Line Tension Measurements: An Application of the Quadrilateral

Langmuir 1997, 13, 3035-3042

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Line Tension Measurements: An Application of the Quadrilateral Relation to a Liquid Lens System P. Chen, S. S. Susnar, A. Amirfazli, C. Mak, and A. W. Neumann* Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario, Canada M5S 3G8

Downloaded by UNIV OF PRINCE EDWARD ISLAND on September 12, 2015 | http://pubs.acs.org Publication Date (Web): May 28, 1997 | doi: 10.1021/la961077x

Received November 5, 1996. In Final Form: March 12, 1997X The quadrilateral relation derived in a previous paper1 is a generalization of the classical Neumann relation; it can be derived either from the minimum principle of free energy1 or from the generalized theory of capillarity of Boruvka and Neumann.2 We demonstrate that such a quadrilateral relation can be applied to line tension measurements. The experimental design involves the measurement of two contact angles and the diameter of a liquid lens system. As an example, the line tension of the three-phase contact line formed by dodecane, water, and air is found to be (-1.29 ( 0.21) × 10-6 J/m at the 95% confidence level, which is in the same order of magnitude as those obtained through drop-size dependence of contact angle measurements3,4 but with the opposite sign. The negative line tension observed is found to be independent of the diameter of the three-phase contact line ranging from 1 to 7 mm, which conforms with the theoretical assumption of the formulation of the quadrilateral relation.1 The methodology presented in this paper is believed to be simple and efficient.

I. Introduction Line tension is a one-dimensional analogue of surface tension; it is a well-defined thermodynamic quantity for a three-phase contact line.5 The importance of line tension can be found in many areas of technology including nucleation, condensation, stabilization of emulsions and foams by fine particles, microbial adhesion, etc.6-9 Although line tension was first postulated by Gibbs more than 100 years ago,5 it has not been quantified satisfactorily. The sign and the magnitude of line tension still remain controversial.8-12 Gibbs speculated that such a quantity might have a negative value and gave an example of two soap bubbles adhering to each other.13 Langmuir estimated that line tension for an oil/water/air system was about 6.5 × 10-5 J/m.14 Harkins calculated that the line tension of the edge of a lens of oil containing a single row of molecules should be in the order of (1 to 10) × 10-11 J/m.15 Starov and Churaev16 calculated that the line tension acting on the contact perimeter of two drops would be about -1 × 10-10 * To whom correspondence should be addressed. X Abstract published in Advance ACS Abstracts, May 1, 1997. (1) Chen, P.; Gaydos, J.; Neumann, A. W. Langmuir 1996, 12, 5956. (2) Boruvka, L.; Neumann, A. W. J. Chem. Phys. 1977, 66, 5464. (3) Gaydos, J.; Neumann, A. W. J. Colloid Interface Sci. 1987, 120, 76. (4) Li, D.; Neumann, A. W. Colloids Surf. 1990, 43, 195. (5) Gibbs, J. W. The Scientific Papers of J. Willard Gibbs, I. Thermodynamics; Dover: New York, 1961. (6) Toshev, B. V.; Platikanov, D.; Scheludko, A. Langmuir 1988, 4, 489. (7) Pethica, B. A. J. Colloid Interface Sci. 1977, 62, 567. (8) Gaydos, J.; Neumann, A. W. In Applied Surface Thermodynamics; Neumann, A. W., Spelt, J. K., Eds.; Marcel Dekker: New York, 1996; Chapter 4. (9) Gaydos, J. Ph.D. Thesis, University of Toronto, Toronto, Canada, 1992. (10) Gaydos, J.; Neumann, In Applied Surface Thermodynamics; Neumann, A. W.; Spelt, J. K., Eds.; Marcel Dekker: New York, 1996; Chapter 2. (11) Wallace, J. A.; Schu¨rch, S. Colloids Surf. 1990, 43, 207. (12) Wallace, J. A.; Schu¨rch, S. J. Colloid Interface Sci. 1988, 124, 452. (13) Gibbs, J. W. The Scientific Papers of J. Willard Gibbs, I. Thermodynamics; Dover: New York, 1961; p 296. (14) Langmuir, I. J. Chem. Phys. 1933, 1, 756. (15) Harkins, W. D. J. Chem. Phys. 1933, 5, 135. (16) Starov, V. M.; Churaev, N. V. Colloid J. USSR 1983, 45, 852.

S0743-7463(96)01077-3 CCC: $14.00

J/m. Further, Churaev et al.17 produced calculations for the line tension of a thin film on a bulk liquid, and their results were in the order of (-1 to -10) × 10-10 J/m. They suggested that the sign of line tension may change from negative to positive as the profile of the transition zone between the wetting film coating a liquid surface and the meniscus of the bulk liquid changes from gently to steeply sloping.17 Clearly, there is no (theoretical) consensus on the value of line tension, although there is a preponderance for values of the order of 10-10 J/m. Experimental determination of line tension has proven to require considerable ingenuity in experimental design, because of the small magnitude of this quantity.11,12 Torza and Mason18 found line tension to be from (-6 to 58) × 10-9 J/m by measuring the shapes of three singlets occurring in three-phase emulsions. Ivanov, Kralchevsky, and Nikolov19-22 experimentally determined line tension to be (-12 to 3) × 10-8 J/m for small bubbles in solutions of surfactant. Gaydos and Neumann3 found line tension to be (1 to 3) × 10-6 J/m by measuring the drop-size dependence of contact angles on a solid surface. Li and Neumann4 determined line tension to be (1 to 6) × 10-6 J/m by using an improved technique from that of ref 3. For liquid drops placed on a liquid/liquid interface, modified by a monolayer, Schu¨rch et al. found the line tension to be in the order of (1 to 2.4) × 10-8 J/m.11,12 Again, divergent values of line tension are reported by various authors. From the above information, it is apparent that further studies are needed to provide a better understanding of line tension and the parameters that affect its magnitude and sign. One of the problems associated with the experiments mentioned above is that the chosen threephase line system is often complicated, and the theoretical model used in the data analysis is not always straightforward. For example, in the measurement of the drop(17) Churaev, N. V.; Starov, V. M.; Derjaguin, B. V. J. Colloid Interface Sci. 1982, 89, 16. (18) Torza, S.; Mason, S. G. Kolloid-Z. Z. Polym. 1971, 146, 593. (19) Ivanov, I. B.; Kralchevsky, P. A.; Nikolov, A. D. J. Colloid Interface Sci. 1986, 112, 97. (20) Kralchevsky, P. A.; Ivanov, I. B.; Nikolov, A. D. J. Colloid Interface Sci. 1986, 112, 108. (21) Kralchevsky, P. A.; Nikolov, A. D.; Ivanov, I. B. J. Colloid Interface Sci. 1986, 112, 132. (22) Nikolov, A. D.; Kralchevsky, P. A.; Ivanov, I. B. J. Colloid Interface Sci. 1986, 112, 122.

© 1997 American Chemical Society

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The experimental design presented in this paper will be valid essentially for all fluid, three-phase systems; it will allow line tension measurements to be performed on both pure liquid lens systems and systems containing surfactant or a monolayer. Here, we choose a system consisting of a dodecane liquid lens floating on a water/air surface as an example. From the experiment, the line tension value is determined by a straightforward utilization of the quadrilateral relation. II. Theoretical Background


From a thermodynamic point of view, the Neumann triangle relation is a mechanical equilibrium condition for a contact line that is formed by intersecting, adjacent interfaces of a multisurface thermodynamic system. The contact line quadrilateral relation is a modification of the classical Neumann triangle relation, which includes line tension in addition to surface (interfacial) tensions. In vector form, the quadrilateral relation can be written as1,24

γ(1)m1 + γ(2)m2 + γ(3)m3 + σκ ) 0

Figure 1. (a) Schematic of a side view of three interfaces intersecting at a three-phase contact line; γ(1), γ(2), and γ(3) denote the surface (interfacial) tensions, and σκ represents the line tension σ contribution to the force balance at a point of the contact line. (b) A quadrilateral composed of three surface tensions γ(1), γ(2), and γ(3) and a line tension contribution σκ.

size dependence of contact angle on a solid surface, both negative and positive line tension have been claimed.23 One possible explanation for the discrepancies observed in the measured values of line tension is the imperfection of the solid surfaces used in the experiments: surface roughness, heterogeneity (and hence contact angle hysteresis), surface deformability, and reactivity can all alter the contact angles from the values on the corresponding ideal solid surfaces. For three-phase line systems involving surfactant or monolayer formation, the equilibration of surface phases and further line phases becomes an issue and can often complicate measuring equilibrium line tension.21 Recently, we formulated a quadrilateral relation1,24 for three-phase contact line systems; this is a generalization of the classical Neumann triangle relation where line tension effects are included. We further developed a graphical representation of such a quadrilateral relation: three surface tensions associated with three intersecting interfaces and a line tension at the intersection form the four sides of the quadrilateral (Figure 1). From this quadrilateral, we can derive an equation of tensions for specific experimental situations by simple geometrical analysis.1 In this paper, we demonstrate an application of this contact line quadrilateral relation to liquid/liquid/fluid systems. The choice of liquid/liquid/fluid systems can avoid the problems relating to the solid surface preparation in the drop-size dependence of contact angle experiment. (23) Lin, F. Y. H.; Li, D.; Neumann, A. W. J. Colloid Interface Sci. 1993, 159, 86. (24) Gaydos, J.; Boruvka, L.; Rotenberg, Y.; Chen, P.; Neumann, A. W. In Applied Surface Thermodynamics; Neumann, A. W., Spelt, J. K., Eds.; Marcel Dekker: New York, 1996; Chapter 1.

(1)

where γ(1), γ(2), and γ(3) are the interfacial tensions for three interfaces; m1, m2, and m3 are the unit tangents to the three interfaces and represent the directions of the three interfacial tensions, σ is the line tension of the threephase contact line, and κ is the curvature vector of the contact line which represents the direction of the line tension (Figure 1). Equation 1 can be obtained from a reduction of the most generalized Neumann triangle relation given in the generalized theory of capillarity of Boruvka and Neumann.2 This equation can also be derived through a minimization of the free energy1 when only line tension, in addition to surface tensions, is considered for the three-phase contact at equilibrium. If the interfacial tensions under consideration are normal to the tangent direction of the three-phase contact line, the resulting quadrilateral (Figure 1b) will be in a single plane since the curvature vector κ is also normal to the tangent direction of the contact line. For most liquid or fluid systems, such a condition, that interfacial tension is normal to the contact line, is always satisfied. From the graphical representation (Figure 1b), one can easily obtain several scalar forms of the quadrilateral relation, such as equations involving cosines of contact angles.1 Here, we only give the scalar form of the quadrilateral relation that suits our present experimental design. The concept of this design is to use a hanging liquid lens: a three-phase contact line is formed by lowering the pendant liquid drop so that it touches the lower liquid (Figure 2). This contact line represents the intersection of three interfaces among the two immiscible liquids and the ambient gaseous phase (air in the present experiment). Such a design avoids the difficulty of liquidliquid engulfing; it also provides control of the extent of the contact between the two liquids and hence improves the stability of the system. This stability is necessary since an immobile hanging liquid lens is required so that the three-phase contact angles can be detected with high accuracy. To calculate line tension through the quadrilateral relation, three interfacial (surface) tensions are needed, which does not present a problem since liquid-fluid interfacial tensions can be routinely measured to a high degree of accuracy by an appropriate methodology, such as axisymmetric drop shape analysis (ADSA).25-28 In (25) Rotenberg, Y.; Boruvka, L.; Neumann, A. W. J. Colloid Interface Sci. 1983, 93, 169. (26) Cheng, P.; Li, D.; Neumann, A. W. Colloids Surf. 1990, 43, 151.


Line Tension Measurements

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Figure 2. Schematic of a pendant/floating liquid lens in contact with a second liquid. γ(1) and γ(2) are the surface tensions of the pendant liquid and the second liquid; γ(3) is the interfacial tension between the two liquids; σκ represents the line tension contribution to the quadrilateral relation. Contact angles θ1 and θ2 are measured in the experiment, which can be used to calculate necessary contact angles in eq 3 to obtain line tension: The angles in the quadrilateral relation can be obtained from θκ1 ) θ1, θ2κ ) π - θ2, and θ12 ) π - θ1 + θ2.

addition, two contact angles at the three-phase contact line are also required. A convenient way to obtain them is to measure the contact angles θ12 and θκ1 directly, through a physical observation of the slopes of the two interfaces between the two liquids and the air. It is noted that only two contact angles are needed among a total of four contact angles at the three-phase contact line, and the contact angles spanned by the dodecane/water interface (in our present case) do not have to be measured; the difficulty to measure all of the contact angles is then eliminated. With the parameters chosen above, one can readily write a scalar form of the quadrilateral relation from Figure 1b1

[γ(3)]2 ) (σκ)2 + [γ(1)]2 + [γ(2)]2 + 2σκ γ(1) cos θκ1 + 2γ(1)γ(2) cos θ12 + 2γ(2)σκ cos θ2κ (2) where the sum of the three contact angles θκ1 + θ12 + θ2κ ) 2π. The above eq 2 is quadratic with respect to the line tension σ, and its solution may be easily obtained once the curvature κ of the three-phase line is known. The radius of this curvature may be measured by determining the diameter of the liquid lens. This, along with the detection of the two slopes of the interfaces, may be achieved by imaging the cross section of the liquid lens. Solving eq 2 for the line tension σ, one obtains

σκ ) -[γ(1) cos θκ1 + γ(2) cos θ2κ] ( ([γ(1) cos θκ1 + γ(2) cos θ2κ]2 - {[γ(1)]2 + [γ(2)]2 + 2γ(1)γ(2) cos θ12 [γ(3)]2})1/2 (3) Two solutions for the line tension are possible, and it will be shown that one of the solutions can be rejected by comparing contact angles predicted by the classical Neumann triangle relation and those calculated through the quadrilateral relation (see below). It is noted that in the actual experiment, we measure the two contact angles θ1 and θ2, corresponding to the slopes of the air/hanging liquid interface and of the air/ second liquid interface (Figure 2). These two angles are related to the contact angles in eq 3 by θκ1 ) θ1, θ2κ ) π - θ2, and θ12 ) π - θ1 + θ2. III. Experimental Section Materials. The pair of immiscible liquids used were dodecane and water. Dodecane with a purity of 99+% was obtained from (27) Cheng, P.; Neumann, A. W. Colloids Surf. 1992, 62, 297. (28) Kwok, D. Y.; Hui, W.; Lin, R.; Neumann, A. W. Langmuir 1995, 11, 2669.

Figure 3. Schematic of the experimental setup. The pendant drop of dodecane is supported by a stainless steel needle. Upon lowering the needle with a stepper motor, a three-phase contact line is formed among dodecane, water, and air. Aldrich Chemical Co., Inc., USA. Water was distilled; its purity was checked by the surface tension measurement which, at 25 °C, resulted in 72.14 ( 0.04 mJ/m2 at the 95% confidence level (the experimental method employed is described below), comparing well to the standard pure water surface tension. The two liquids were saturated with each other for 24 h before the experiment. Interfacial (Surface) Tensions. The surface tensions of the two liquids and the interfacial tension between them were measured by axisymmetric drop shape analysis (ADSA).25-28 The ADSA technique determines liquid/fluid interface tensions and contact angles from the shape of axisymmetric menisci, i.e., from pendant drops and sessile drops. The principle of such a technique is to fit the experimental drop profile to a theoretical one given by the Laplace equation of capillarity, and the interfacial tension is generated as a fitting parameter through a nonlinear regression procedure. The ADSA program requires several coordinate points along the drop profile, the value of the density difference across the interface, and the value of the local gravitational constant as input. The details of this methodology and its experimental design can be found elsewhere.25-28 To maintain the mutual saturation of the liquids throughout the surface/interfacial tension measurements, the experimental setup of ref 28 is used. Briefly, a quartz cuvette (4.5 × 2.5 × 8 cm) contained the two immiscible liquids where the heavier one sank in the bottom. The drops were created at the tip of a Teflon tube inserted in the appropriate liquid phase. Such a setup allows measurement of surface tension by creating an air bubble in each of the two liquids in the same glass cuvette; the interfacial tension is measured by forming a water drop inside the dodecane phase, while the two liquids are in contact to maintain saturation. Since the density of each of the saturated liquids is likely affected by mutual solubility, an accurate measurement of the density is also required. The density measurement was performed on a digital density meter (Anton Paar DMA 45) at room temperature, 25 °C. The results were 0.7451 g/cm3 for dodecane and 0.9970 g/cm3 for water. With these density values, the interfacial tension between dodecane and water was 52.09 ( 0.04 mJ/m2, the surface tension between dodecane and the air was 23.53 ( 0.06 mJ/m2, and the surface tension between water and the air was 71.93 ( 0.02 mJ/m2. The statistical analysis was based on the sample size n ) 100, and the error limits were given at the 95% confidence level. All these measurements, along with the line tension experiment (see below), were performed at room temperature 25 °C. The values of these interfacial and surface tensions are comparable to the data published in ref 28 where both decane and hexadecane saturated with water were used; the values for dodecane/water are intermediate between those for decane and hexadecane/water systems. Experimental Design. A schematic of the experimental setup is shown in Figure 3. The dodecane drop was created at the tip of a stainless steel needle (Chromatographic Specialties, Inc., Canada) with an outer diameter of 0.3 mm and an inner diameter of 0.15 mm; the needle was connected to a Hamilton gastight syringe (Chromatographic Specialties) which was driven by a programmable stepper motor (Model 18705, Oriel Corp. Stratford, CT). The syringe was supported by a micromanipulator (Leica, ON, Canada) which could be used to position the pendant/ floating dodecane pendant drop with precision. When the needle

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Figure 4. An image acquired for the cross section of a pendant/ floating liquid lens of dodecane in contact with water. was lowered, a dodecane liquid lens was formed on the center of the water surface. The size of the three-phase contact line between dodecane, water, and air could be adjusted by raising or lowering the dodecane drop, by varying the initial size of the drop, or by adding or removing liquid from the drop. The typical drop diameter range used in the present experiment was between 1 and 7 mm. The water was contained in a 2 cm diameter by 2 cm in height glass cylinder with a circular opening which ensures that the liquid lens formed was also circular. To observe the water-air interface from a side view of the pendant/floating liquid lens system, the water level in the glass cylinder had to be above the rim of the cylinder; i.e., the cylinder was overfilled with water.29-31 To prevent possible contamination from airborne impurities, the system was enclosed in a quartz cuvette which was sitting on a leveling stage. The pendant/floating liquid lens was illuminated with a white light source (Model V-WLP 1000, Newport Corp., Fountain Valley, CA) shining through a heavily frosted glass diffuser. The image of the cross section of the liquid lens was obtained by a microscope (Leitz Apozoom, Leica) linked to a monochrome charge-coupled device video camera (Cohu 4810, Infrascan, Inc., BC, Canada). The video signal of the drop was transmitted to a digital video processor (Parallax XVideo board) which performed the framegrabbing and digitization of the image to 640 × 480 pixels with 256 gray levels. A workstation (Sun SPARCstation 10, Sun Microsystems, ON, Canada) was used to acquire the images from the digitization board. An image analysis scheme was then followed (see below). The entire setup, except for the workstation, was placed on a vibration-free table (Technical Manufacturing Corp., Peabody, MA) to isolate the system from external disturbances. The experiments were performed at room temperature, 25 °C. A typical image obtained is shown in Figure 4. With the motorized syringe, both static and dynamic modes were employed in the experiment. In the static mode, the liquid lens maintains its size by fixing the volume of the dodecane drop created at the needle tip. For each static liquid lens, images were captured at 0.5 s intervals initially and progressively less rapidly, up to 1 min intervals, near the end of the run at 6 min. In the dynamic mode, the dodecane drop size is varied by changing the drop volume through the motor-driven syringe. Within a 1 min cycle, the diameter of the three-phase contact line changed from 4.5 to 7 mm. During this size change, images were taken at 0.3 s intervals throughout. Image Analysis. Having acquired the drop image, an image analysis scheme is needed to extract the drop profile and to obtain the profile coordinates of interfaces. The goal in this procedure is to obtain the diameter of the three-phase contact line (i.e., the diameter of the liquid lens) and two contact angles (θ1 and θ2) at the intersecting point of the three interfaces (Figure 2). A program in C language was written to process the digital image of the liquid lens (Figure 4). First, edge detection was performed to extract the profile coordinates of the interfaces. The Sobel edge detection scheme was chosen because of its relative insensitivity to noise and the ability to detect edges well in situations other than horizontal and vertical.32,33 After applying (29) Kwok, D. Y.; Li, D.; Neumann, A. W. In Applied Surface Thermodynamics; Neumann, A. W.; Spelt, J. K., Eds.; Marcel Dekker: New York, 1996; Chapter 9. (30) Neumann, A. W. Adv. Colloid Interface Sci. 1974, 4, 105. (31) Neumann, A. W. PhD Thesis, University of Mainz, Germany, 1962. (32) Pratt, W. K. Digital Image Processing; Prentice-Hall: Englewood Cliffs, NJ, 1979. (33) Rosenfeld, A.; Kak, A. C. Digital Image Processing; Academic Press: New York, 1982.

Figure 5. A drop profile acquired from an image (as in Figure 4) through an image detection scheme. The distance between A and A′ is 6.69 mm, representing the diameter of the liquid lens. The bottom picture is an enlargement of the portion enclosed by a rectangle in the top picture, where 30 pixel points along each of the interfaces AB and AC, starting from the intersection point A, are used in the linear curve-fit. the Sobel edge operator, we obtained a secondary image called a gradient image, from which the edges (corresponding to interfaces) are identified as the arrays of pixels with the steepest intensity (gray-level) gradient. A resulting interface profile is shown in Figure 5. The next step is to find the coplanar points A and A′ in Figure 5; these intersection points correspond to the points at the threephase contact line. The strategy to find the left intersection point A, for instance, is to march along the interface profile from the left side (point C in Figure 5) and search for an abrupt change in the slope of the profile. The monitoring process involves fitting a straight line to the first 10 coordinate points, starting from point C, and then skipping the 10 coordinate points and fitting another line to the next 10 points. If there is a significant difference in the slope of the two fitted lines, say 10°, we know that the intersection point A is located between the two fitted lines. The coordinate of the intersection between these two fitted lines should give the coordinate for the desired point A with adequate accuracy. The same procedure is repeated for finding the right intersection point A′ at the three-phase contact line, but this time starting from the right side of the interface profile (point C′ in Figure 5). The distance between the two intersection points A and A′ gives the diameter of the liquid lens (i.e., the diameter of the three-phase contact line). Finally, to find the two contact angles at each of the two intersection points A and A′, we fit a straight line to the first 30 coordinate points, starting from the intersection point A for instance, along each of the two interfaces that are intersecting at A (Figure 5b). From the slopes of the fitted lines, the two contact angles θ1 and θ2 in Figure 2 are obtained. The same procedure is repeated for the other intersection point A′. The average values for θ1 and θ2 are used in eq 3 to calculate line tension. The reasons for using a straight line fit to 30 coordinate points for finding the contact angles at the intersection points are as follows: Ideally, one would wish to find the tangent to the interface at the intersection, and from this tangent the contact angle would be calculated. One of the methods to find the tangent is to fit a few pixel points at or near the intersection to a straight line along the interface; the slope of the fitted line will represent the tangent. However, as can be seen from Figure 5, it is not possible to use only a small number, say 5 or fewer pixel points,



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Figure 6. The slope of the fitted straight line to the interface AB in Figure 5 against the number of pixel points, which indicates a steady value of the slope, also with a relatively small fluctuation, in the region of 20-40 pixel points. The errors associated with each slope are the standard errors in the linear curve-fit.

Figure 7. Slope of the fitted straight line to the interface AC in Figure 5 against the number of pixel points, which indicates a steady value of the slope, also with a relatively small fluctuation, in the region of 20-40 pixel points. The errors associated with each slope are the standard errors in the linear curve-fit.

because the resolution of the profile is only to the closest pixel. This results in Figure 5, e.g., to the left of point A, in horizontal strings of pixels. To obtain adequate averaging a number of pixels significantly larger than five must be used. By inspection of the interface profile in Figure 5, we also see that each interface is actually curved; hence, using too many pixel points would also lead to errors. To seek an optimal number of pixel points used in the linear curve-fit, we plot linear curve-fit results against the number of pixel points for a typical drop profile (Figures 6 and 7). Figure 6 shows the slope of the fitted straight line to the pixel points along the interface AB, starting from the intersection A, as in Figure 5. It is seen that error limits are very large when using too few pixel points (less than 10); as more pixel points (more than 40) are used, the slope is declining progressively. This latter observation is also perceived in the curvature along the interface AB in Figure 5. Only within the region where between 20 and 40 pixel points are used, are constant slopes, also with relatively small error limits, observed. The contact angle variation corresponding to the slope fluctuation within the 20 to 40 pixel point region is found to be less than 0.3°, after converting the slope to the angle. As will be shown later, such a small contact angle change does not affect the results presented in this paper. Therefore, 30 pixel points were chosen for the linear curve-fit. A similar pattern is found in Figure 7, where the slope of the fitted straight line to the pixel points along the interface AC of Figure 5, starting from the point A, is plotted against the number of the pixel points. Again, a relatively steady slope is found in the region of 20-40 pixel points. When more pixel points are used in the curve-fit, the slope starts to increase progressively; this is also perceived in the curvature along the interface AC in Figure 5. As a consequence from observing Figures 6 and 7, 30 pixel points were chosen in the linear curve-fitting procedure. Another reason for using a 30 pixel point linear curve-fit is that the 30 pixel point fit provides excellent linear correlation coefficients for each of the two intersecting interfaces (AB and AC in Figure 5), typically, better than 0.995 for the interface AC and 0.9 for the interface AC along with extremely high confidence levels (>99.9%). It might be suggested to fit a higher order polynomial to the interface profile and then to calculate the slope of the fitted polynomial at the intersection point; however, we found that the higher order polynomial fit was too sensitive to the pattern of pixels (see Figure 5) in the profile coordinates of the interface, resulting in contact angle fluctuation of well above 1° when a slightly different number of pixel points (say, 32 instead of 30) was used for a third- or fourth-order polynomial fit. Therefore, the linear curve-fit to 30 pixel points along the interface profile seemed optimal. Line Tension Calculation. With the three interfacial (surface) tensions, the two contact angles, and the radius of curvature of the three-phase contact line, the line tension can

be calculated by using eq 3, for each image acquired. In eq 3, γ(1), γ(2), and γ(3) correspond to the interfacial tensions of dodecane/ air, water/air, and dodecane/water, respectively. It is noted that it is necessary to make a choice for the sign in front of the square root on the right-hand side of eq 3. However, we had no a priori knowledge of line tension. Both of the signs were then used to calculate the line tension value initially. The next step was to determine which value of line tension was reasonable. Having the line tension value, one could calculate all the contact angles including θ3κ (Figure 2). Projecting the quadrilateral relation into two orthogonal directions, one of which follows the horizontal line tension direction, yields1

cos θ3κ ) -

1 [σκ + γ(1) cos θκ1 + γ(2) cos θ2κ] γ(3)

(4)

and

sin θ3κ )

1 (1) [γ sin θκ1 - γ(2) sin θ2κ] γ(3)

(5)

Thus, the contact angle θ3κ can be determined. The resulting θ3κ values are expected to be drastically different for the two line tension values. On the other hand, we know that the existence of line tension does not change contact angles by large amounts from those determined from the classical Neumann triangle relation.3,4 Therefore, by using the three surface tensions obtained above, we could calculate the contact angles with the classical Neumann relation and compare the results with those calculated from eqs 4 and 5. The one of the two θ3κ values which significantly differs from the prediction of the classical Neumann triangle relation will be rejected; see below.

IV. Results To determine the appropriate sign in eq 3, by using the data extracted from an image as shown in Figure 4 for the air/dodecane/water system, we calculated the two line tension values approximately as -1 × 10-6 and 8 × 10 -5 J/m corresponding to the negative and positive signs preceding the square root term on the right-hand side of eq 3, respectively. By applying eqs 4 and 5, we obtained θ3κ values of 13° and 167°, corresponding to the minus and plus signs. On the other hand, suppose the line tension can be neglected, i.e., the classical Neumann triangle relation can be used. We calculate the contact angle θ13 () θ3κ + θκ1 in Figure 2) between the dodecane/air interface and the dodecane/water interface, yielding a value of 39°. With

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Chen et al. Table 1. Line Tension versus Diameters of the Liquid Lens diameter (mm)

line tension

95% conf limits

1.84 2.31 3.47 4.55

-1.43 -1.03 -1.39 -1.16

0.03 0.16 0.05 0.15

6.91 3.82a

-1.42 -1.29a

0.11

0.21b


a

Average. b 95% confidence limit for the average.

Figure 8. The diameter, two contact angles corresponding to the slopes of the dodecane/air interface and water/air interface (relating to the contact angles in Figure 2 by θ1 ) θκ1 and θ2 ) θκ1 + θ12 - π), and line tension plotted as a function of time up to 6 min. The values of the diameter and line tension are entered into Table 1 for comparing to other liquid lenses of different diameters.

the contact angle θ1 ()θκ1 ∼ 28°) observed from the slope of the dodecane/air interface, we obtain θ3κ ≈ 39 - 28 ) 11°, which is close to θ3κ ≈ 13° for the minus sign in eq 3. Because the contact angle θ3κ ) 167° corresponding to the plus sign is drastically different from the prediction from the classical Neumann triangle relation, this choice of sign is then rejected. Consequently, the negative sign was used in calculating line tension from eq 3 for the present system. It is worth noting that in using eq 3 to calculate line tension, only two contact angles, θ1 and θ2, are needed; see Figure 2. The contact angles spanned by the dodecane/ water interface do not have to be measured; hence, the difficulty to determine the position of this interface is avoided. Figure 8 illustrates the results of diameter, two contact angles θ1 and θ2 (cf. Figure 2), and line tension calculated from eq 3. The data are shown as a function of time in a static experiment where the dodecane drop volume was kept constant. Within the 6 min span, there is no obvious trend of change in the data. Small fluctuations occur in each of the four graphs in this figure, which correspond to less than 10% error limits at the 95% confidence level. Particularly, the error limits for the two contact angles θ1 and θ2 are better than (0.1°, based on the analysis of more than 30 images of a static drop and the use of the Student t distribution. It is noted that all the line tension values are negative, resulting in an average of (-1.42 ( 0.11) × 10-6 J/m at the 95% confidence level for the liquid lens with a diameter of 6.91 mm. These values are also entered into Table 1 for comparison with the results from other liquid lenses of different diameters. With a change in the dodecane drop volume, the diameter (and hence the curvature) of the three-phase

Figure 9. The diameter, two contact angles corresponding to the slopes of the dodecane/air interface and water/air interface, and line tension versus time in a dynamic mode, where the drop size is controlled by adjustment of the drop volume of dodecane. The intercept at time zero of the fitted straight line to the line tension values yields a line tension of (-1.39 ( 0.05) × 10-6 J/m (0.05 is the standard error limit in the linear regression), comparable to the values in the static experiment (Table 1).

contact line changes. In the static mode, line tensions were monitored as shown in Table 1. Again, the values of line tension for all five runs, with diameters ranging 1-7 mm, are negative, on the order of 10-6 J/m. From the table, no apparent trend in line tension is observed with the drop size variation. Averaging these five line tensions results in (-1.29 ( 0.21) × 10-6 J/m at a 95% confidence level (Table 1). Figure 9 illustrates line tension as a function of time in a dynamic experiment, during which the diameter of the three-phase contact line was varied in a periodic mode from 4.5 to 7 mm. Each cycle took approximately 1 min, and two cycles are presented in Figure 9. Although the change in the diameter induces a periodic variation in the contact angles, especially for θ1, the line tension essentially remains constant. The steady, slight decrease in the line tension is attributed to a gradual accumulation of impurities (e.g., from the air) during the experiment. Applying linear regression to the line tension values yields an intercept value for the line tension at (-1.40 ( 0.05) ×


Langmuir, Vol. 13, No. 11, 1997 3041


10-6 J/m (0.05 is the standard error from the linear regression), which is comparable to the line tension value from the static experiment (Table 1). This intercept value of line tension represents the value at time zero when the experiment starts, and hence it is assumed to be least subject to impurities. An interesting feature in Figure 9 is the drop-size dependence of the contact angle, i.e., the effect of diameter variation on the contact angle. Intuitively, one may adopt a similar analysis used in the drop-size dependence of contact angle for a sessile drop on a solid surface,3,4 i.e., to use the modified Young equation to analyze the dropsize dependence in Figure 9. The fact that the increase in the contact angle (θ1 in Figure 9) corresponds to the increase in the drop size (represented by the diameter in Figure 9) indicates a negative value for the line tension. This confirms that the line tension in the present system is negative. V. Discussion By using the automatic image analysis scheme, we are able to analyze a large amount of data, which all result in a negative line tension of the order of 10-6 J/m (Table 1, Figures 8 and 9). To confirm this image analysis scheme, we also performed manual digitization for an image as shown in Figure 4. In this manual digitization, we first chose ten pixel points along an interface, starting from an intersection point and, by linear curve-fitting to these ten points with a least-squares procedure, found the slope of the interface. Then, we converted the slope into a contact angle; we then measured the distance between the two intersection points which represents the diameter of the liquid lens. Finally, with the contact angles and the diameter found, we calculated the line tension by using eq 3. The resulting values are comparable to the values reported in Table 1 for three runs of the experiment in a static mode; except that the scatter in the line tension value is from (-1 to -3) × 10-6, which is larger than that of the automatic image analysis scheme (Table 1). This indicates that the automatic image analysis scheme can not only process a large amount of data points in a relatively short period of time but also produce better accuracy than a manual digitization scheme. In Figure 5, which results from the automatic image analysis scheme, one sees that the interface AC contains two steps within the first 30 pixel points from the intersection point A. There are 5, 15, and 9 pixel points in consecutive horizontal sections; the slope of a fitted straight line will then be affected by the positions of these two steps. This is reflected in Figure 9, where the scatter of the contact angle bands is observed. The line tension values show a corresponding scatter. Another possible source of error might be the surface and interfacial tension determination. The ADSA technique is able to determine surface and interfacial tensions typically to (0.04 mJ/m2 at the 95% confidence level, with the present setup. These error limits will translate into an error of the order of 10-8 J/m in the line tension value, which is 2 orders of magnitude smaller than the values of the line tension found. We also used the surface/interfacial tension values reported28 for several other alkane/water systems to interpolate the values for the dodecane/water system. The resulting surface/interfacial tensions are 23.71, 71.55, and 51.51 mJ/m2 for dodecane/air, water/air, and dodecane/ water interfaces. With these values, we recalculated the line tension corresponding to Figure 8, yielding (-1.23 ( 0.29) × 10-6 J/m. This indicates that the errors from the surface/interfacial tensions are minimal, not resulting in substantial change in the line tension value.

Figure 10. Quadrilateral diagram with four sides representing three surface (interfacial) tensions (dodecane/air, water/air, and dodecane/water) and a line tension contribution (dodecane/ water/air three-phase contact line) in proportion, using the contact angles obtained from the actual experiment (corresponding to Figure 4). Because of the minuteness of the line tension relative to the surface tension, the resulting quadrilateral diagram looks like a classical Neumann triangle (top portion of the figure). With a magnification of 100 times, the contribution of this small, negative line tension can be seen, which constitutes the fourth side of the quadrilateral (bottom portion of the figure). The negative line tension contribution to the quadrilateral diagram results in crossing of the two surface tension sides, although the topology of the quadrilateral remains intact.

The line tension obtained in the present experimental design is -1.29 × 10-6 J/m (Table 1, with the diameter of the liquid lens ranging from 1 to 7 mm). This line tension has the same order of magnitude as that resulting from the drop-size dependence of contact angle measurement on a solid, although with the opposite sign.3,4 However, this value differs from others11,12,15-22 by orders of magnitude, where an order of 10-10 J/m is typically reported. From the perspective of the results reported here, such a result would mean that line tension would not be detectable, i.e., σ ) 0. In order to demonstrate that our observation of the shape of the liquid lens (Figure 4) cannot be explained in the absence of line tension or with line tension only of an order of 10-10 J/m, the following calculations are performed. First, we calculate the contact angle with the classical Neumann triangle relation, i.e., setting the line tension to zero in eq 2. Then we compare this calculated contact angle with the angle measured in the present experiment. For a typical liquid lens, the classical Neumann relation gives the contact angle θ12 ) 152.9° (cf. Figure 2), but the experimental value θ12 ) π - θ1 + θ2 ) 155.7 ( 0.2° (data from Figure 4) at the 95% confidence level. Obviously, the difference in the contact angle of 2.8° is an order of magnitude larger than the error limits in the experiment, and hence the possibility of zero line tension is ruled out. With the line tension measured at -1.29 × 10-6 J/m (Table 1), we may draw a quadrilateral with four sides representing the three surface (interfacial) tensions and the one line tension term in proportion, and using the contact angles obtained from the actual measurement (Figure 10). As expected, the resulting quadrilateral (top portion of Figure 10) looks essentially the same as a classical Neumann triangle since the line tension value

3042 Langmuir, Vol. 13, No. 11, 1997

is small and the line tension contribution is negligible relative to the surface tension terms in the quadrilateral. However, with a magnification of 100 times, we can show the contribution of the line tension, which, while large compared to some published data, is small compared to the surface tension contribution (bottom portion of Figure 10). It is seen that the negative line tension contribution to the quadrilateral diagram results in crossing of the two surface tension sides, although the topology of the quadrilateral remains intact. Summary


The quadrilateral relation, as a generalization of the classical Neumann triangle relation, is applied to a liquid lens system. The line tension of the three-phase contact line formed by dodecane/water/air three phases is determined to be (-1.29 ( 0.21) × 10-6 J/m at a 95% confidence level, with the diameter of the liquid lens ranging from

Chen et al.

1 to 7 mm. This value of line tension is supported by both static and dynamic measurements on the present liquid lens system. As assumed in the formulation of the quadrilateral relation,1,24 there is no drop-size dependence of line tension observed with the diameter of the liquid lens ranging from 1 to 7 mm. The experimental design presented is able to provide relatively stable and reproducible line tension measurements, and this design in principle works for all three-phase contact line systems involving only fluid phases. Acknowledgment. This research was supported by the Natural Sciences and Engineering Research Council of Canada (Grant No. A8278), an Ontario Graduate Scholarship (P.C.), and a University of Toronto Open Fellowship (P.C.). LA961077X

Line Tension Measurements: An Application of the Quadrilateral

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