecane on Teflon AF 1600 analyzed by ADSA-NA and TIFA-AI. The errors given are the standard deviations. T is the average computation time per image in seconds. tolerance ADSA-NA TIFA-AI θ adv T(s) θ adv T(s) ± ± dodecane ± ± (RunA) ± ± ± ± ± ± ± ± tetradecane ± ± (RunA) ± ± ± ± ± ±

89 75 Table 4.3: 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 values. The errors given for ADSA-NA and TIFA-AI are the standard deviations, and the errors given for Ref. [21] are the 95% confidence limits. The values for Ref. [21] had been averaged over five runs. ADSA-NA TIFA-AI Ref. [21] Spline Sigmoid decane RunA ± ± ± ± 0.18 RunB ± ± ± 0.02 dodecane RunA ± ± ± ± 0.32 RunB ± ± ± 0.03 tetradecane hexadecane RunA ± ± ± 0.02 RunB ± ± ± 0.03 RunC ± ± ± 0.01 RunA ± ± ± 0.02 RunB ± ± ± 0.02 RunC ± ± ± ± ± 0.16

90 76 Table 4.4: 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. [95] decane RunA RunB dodecane RunA RunB tetradecane RunA RunB RunC hexadecane RunA RunB RunC

91 77 Table 4.5: The effect of two different strategies of solid detection on the advancing contact angles θ adv (degrees) of dodecane and tetradecane on Teflon AF The profile and the reflection strategies are illustrated in figures 3.5(a) and 3.5(b), respectively. The errors given are the standard deviations. Liquid Run strategy θ adv dodecane RunA profile ± 0.02 reflection ± 0.02 RunB profile ± 0.02 reflection ± 0.04 tetradecane RunA profile ± 0.03 reflection ± 0.04 RunB profile ± 0.02 reflection ± 0.04 RunC profile ± 0.02 reflection ± 0.02

92 78 Table 4.6: Advancing contact angles (degrees) of sample liquids on Teflon AF 1600 with optimized surface tension. θ adv is the average advancing contact angle, and γ (mj/m 2 ) is the surface tension. The errors given are the standard deviations. decane dodecane tetradecane hexadecane ADSA-NA TIFA-AI Run θ adv γ θ adv γ RunA ± ± ± ± 0.32 RunB ± ± ± ± 0.33 RunA ± ± ± ± 0.12 RunB ± ± ± ± 0.17 RunA ± ± ± ± 0.11 RunB ± ± ± ± 0.12 RunC ± ± ± ± 0.12 RunA ± ± ± ± 0.08 RunB ± ± ± ± 0.19 RunC ± ± ± ± 0.07

93 79 Table 4.7: The effect of tolerance value in the optimization procedure on the advancing contact angles θadv (degrees) of dodecane and tetradecane on Teflon AF 1600 with optimized surface tension analyzed by ADSA-NA and TIFA-AI. θadv is the average advancing contact angle, and γ (mj/m 2 ) is the surface tension. The errors given are the standard deviations. T is the average computation time per image in seconds. dodecane (RunA) tolerance ADSA-NA TIFA-AI θadv γ T(s) θadv γ T(s) ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± tetradecane (RunA) ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

94 80 Table 4.8: Advancing contact angle θ adv (degrees) of dodecane on Teflon AF 1600 analyzed by ADSA-NA using different fixed values of surface tension (γ (mj/m 2 )). The errors given are the standard deviations. dodecane (RunA) γ θ adv ± ± ± ± ± ± ± 0.02

95 Chapter 5 Examination of Image Analysis and Optimization Strategies 5.1 Introduction It was shown in the previous chapter that ADSA-NA enhanced the accuracy of contact angle measurement (by at least a factor 5) compared to the original ADSA using a drop with an apex. To establish that the high accuracy obtained for contact angle measurement by ADSA-NA is not an artifact of the image analysis or the optimization procedure, the numerical components of ADSA-NA are explored in this chapter well beyond the original ADSA. In Section 5.2, the results obtained from one and the same drop image are compared using different edge detection strategies. The determination of the location of the solid Part of this chapter was previously published [97] and reproduced with permission. 81

96 82 surface in the image is also further refined to improve contact angle measurement. The effect of different optimization methods on the accuracy of ADSA-NA measurements are discussed in Section Image Analysis Edge Detector In the previous chapter the Canny edge detector [62] was used to detect the drop profile to pixel resolution. As described before, Canny is a gradient edge detector, i.e. it requires the derivatives of the gray levels of an image. Laplacian of Gaussian (LoG) [98] and Sobel [99] are other well-known examples of gradient edge detection methods. The LoG operator first smooths the image by applying a Laplacian of Gaussian filter. Then it finds the pixels of edges as the locations where the gradient of the gray level of the smoothened image is at the maximum. In other words, LoG locates the edge where the second derivative of the gray level is zero, i.e. the Laplacian changes sign. The Sobel operator first calculates the gray-level gradient for each pixel of the image by applying a 3 3 filter, i.e. convolution mask. Then, the pixels lying on edges are selected as the pixels with the steepest gradient. There has been concern about the sensitivity of gradient edge detectors to noise in the image [81] due to possible amplification of the noise by derivatives used in these methods. Recently, a non-gradient robust edge and corner detector called Smallest Univalue Segment Assimilating Nucleus (SUSAN) [81] was introduced. To investigate the effect of

97 83 different edge detection strategies on the results, the contact angles and surface tensions obtained by ADSA-NA using the three different gradient edge detection methods, i.e. Canny, LoG, and Sobel, are compared with the results obtained by the non-gradient edge detector, SUSAN [81]. SUSAN is a fairly recent edge and corner detection method. Briefly, in the SUSAN method, a circular mask or window of 37 pixels is placed over each pixel in the image. The center pixel is known as the nucleus in SUSAN. The gray level of each pixel within the mask is compared with the gray level of the nucleus. The area of the mask is defined as the sum of a non-linear function of the gray levels of the pixels in the mask that have similar gray level to the nucleus. This area of the mask is known as Univalue Segment Assimilating Nucleus (USAN). The USAN is calculated for every pixel in the image, and the edges in the image are identified as the pixels with a minimum USAN value. The advancing contact angles and surface tensions of four different alkanes (Table 4.1) suspended from a needle on Teflon AF 1600 were measured by ADSA-NA using the four different edge detectors: Canny, LoG, Sobel, and SUSAN. As mentioned in Chapter 4, contact angle measurements were performed for each liquid on one solid surface in two or three different runs. Each run consisted of enlarging and shrinking of a drop formed on a different spot of the solid surface. Advancing contact angles were calculated for the largest drops (around 100 images), for which drop size dependence is insignificant. The results for advancing contact angles with surface tension held fixed during the optimization are shown in Table 5.1. Comparing the advancing contact angles results

98 84 reveals that there is more consistency among the contact angle values obtained with Canny, Sobel, and SUSAN, compared to LoG. In fact, contact angle results obtained with Canny, Sobel, and SUSAN for one and the same image agree to within ±0.1. Comparing the contact angles between different runs of each liquid shows that reproducibility or runto-run scatter for Canny, Sobel, and SUSAN is also better than 0.1. Table 5.2 shows the results for advancing contact angles with surface tension as an optimization parameter. Comparing the advancing contact angles results in Table 5.2 reveals a similar pattern as in Table 5.1, i.e. there is more consistency among the contact angle values obtained with Canny, Sobel, and SUSAN, compared to LoG. Comparing the contact angles between different runs of each liquid shows that the run-to-run scatter for Canny, Sobel, and SUSAN is still less than 0.1. Looking at the contact angle values of different runs of hexadecane obtained with Canny, Sobel, and SUSAN reveals that the solid surface is slightly more hydrophobic on the spot of run B than on the spots of run A and run C, indicating the real effect of the solid surface on the contact angle value. Comparing the surface tension results shows that the measured surface tension value for hexadecane using the LoG method deviates significantly ( 0.2 mj/m 2 ) from the values obtained with the other edge detection methods. It can be seen that surface tension results obtained with Canny, Sobel, and SUSAN for one and the same image agree to within ± 0.1 mj/m 2. Comparing the results for each run in Table 5.1 with the corresponding run in Table 5.2 shows that there is less than 0.1 change in the average advancing contact angle for each run with optimized surface tension compared to the case of fixed surface tension

99 85 obtained with Canny, Sobel, and SUSAN. Similar to the results found in Chapter 4, the scatter of the measured contact angle for every liquid is higher when surface tension is optimized. The typical image-to-image scatter for advancing contact angles in Table 5.2 is about 0.05, which is higher than the scatter of 0.02 for the strategy of using surface tension as input. It should be noted that optimizing the surface tension in the analysis leads to contact angle results with slightly higher scatter and longer processing time, compared to the case of using a known value of surface tension. Apparently this strategy gives more flexibility to the optimization procedure by expanding the parameter space which results in a better fit, i.e. smaller objective function value, between the Laplacian curves and the experimental drop profile. This is important for the cases where the drop profile has some irregularities or the drop is not strictly axisymmetric. Thus in such cases, inaccurate values of surface tension may result. This issue is more complicated for the contact angle because there is no base for the contact angle values, i.e. there are no highly accurate contact angle values available in the literature. Based on the above discussion, to measure the thermodynamic contact angle, it is better to use a known value of surface tension measured accurately beforehand, e.g. with pendant drop experiments. Because of better consistency among the contact angle and surface tension values obtained with Canny, Sobel, and SUSAN in Tables 5.1 and 5.2, it is concluded that the LoG method is not a good choice for edge detection in ADSA-NA, and it was not considered further. To keep the study manageable, one of the edge detectors, i.e. Canny, Sobel, and SUSAN, had to be chosen for further study. It was shown that the Sobel edge detector is inferior to Canny for noisy or blurry images [60]. Since Canny has been

100 86 tested and evaluated more extensively than SUSAN for measuring interfacial properties, Canny remains the preferred choice for edge detection in ADSA-NA. Early in this thesis, sigmoid function and spline function were used to locate drop profiles with sub-pixel resolution (see Section 4.3.1). Here, the effect of the two different sub-pixel resolution methods on the surface tension measurement is investigated. Table 5.3 shows the advancing contact angles with surface tension as an optimization parameter using the two different sub-pixel resolution methods. It can be seen that the surface tension values obtained with the two sub-pixel resolution methods agree to better than 0.1 mj/m 2 for most sample runs. Here, the spline method is preferred because it requires less computation time Solid Surface Location In the previous chapter, it was shown that locating the solid surface is a significant source of uncertainty in contact angle measurement. The profile strategy so far was preferred over the reflection strategy for detecting the contact points at the solid-liquid interface because of slightly lower scatter in contact angles, i.e. smaller image-to-image standard deviation of contact angles. As described in Chapter 3, the accuracy of the profile strategy by its design is limited to ±0.5 pixels for the determination of the solid location. As mentioned before, the accuracy of the reflection strategy is affected by the quality of the reflection of the drop profile. The reflection of the profile is not always a smooth profile because the region around the corner at the solid-liquid interface is usually fuzzy (Figure 5.1). However, the reflection strategy has the potential of determining the

101 87 solid location with sub-pixel resolution. Therefore it is worthwhile to pursue improving this strategy for imperfect reflection of drop profiles. The reflection strategy was improved by excluding outlier pixels (Figure 5.1) from the reflection of the drop profile and the drop profile, i.e. eliminating the pixels that do not lie close to the fitted straight lines. A procedure called Peirce s criterion [ ] was used to detect and delete the outliers. Figure 5.2 illustrates the procedure of detecting outliers by means of the Peirce s criterion. In the Peirce s procedure, first, the maximum allowable deviation of a data point from the data mean is calculated. The maximum allowable deviation is a linear function of the standard deviation of the data set: x i x m max = Rσ (5.1) where x i is a measured data point, x m is the mean of the data set, σ is the sample standard deviation of the data set, and R is the Peirce s ratio. The value of R is a function of the number of data points and the number of outliers, which can be obtained from Reference [100]. It is noted that in our study, the measured data points are the normal distances of the 10 selected pixels of the reflection of the drop profile or the drop profile from the fitted straight line. In the Peirce s procedure, first, it is assumed that there is one outlier in the data set. Then, the corresponding maximum allowable deviation is calculated. In the next step, the deviations of all data points to the data mean is calculated. If this deviation for any data point is larger than the maximum allowable deviation, that point is detected as an outlier and eliminated from the data set. The number of detected

102 88 outliers is counted, and a new number of outliers is assumed to be the number of detected outliers plus one. Then, a new value of the maximum allowable deviation corresponding to the new number of outliers is calculated, keeping the original values of the mean and standard deviation, and the original number of data points. The above procedure of detecting the outliers and calculating the new value of maximum allowable deviation is repeated until no more data points need to be eliminated. Table 5.4 shows advancing contact angles of dodecane and tetradecane on Teflon AF 1600 obtained with the modified reflection strategy, the original reflection strategy, and the profile strategy. It is noted that the surface tension values were taken as input, i.e. not optimized, for this analysis. It can be seen that there is a discrepancy less than 0.1 between the contact angles obtained with the modified reflection and the original reflection strategies. However, the modified reflection strategy shows lower scatter, i.e. smaller image-to-image standard deviation. Overall, the contact angles obtained with the three different strategies agree to within 0.1. Because of its lower scatter compared to the original reflection strategy and its location of the solid level with sub-pixel resolution, the modified reflection strategy is the preferred choice to use in ADSA-NA. It was mentioned in Section that there is better consistency among the contact angle values obtained with Canny, Sobel, and SUSAN, compared to LoG. One of the reasons for the relatively poor performance of LoG is connected to the determination of locating the solid surface. Table 5.5 shows the difference between the calculated solid surface locations by Canny on the one hand and LoG, SUSAN, and Sobel on the other hand. Indeed, it can be seen that the LoG method gives a larger difference for the

103 89 calculated solid surface location for each run compared to the locations obtained with SUSAN and Sobel. It is noted that second derivatives of the gray levels in the image are used in the LoG method which makes it less robust in a noisy area of the image, e.g. the corner area of a drop and its reflection, compared to the gradient methods using the first derivative of gray levels, i.e. Canny and Sobel. 5.3 Optimization Procedure As mentioned in Chapter 3, the Nelder-Mead simplex method [88] was implemented in the optimization procedure in ADSA-NA because of the computational simplicity in implementation. There are other powerful and well-known optimization methods such as Levenberg-Marquardt and Newton-Raphson. Levenberg-Marquardt requires the calculation of the first derivatives of the objective function, and Newton-Raphson requires the calculation of the first and the second derivatives of the objective function. It had been shown that Newton-Raphson was faster than Levenberg-Marquardt for sessile and pendant drops analyzed by the original ADSA, but it was more sensitive to the choice of initial values [55]. In that study, Newton-Raphson was the choice for the optimization; in the case of divergence, Levenberg-Marquardt was used instead of Newton-Raphson. There is also another study by Alvarez et al. [73] claiming that the Nelder-Mead simplex method measures the surface tension of near spherical pendant drop shapes with better accuracy than the gradient-based optimization methods, i.e. Levenberg-Marquardt and Newton-Raphson. To test this claim and to investigate the effect of different optimization

104 90 methods in ADSA-NA, the Levenberg-Marquardt and Newton-Raphson methods were implemented in ADSA-NA and will be discussed and compared with the Nelder-Mead simplex method. First, it is required to derive the system of equations for each optimization method. The objective function, E, in ADSA-NA (Equation 3.25) is defined as the sum of the squares of the normal distances between all detected drop profile points, i.e. the detected points with sub-pixel resolution, and the theoretical profile. The objective function, E, is a function of a set of optimization parameters, a, with elements a k, k = 1,..., M. Here, a is the vector of six parameters (M = 6) or any subset of it: a = {b, c, R 0, θ 0, x 0, α} (5.2) where c = ( ρ)g γ (5.3) and b is the mean curvature of the drop profile at the solid surface, c is the capillary constant, R 0 is the contact radius of the drop, θ 0 is the contact angle, x 0 is the horizontal offset between the coordinate systems (experimental vs. theoretical), α is the rotation angle, γ is the liquid-fluid interfacial tension, g is the gravitational acceleration, and ρ is the density difference between the liquid and vapor. The goal of the optimization is to calculate the values of the optimization parameters a k that minimize E, thus giving the best fit between the experimental profile and a

105 91 Laplacian curve. The necessary conditions for a minimum in the value of E are: E a k = N i=1 w i e i a k = 0, k = 1,..., M (5.4) Equation 5.4 forms a set of non-linear equations which requires an iterative solution. The iterative procedure can be expressed as: a i+1 = a i a i (5.5) where a i is the vector of unknown variables (optimization parameters) at the ith iteration step, and a i is a correction vector. If the Newton-Raphson method is used, a i is evaluated from the solution of the following linear system: H(a i ) a i = J(a i ) (5.6) where H(a i ) is the Hessian matrix, and J(a i ) is the Jacobian vector. The components of the vector J(a i ) are the first derivative terms obtained from Equation 5.4 and evaluated at the ith step. The components of the Hessian matrix, H(a i ), are computed as: 2 E a k a l = N i=1 w i 2 e i, k, l = 1,..., M (5.7) a k a l It is noted that the details of the formulations for the Hessian matrix, and the Jacobian vector can be found in Appendix A.

106 92 If Levenberg-Marquardt is used, a i is evaluated from the solution of the following linear system: (J(a i ) T J(a i ) + λdiag(j(a i ) T J(a i ))) a i = J(a i ) T D(a i ) (5.8) where J(a i ) is the Jacobian vector evaluated at the ith step, λ is the damping factor, and D(a i ) is a vector of minimum distances, i.e. d i in Equation It is noted that λ is adjusted at each iteration. If reduction of E is rapid, a smaller value of λ is used, whereas if an iteration gives insufficient reduction in the residual, λ is increased. The contact angles and surface tensions of four sample alkanes were determined with the two additional optimization methods. The results for advancing contact angles with surface tension held fixed during the optimization are shown in Table 5.6, and the results for advancing contact angles with surface tension as an optimization parameter are shown in Table 5.7. It is noted that the results are shown only with six significant figures, however, all three optimization methods converge to the same solution with eight significant figures for one and the same image. The only difference is the computation time of an image analyzed with each optimization method. Levenberg-Marquardt is the fastest method, and the Nelder-Mead simplex is the slowest method. For instance, on a computer with a 2.66 GHz CPU and 3.00 GB of RAM, the average computation times of dodecane for run A (around 100 images) were about 45 s, 25 s, and 5 s using Nelder- Mead simplex, Newton-Raphson, and Levenberg-Marquardt, respectively. Because of its much lower processing time, Levenberg-Marquardt is now the preferred choice for the

107 optimization method in ADSA-NA. 93

108 94 number of pixel points in vertical direction 660 modified reflection strategy C 675 outlier number of pixel points in horizontal direction Figure 5.1: Modified reflection strategy for determining the contact points at the solidliquid interface of a dodecane drop on Teflon AF The original reflection strategy detected the contact point (point B in Figure 3.5(b)) as (877.6,670.1), and the modified reflection strategy detected the contact point (point C) as (877.0,670.2), for this specific drop image. Student Version of MATLAB

109 95 x m Calculate mean ( ) and sample standard deviation ( ) of data values ( ) x i,i=1,... n Assume no. of outliers ( N O ) = 1 N O =N O 1 Obtain R from the Peirce's table (Reference [96]) Calculate max. allowable deviation: R Eliminate the outliers from the the data set x i x m Calculate data points for all Count the number of outlier points ( ) N O Yes Is there any point that x i x m R End No Figure 5.2: Flowchart of identifying outliers using Pierce s criterion

110 96 Table 5.1: Advancing contact angles (degrees) of four sample alkanes on Teflon AF 1600 with fixed surface tension analyzed by ADSA-NA using different edge detections methods. The errors given are the standard deviations. The surface tension values measured by ADSA using pendant drop experiments (Table 4.1) were used as inputs. decane dodecane tetradecane hexadecane Canny LoG SUSAN Sobel RunA ± ± ± ± 0.04 RunB ± ± ± ± 0.03 RunA ± ± ± ± 0.02 RunB ± ± ± ± 0.02 RunA ± ± ± ± 0.02 RunB ± ± ± ± 0.02 RunC ± ± ± ± 0.02 RunA ± ± ± ± 0.03 RunB ± ± ± ± 0.02 RunC ± ± ± ± 0.02

111 Table 5.2: Advancing contact angles (degrees) of four sample alkanes on Teflon AF 1600 with optimized surface tension analyzed by ADSA-NA using different edge detection methods. The errors given are the standard deviations. Canny LoG Sobel SUSAN Run θadv γ θadv γ θadv γ θadv γ RunA ± ± ± ± ± ± ± ± 0.09 RunB ± ± ± ± ± ± ± ± 0.11 decane RunA ± ± ± ± ± ± ± ± 0.12 RunB ± ± ± ± ± ± ± ± 0.09 dodecane RunA ± ± ± ± ± ± ± ± 0.08 RunB ± ± ± ± ± ± ± ± 0.13 RunC ± ± ± ± ± ± ± ± 0.13 tetradecane RunA ± ± ± ± ± ± ± ± 0.13 RunB ± ± ± ± ± ± ± ± 0.13 RunC ± ± ± ± ± ± ± ± 0.13 hexadecane 97

112 98 Table 5.3: Advancing contact angles (degrees) of sample liquids on Teflon AF 1600 with optimized surface tension analyzed by ADSA-NA using the two different sub-pixel resolution methods. θ adv is the average advancing contact angle, and γ (mj/m 2 ) is the surface tension. The errors given are the standard deviations. Canny was used as the edge detector. decane dodecane tetradecane hexadecane Spline Sigmoid Run θ adv γ θ adv γ RunA ± ± ± ± 0.14 RunB ± ± ± ± 0.12 RunA ± ± ± ± 0.13 RunB ± ± ± ± 0.13 RunA ± ± ± ± 0.10 RunB ± ± ± ± 0.16 RunC ± ± ± ± 0.13 RunA ± ± ± ± 0.16 RunB ± ± ± ± 0.14 RunC ± ± ± ± 0.15

113 99 Table 5.4: The effect of different strategies of solid detection (using the Canny edge detection) on the advancing contact angles θ adv (degrees) of dodecane and tetradecane on Teflon AF The errors given are the standard deviations. Liquid Run Strategy θ adv dodecane RunA modified reflection ± 0.02 reflection ± 0.02 profile ± 0.02 RunB modified reflection ± 0.03 reflection ± 0.04 profile ± 0.02 tetradecane RunA modified reflection ± 0.02 reflection ± 0.04 profile ± 0.03 RunB modified reflection ± 0.02 reflection ± 0.04 profile ± 0.02 RunC modified reflection ± 0.02 reflection ± 0.02 profile ± 0.02

114 100 Table 5.5: Difference between the calculated solid surface locations by Canny on the one hand and LoG, SUSAN, and Sobel on the other hand. The difference is in pixels. The minus sign means the calculated solid location is lower than the one calculated by Canny. The modified reflection strategy was used for detecting the solid surface location. decane dodecane tetradecane hexadecane LoG SUSAN Sobel RunA RunB RunA RunB RunA RunB RunC RunA RunB RunC

115 101 Table 5.6: Advancing contact angles (degrees) of four sample alkanes on Teflon AF 1600 with fixed surface tension analyzed by ADSA-NA using different optimization methods. The errors given are the standard deviations. The surface tension values measured by ADSA using pendant drop experiments (Table 4.1) were used as inputs. decane dodecane tetradecane hexadecane Levenberg-Marquardt Newton-Raphson Nelder-Mead RunA ± ± ± RunB ± ± ± RunA ± ± ± RunB ± ± ± RunA ± ± ± RunB ± ± ± RunC ± ± ± RunA ± ± ± RunB ± ± ± RunC ± ± ±

116 Table 5.7: Advancing contact angles (degrees) of four sample alkanes on Teflon AF 1600 with optimized surface tension analyzed by ADSA-NA using different optimization methods. The errors given are the standard deviations. Levenberg-Marquardt Newton-Raphson Nelder-Mead Run θadv γ θadv γ θadv γ RunA ± ± ± ± ± ± RunB ± ± ± ± ± ± decane RunA ± ± ± ± ± ± RunB ± ± ± ± ± ± dodecane RunA ± ± ± ± ± ± RunB ± ± ± ± ± ± RunC ± ± ± ± ± ± tetradecane RunA ± ± ± ± ± ± RunB ± ± ± ± ± ± RunC ± ± ± ± ± ± hexadecane 102

117 Chapter 6 Optimization of Experimental Geometry of ADSA-NA 6.1 Introduction Although the main purpose of this study is to improve contact angle measurement, the simultaneous measurement of contact angle and surface tension is of interest as well. There are some situations in which the simultaneous measurement of contact angle and surface tension is necessary, for instance in studies of the effect of adsorption of surfaceactive substances on solid-liquid and liquid-vapor interfacial tensions [ ] or in studies of the work of adhesion in molten alloy-ceramic substrate systems [ ]. Among the methods of simultaneous contact angle and surface tension measurement, the combination of the capillary rise method and the Wilhelmy method [8], as well as Part of this chapter was previously published [97] and reproduced with permission. 103

118 104 drop shape techniques [14], are capable of simultaneous measurement. While the joint use of the capillary rise method and the Wilhelmy method to simultaneously measure contact angle and surface tension can be very accurate, these methods have several drawbacks. They are not easy to handle experimentally, and they require relatively large amounts of liquid [14]. Furthermore, the plate used in these methods must have the same composition and morphology on both sides. Drop shape techniques such as ADSA may also be used for simultaneous measurement of surface tension and contact angle using a sessile drop constellation [14]. However, the accuracy of the surface tension, i.e. the deviation from the true value, obtained from a sessile drop by means of ADSA was far below the accuracy that can be obtained from a pendant drop: 1.0 mj/m 2 accuracy at best using sessile drops versus 0.01 mj/m 2 accuracy using pendant drops [14]. It was shown in Chapter 4 that the accuracy of the surface tension obtained by ADSA- NA was improved significantly (by at least a factor of 5) compared to that obtained from sessile drops using the standard ADSA, without actual effort to do so. However, ADSA-NA is still not as accurate as a surface tension measurement by a pendant drop suspended from an inverted sharp-edged pedestal, e.g. a stainless steel holder [14]. It was shown [14, 25, 109] that the accuracy of surface tension measurement is sensitive to the geometry of the drop, e.g. volume and height. In this chapter, the effect of controllable experimental factors such as drop height and volume on the accuracy of surface tension measurement is investigated, and practical experimental guidelines for obtaining accurate results for surface tension measurement using the ADSA-NA setup are established. In the sessile drop constellation with a stainless steel capillary protruding into the

119 105 drop (Figure 1.2), the liquid may spread over the outside wall of the capillary. Therefore, the drop may easily become asymmetric if the liquid wetted the wall of the capillary non-uniformly, e.g. due to possible impurities on the capillary. The asymmetry tends to be more pronounced for drops of relatively high surface tension liquids, e.g. water. As a result, forming axisymmetric drops for such liquids is difficult using a capillary. It should be noted that the drop axial symmetry also depends on the solid substrate. The other disadvantage of using a capillary is that it is not possible to control the drop height. For the sake of the investigation of experimental factors in the accuracy of surface tension measurement, the capillary was replaced with an inverted sharp-edged pedestal (Figure 6.1) to render the drop height a controllable experimental parameter and to assure axisymmetry of the drops. It is noted that the sharp-edged pedestal was used before in pendant drop experiments [14] and in constrained sessile drop experiments [109] to prevent spreading of the liquid on the outer surface of the pedestal. Since the pedestal width may be an important factor in the accuracy of surface tension measurement [14], measurements were performed here using three different pedestal widths: 1.6 mm, 2.8 mm, and 3.8 mm. In this chapter, the effect of four different experimental factors on the accuracy of surface tension measurement is investigated: drop volume, drop height, pedestal width, and drop asymmetry. This investigation is performed using two different types of sessile drop experiments. In the first type, which is called a constant height experiment, the drop front was advancing by increasing the drop volume while the vertical location of the pedestal was fixed. Since the location of the pedestal is fixed in this type of experiments,

120 106 it is possible to isolate the effect of drop volume. In the second type of experiments, called constant volume experiments, a sessile drop was formed and images of the drop were captured with the pedestal raised to different heights while the drop volume was kept unchanged. The same procedure was then repeated for a larger drop. Since the image of the drop is captured at different vertical positions of the pedestal in constant volume experiments, it is possible to study the effect of drop volume and drop height together. It should be emphasized that for each type of the experiments, the relative importance of one or more factors is illustrated with the aim of showing a trend in the accuracy of the results, not to provide a final answer with the best accuracy. Four different sample liquids were used: DMCPS, dodecane, cyclododecatriene, and water. The liquids are listed in Table 6.1. These liquids were chosen because of their availability, low vapour pressure, and non-toxicity. The literature values of the surface tensions of the liquids obtained from references [91, 110] and the surface tension values measured using pendant drop experiments analyzed by ADSA [14] are also shown in Table 6.1. The errors given for the measured values of surface tension are 95% confidence limits. It is noted that no confidence limits for the values of surface tension are reported in references [91, 110]. For highly purified liquids, i.e. water and dodecane, the measured values of surface tension by ADSA-pendant drop show good agreement with literature values. For DMCPS and cyclododecatriene, the presence of impurities presumably is the reason for the discrepancy between the measured values of surface tension by ADSA-pendant drop and the literature values. It should be emphasized that we are not interested in generating new literature values, but rather in optimizing the

121 107 surface tension measurement which may or may not be correct. The last column in Table 6.1 contains the final results of ADSA-NA for the surface tension of the sample liquids implementing all the experimental guidelines that will be shown and discussed in this chapter. It can be seen that there is an error of less than 0.1 mj/m 2 between the surface tension values measured by ADSA using pendant drop experiments and the surface tension values measured by ADSA-NA. 6.2 Effect of Volume at Constant Drop Height The surface tensions of the four sample liquids (DMCPS, dodecane, cyclododecatriene, and water) using the three different pedestal widths were measured in constant height experiments. For each surface used in the experiments, two different drops were formed on fresh spots of the surface and enlarged by pumping the liquid through the pedestal. Each run consisted of approximately 100 images captured at different drop volumes. The ranges of drop volumes in these experiments for DMCPS, dodecane, cyclododecatriene, and water were 5 35 µl, 8 50 µl, µl, and µl, respectively, and the ranges of drop heights were mm, mm, mm, and mm, respectively. Figures show the images of different drop sizes formed with different pedestal widths for each sample liquid captured in constant height experiments. Table 6.2 shows the surface tensions of the sample liquids for all runs obtained from constant height experiments. It can be seen in Table 6.2 that the total runs performed for dodecane with the pedestal width of 1.6 mm outnumber the runs for dodecane with the other pedestal

122 108 widths and also the runs for the other liquids. The reasons are the ready availability of dodecane compared to the other liquids and the erroneous initial anticipation that the pedestal width of 1.6 mm would be large enough to obtain accurate results for the surface tensions of different liquids. Comparing the measured surface tension of each run in Table 6.2 with the value of surface tension obtained with pendant drop experiments (Table 6.1) reveals that the average errors of the measured surface tension with the largest pedestal (3.8 mm) for DMCPS, dodecane, cyclododecatriene, and water are 0.1 mj/m 2, 0.2 mj/m 2, 0.1 mj/m 2, and 0.8 mj/m 2, respectively. It should be emphasized that the surface tension results shown in Table 6.2 are preliminary. They do not represent the best values that can be measured by ADSA-NA. They are used to show the effect of volume alone on the accuracy of the measured surface tension. Since the location of the pedestal is fixed in this type of experiment, it is possible to isolate the effect of drop volume. To this end, a linear regression was performed for the surface tension error versus the volume for each run in Table 6.2. The regression equation was defined as: γ γ 0 = m V + n (6.1) where γ is the measured surface tension (mj/m 2 ), γ 0 is the true value of surface tension (mj/m 2 ) obtained with pendant drop experiments (Table 6.1), and V is the drop volume in µl; m (10 6 Kg/m 3 s 2 ) and n (mj/m 2 ) are the coefficients of the linear regression. Table 6.3 shows the calculated coefficient m for all runs of each sample liquid. For

123 109 convenience, the average of the calculated coefficient m for all runs of each sample liquid for each pedestal size is shown in Table 6.4. It can be seen that the sign of m is not consistently positive or negative. This suggests that (perhaps counter to expectation) there is no significant, general effect of drop volume on the accuracy of surface tension measurement. 6.3 Effects of Drop Height, Drop Volume, Pedestal Width, and Drop Asymmetry This section concerns the constant volume experiments. These experiments were repeated for different volumes and pedestal widths, allowing for investigation of the effect of four different experimental factors on the accuracy of surface tension measurement: drop volume, drop height, pedestal width, and drop asymmetry. Figure 6.6 shows the images of a drop of dodecane captured at different vertical positions of the pedestal in the constant volume experiments. The ranges of drop volumes in the constant volume experiments for DMCPS, dodecane, cyclododecatriene, and water were 5 32 µl, 4 39 µl, 3 51 µl, and 6 48 µl, respectively, and the ranges of drop heights were mm, mm, mm, and mm, respectively. For each drop, the maximum pedestal height was limited either by the drop detaching or else by the fixed field of view of the microscope. Although it would be interesting to investigate the critical drop height causing drop detachment in the ADSA-NA configuration, this is beyond the scope of this thesis. Studies [24, 111] have been performed

124 110 for the stability analysis of similar drop configurations. However, the stability analysis for the drop configuration used by ADSA-NA would be more complicated because additional parameters, e.g. contact angle and contact radius, would need to be considered. In addition, contact angle hysteresis would increase the complexity of the analysis. As an example, Table 6.5 shows the surface tension of different drops of dodecane formed with different pedestal widths with different volumes and different pedestal heights. Comparing the values of the measured surface tension with the value of the surface tension measured with pendant drop experiments for dodecane in Table 6.1 shows that bigger drop heights yield lower surface tension errors. To illustrate more generally the effect of height on the accuracy of the measured surface tension, surface tension versus drop height is shown in Figures for all drops of DMCPS, dodecane, cyclododecatriene, and water. It can be seen that bigger drop heights yield more accurate surface tension values. Figures also show that larger heights can be reached with larger pedestal widths. ADSA-NA assumes that drops are axisymmetric. Although the experiments were performed carefully to avoid experimental errors, some drop asymmetry is unavoidable. While asymmetry is not a controllable experimental parameter, it is anticipated that the large number of images of different drops allows the effect of drop asymmetry on the errors in surface tension to be compared with the effects of the other parameters. The asymmetry of a drop was defined as: x DR x P R x DL x P L (x DR x DL )/2 100 (6.2)

125 111 where x DR, x P R, x DL, and x P L are the x coordinates of the lower right corner of the drop, the pedestal right corner, the lower left corner of the drop, and the pedestal left corner, respectively. It should be noted that the calculated asymmetry only considers the projected area of the drop captured by the camera. Based on Equation 6.2, the asymmetry for most of the drops in the constant volume experiments was less than 2%, and the asymmetry for most of the drops in the constant height experiments was less than 1%. The reason for the larger asymmetry in the constant volume experiments compared to the constant height experiments is that raising up the pedestal in the constant volume experiments may induce vibration in the drop which may lead to uneven retreating of the drop front and hence more asymmetry compared to the case in the constant height experiments where the pedestal location is fixed while the drop front is advancing. To investigate the effects of volume, height, drop asymmetry, and pedestal width on the accuracy of surface tension measurement, a linear regression was performed for the surface tension error versus these parameters for every drop obtained in the constant volume experiments. It is noted that this linear regression is more general than the linear regression in Equation 6.1 that was performed for investigating the effect of volume only. The regression equation was defined as: γ γ 0 = a V/W 3 s + b h/w s + c asy + d W/W s + e (6.3) where γ is the measured surface tension (mj/m 2 ), γ 0 is the true value of surface tension (mj/m 2 ), V is the drop volume (µl), h is the drop height (mm), asy is the asymmetry of

126 112 the drop, W is the pedestal width (mm), and W s is the smallest pedestal width (1.6 mm). Since all the regression variables are non-dimensionalized, the units of all the coefficients of the regression function are mj/m 2. Table 6.6 shows the coefficients (a, b, c, and d) of the regression function for surface tension error fitted to volume, height, asymmetry, and pedestal width for all drops of the sample liquids obtained. Comparing the values of the regression coefficients (Equation 6.3, Table 6.6) for each liquid reveals that the coefficient b is always negative and larger than the other coefficients, and has a 95% confidence interval that does not include zero. This indicates that the height of a drop is the most important factor, among those considered, for the accuracy of the measured surface tension. Bigger drop heights yield lower surface tension errors. The signs of the other coefficients are not consistently positive or negative, and the 95% confidence intervals usually include zero. This implies that there is no significant effect of drop volume, asymmetry, or pedestal width on the accuracy of surface tension measurement. However, forming drops with larger drop heights requires using larger pedestal widths so that pedestal width still has an indirect effect on the accuracy of surface tension measurement. The importance of drop height for surface tension measurement accuracy can be understood as follows. Drop shape methods fit the drop profile to the Laplace equation: κ(z) = κ(0) + ρg γ z (6.4) where κ(z) is twice the mean curvature of the liquid-vapour interface, and κ(0) is the

127 113 mean curvature of the interface at z=0. Thus, the measurement of surface tension by ADSA-NA can be regarded as a linear regression between κ as estimated from the experimental image, and z. The surface tension is obtained from the slope of the best fit line, with ρ and g known. The height of the drop is the range of z over which the regression is performed. A larger range gives a more certain fit, and hence a more accurate measurement of γ. The effect of drop height can also be demonstrated computationally, by integrating the Laplace equation to obtain theoretical drop profiles. Figure 6.11 shows pairs of calculated profiles with boundary conditions corresponding to unattached drops, drops suspended from a needle, and drops suspended from a pedestal. In each pair, the open symbols correspond to a 10% higher surface tension relative to the solid symbols. It is seen that taller profiles are more sensitive to a change in surface tension, i.e., it should be possible to measure the surface tensions of taller drops more accurately. In practice, taller drops can be formed by suspending them from a large wetting pedestal. It was mentioned previously that the best measurement for water surface tension in the data shown in the constant height experiments (Table 6.2) had a discrepancy of 0.3 mj/m 2 with the value measured using pendant drop experiments (Table 6.1). The reason for such a large discrepancy was that the drop height was not big enough in those experiments. Thus, to obtain an accurate value of surface tension for water using the ADSA-NA constellation, an additional constant volume experiment was performed for water using a still larger pedestal width of 6.0 mm. To reach bigger drop heights with this new pedestal, the magnification of the microscope was decreased from 5.8 to 2.9

128 114 to expand the field of view. Different drops of water were formed on Teflon AF 1600 with different volumes and different heights. Figure 6.12 shows examples of different drops of water formed on Teflon AF 1600 using a pedestal width of 6.0 mm. The range of drop heights was mm, and the range of drop volumes was µl in these experiments. Table 6.7 shows the results of ADSA-NA for these experiments. According to Table 6.7, the average value of surface tension is ± 0.04 mj/m 2 at 24 C, in good agreement with the value measured by ADSA using pendant drop experiments (72.39 ± 0.01 mj/m 2, Table 6.1). 6.4 Objective Function Curvature at Its Minimum and the Accuracy of Surface Tension Measurement It is apparent that the surface tension values obtained from ADSA-NA vary widely in accuracy, i.e. the output value of surface tension may or may not be the true value of the surface tension. Therefore, a criterion is needed to judge the quality of measurement, or, even more importantly, to guaranty a certain accuracy for a given image. Extensive experience has shown that a sharper minimum of the objective function (Equation 3.13) or, possibly, a lower value of the objective function at its minimum represent better accuracy for the output value of surface tension. Since the objective function and hence its minimum value depends on the number of drop profile points, the sharpness of the ob-

129 115 jective function should be considered. To illustrate, Figure 6.13 shows a general example of the objective function value versus hypothetical surface tension for two different drops of dodecane with the pedestal width of 1.6 mm. It can be seen that the curvature at the minimum point is higher for the drop with the smaller deviation from the correct value. To investigate the relation between the output value of surface tension and the curvature of the objective function at its minimum, this curvature was calculated by fitting a parabola to the region around the minimum point for all drops of the sample liquids used in the constant volume experiments. Figures illustrate the output value of surface tension versus the curvature at the minimum point of the objective function for all drops of DMCPS, dodecane, cyclododecatriene and water. As expected, higher curvatures indeed yield lower surface tension errors. It can be seen in Figure 6.17 that for most drops of water the output value of surface tension has a discrepancy of more than 0.2 mj/m 2 with the correct value. The reason for such a large discrepancy is that the drop height was not large enough. Correspondingly, the curvature for the water drops is significantly smaller compared to the other sample liquids. Comparing the curvature values with the corresponding output value of surface tensions for all sample liquids reveals that for curvatures larger than m 6 /J 2, the error of the output value of surface tension falls within ±0.2 mj/m 2. To generalize this criterion, the curvature at the minimum point of the objective function versus non-dimensional drop height (drop height divided by capillary length) is shown in Figure 6.18 for all drops of the sample liquids with different volumes, heights, and pedestal widths. For clarity, a parabola was fitted to the curvature values of all drops for each liquid. These parabolas are shown in Figure

130 It can be seen in Figure 6.18 that bigger drop heights yield higher curvature values and hence lower surface tension errors. Figure 6.18 also shows that bigger drop heights are required for liquids with higher surface tensions to reach low errors for the output value of surface tension. According to Figure 6.18, a curvature value of m 6 /J 2 corresponds to a value of 1.8 for the non-dimensional drop height, suggesting this as a minimum non-dimensional drop height in order to reach an error of less than 0.2 mj/m 2. The non-dimensional drop height of 1.8 for DMCPS corresponds to a drop height of 2.5 mm. For dodecane, cyclododecatriene, and water, this drop height is 3.3 mm, 3.5 mm, and 4.9 mm, respectively. It is noted that a pedestal width of 4 mm should be sufficient to form drops tall enough to reach an accuracy of ±0.2 mj/m 2 for surface tension measurement for most common liquids, i.e. liquids with surface tension in the range of mj/m 2, and wider pedestals may be used for high surface tension liquids. Although ADSA-NA was tested for liquids with surface tensions in the range of mj/m 2, it can also be used to measure very low interfacial tensions, e.g mj/m 2 as may occur in liquid/liquid systems. It is noted that pendant drop experiments using common pedestal sizes in the range of millimeters would not work accurately for a liquid/liquid system with a very low interfacial tension and small density difference, e.g gr/cm 3, in normal gravity [112]. In fact, a pendant drop for such a system is not stable [111], i.e. it would detach from the pedestal. However, the ADSA-NA setup allows formation of a stable drop for such a system. As an example, ADSA-NA was applied to an image of a sessile drop suspended from a capillary in a liquid/liquid system with ρ = g/cm 3. The drop image was analyzed by ADSA-NA, and an interfacial tension was

131 117 measured near mj/m 2. It is noted that the drop image in question was obtained from a study outside this thesis.

132 118 1 mm pedestal edge Figure 6.1: Image of a dodecane drop on a Teflon AF 1600 coated surface formed by injecting the liquid through an inverted vertical stainless steel pedestal. Student Version of MATLAB

133 119 (a) (b) (c) Figure 6.2: Images of different drops of DMCPS on Teflon AF 1600 coated surfaces obtained from constant drop height experiments. Drop (a) was formed using a pedestal width of 1.6 mm with a volume of 5 µl. Drop (b) was formed using a pedestal width of 2.8 mm with a volume of 20 µl. Drop (c) was formed using a pedestal width of 3.8 mm with a volume of 35 µl.

134 120 (a) (b) (c) Figure 6.3: Images of different drops of dodecane on Teflon AF 1600 coated surfaces obtained from constant drop height experiments. Drop (a) was formed using a pedestal width of 1.6 mm with a volume of 8 µl. Drop (b) was formed using a pedestal width of 2.8 mm with a volume of 29 µl. Drop (c) was formed using a pedestal width of 3.8 mm with a volume of 50 µl.

135 121 (a) (b) (c) Figure 6.4: Images of different drops of cyclododecatriene on Teflon AF 1600 coated surfaces obtained from constant drop height experiments. Drop (a) was formed using a pedestal width of 1.6 mm with a volume of 12 µl. Drop (b) was formed using a pedestal width of 2.8 mm with a volume of 35 µl. Drop (c) was formed using a pedestal width of 3.8 mm with a volume of 58 µl.

136 122 (a) (b) (c) Figure 6.5: Images of different drops of water on Teflon AF 1600 coated surfaces obtained from constant drop height experiments. Drop (a) was formed using a pedestal width of 1.6 mm with a volume of 11 µl. Drop (b) was formed using a pedestal width of 2.8 mm with a volume of 45 µl. Drop (c) was formed using a pedestal width of 3.8 mm with a volume of 78 µl.