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Langmuir 2000, 16, 2024-2031

Measurements of Line Tension for Solid-Liquid-Vapor Systems Using Drop Size Dependence of Contact Angles and Its Correlation with Solid-Liquid Interfacial Tension A. Amirfazli,† S. Ha¨nig,‡ A. Mu¨ller,‡ and A. W. Neumann*,† Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Rd., Toronto, Ontario, M5S-3G8, Canada, and Department of Chemistry, University of Greifswald, Soldmannstrasse 16, D-17487, Greifswald, Germany Received May 18, 1999. In Final Form: October 4, 1999 According to the modified Young equation, one of the direct ways of measuring line tension for solidliquid-vapor systems takes advantage of the dependence of the contact angle on the radius of curvature of the three-phase line. This approach was used to determine line tension measurements for six organic liquids on two different self-assembled monolayer (SAM) surfaces (mixed monolayers of methyl and carboxylic acid terminated alkanethiol on gold at two different composition levels). Low-rate advancing contact angles were measured using Axisymmetric Drop Shape Analysis-Profile (ADSA-P). The general trend observed for each system was that the contact angle decreases as the radius of the three-phase line for the sessile drop increases from approximately 1 to 5 mm. It was found that the drop size dependence of contact angles can be interpreted as being due to a positive line tension. The line tension values are all in the order of 10-6 J/m. Also a correlation was observed between the line tension and the solid-liquid interfacial tension for a series of liquids on each of the two surfaces. This observation conforms to the predictions from a phase rule discussion for capillary systems.

Introduction From a mechanical point of view, line tension is defined as the operative force along the so-called three-phase line. A three-phase line is the intersection of three interfaces, e.g., the periphery of the contact circle of a liquid drop that is placed on a solid surface and is surrounded by a vapor phase. Similar to surface tension, i.e., the tensile force encountered where two bulk phases meet, line tension is a well-defined thermodynamic property. However, unlike surface tension, it is not a well-quantified property; experimental values in the literature range from 10-11 to 10-5 N.1-15 The forces reported in some studies are tensile and in others, compressive, i.e., line tension values are reported with positive and negative signs, respectively. A positive line tension operates to constrict the length of the three-phase line, whereas a negative * To whom correspondence should be addressed. † University of Toronto. ‡ University of Greifswald. (1) Vesselovsky, V. S.; Pertzov, V. N. Zh. Fiz. Khim. 1936, 8, 245. (2) Torza, S.; Mason, S. G., Kolloid Z. Z. Polym. 1971, 246, 593. (3) Good, R. J.; Koo, M. N. J. Colloid Interface Sci. 1979, 71, 283. (4) Platikanov, D.; Nedyalkov, M.; Scheludko, A. J. Colloid Interface Sci. 1980, 75, 620. (5) Platikanov, D.; Nedyalkov, M.; Nasteva, V. J. Colloid Interface Sci. 1980, 75, 620. (6) Kralchevsky, P. A.; Nikolov, A. D.; Ivanov, I. B. J. Colloid Interface Sci. 1986, 112, 132. (7) Gaydos, J.; Neumann, A. W. J. Colloid Interface Sci. 1987, 120, 76. (8) Li, D.; Neumann, A. W. Colloids Surf. 1990, 43, 195. (9) Drelich, J.; Miller, J. D.; Hupka, J. J. Colloid Interface Sci. 1993, 155, 379. (10) Drelich, J.; Miller, J. D. J. Colloid Interface Sci. 1994, 164, 252. (11) Duncan, D.; Li, D.; Gaydos, J.; Neumann, A. W. J. Colloid Interface Sci. 1995, 169, 256. (12) Aveyard, R.; Clint, J. H. J. Chem. Soc., Faraday Trans. 1995, 91, 175. (13) Gu, Y.; Li, D.; Cheng, P. J. Colloid Interface Sci. 1996, 180, 212. (14) Chen, P.; Susnar, S. S.; Amirfazli, A.; Mak, C.; Neumann, A. W. Langmuir 1997, 13, 3035. (15) Amirfazli, A.; Kwok, D.; Gaydos, J.; Neumann, A. W. J. Colloid Interface Sci. 1998, 205, 1.

line tension tends to expand it. In theoretical studies, most of the estimates for the magnitude of line tension verge on the lower limit of the experimental range.16-21 Theoreticians often assert that line tension can have either a positive or a negative sign. There are practical as well as theoretical reasons to measure line tension. First, the magnitude of line tension determines whether it is a relevant factor in technological applications such as dropwise condensation in heat transfer, stabilization of emulsions and foams by fine particles, and cell adhesion to implants. If the line tension is relatively small, about 10-11 N, then its practical role would be limited. Second, models used in theoretical calculations often involve simplifications and hence uncertainty.22 For instance, depending on how the contribution of short-range forces to the free energy of the three-phase line is considered in the statistical mechanics approach, different signs for line tension are obtained (see ref 19 and the references therein). Thus, by comparing experimental results with theoretical predictions, the appropriateness of the simplifying assumptions and the models describing the molecular interactions could be verified and/or perhaps refined. The wide range of experimental results for line tension is viewed by some as irreconcilable; it is assumed that accepting one value would preclude others. However, some of the inconsistent line tension results published may simply reflect the difference between the systems studied. For instance, it is plausible that line tension values obtained from one system, e.g., solid-liquid-vapor, would (16) Gershfeld, N. L.; Good, R. J. J. Theor. Biol. 1967, 10, 1. (17) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Oxford Science Publications: New York, 1984; p 240. (18) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827. (19) Indekeu, J. O. J. Mod. Phys. B. 1994, 8, 309. (20) Tarazona, P.; Navascues, G. J. Chem. Phys. 1981, 75, 3114. (21) Vignes-Adler, M.; Brenner, H. J. Colloid Interface Sci. 1985, 103, 11. (22) Toshev, B. V.; Platikanov, D.; Scheludko, A. Langmuir 1988, 4, 489.

10.1021/la990609h CCC: $19.00 © 2000 American Chemical Society Published on Web 12/10/1999

Measuring Line Tension for Solid-Liquid-Vapor Systems

differ from the results of another system, e.g., liquidfluid-fluid, because of differences in the molecular structures/interactions in the three-phase zone23,24 (similar to the range of surface tension values). The primary goal of this study is to provide further evidence to support our findings in ref 15. The magnitude of line tension in ref 15 was found to be in the order of 10-6 N with a positive sign. Artifacts that may affect the drop size dependence of the contact angles including contortions of the three-phase line will also be discussed. Finally, we will explore experimentally a possible correlation between line tension and the solid-liquid interfacial tension. Various test liquids in a range of surface tension are used on two different solid surfaces, and numerous contact angles are measured using Axisymmetric Drop Shape Analysis-Profile (ADSA-P).15,25 Theoretical and Experimental Considerations The starting point for the determination of line tension is to establish the mechanical equilibrium condition at the three-phase line. From a thermodynamic perspective, the mechanical equilibrium conditions for systems with bulk, surface, and line phases can be found by minimizing the total free energy of the system. For a system comprising a sessile drop placed on an ideal solid surface (i.e., rigid, homogeneous, horizontal, and smooth) and surrounded by a gas phase, the mechanical equilibrium condition at the three-phase line will be the modified Young equation.26,27 If the system has a constant line tension (similar to surface tension) and restricted to moderate curvatures, the modified Young equation can be written as follows:

γlv cos θ ) γsv - γsl - σκgs

(1)

where γlv, γsv, and γsl are liquid-vapor, solid-vapor, and solid-liquid interfacial tensions, respectively; θ is the contact angle, σ is the line tension, and κgs is the geodesic curvature of the three-phase line, i.e., the curvature in the plane of the solid phase. For the system under consideration, the sessile drop will be axisymmetric, i.e., the three-phase line is circular; therefore, the geodesic curvature will be equal to the reciprocal of the radius of the three-phase circle (κgs ) 1/R). For a very large drop (i.e., in the limit of Rf∞), the geodesic curvature will be zero, and hence the line tension term will vanish in eq 1. Thus, the classical Young equation is recovered:

γlv cos θ∞ ) γsv - γsl

(2)

where θ∞ is the contact angle corresponding to an infinitely large drop. To reduce the number of parameters needed to obtain the line tension from eq 1, this equation can be combined with eq 2 and after substitution for κgs has the following form26,27:

cos θ ) cos θ∞ -

σ 1 γlv R

(3)

Equation 3 suggests the following two points. First, the contact angle, θ, depends on the drop size, R, if σ * 0. (23) Napiorkowski, M.; Koch, W.; Dietrich, S. Phys. Rev. A. 1992, 45, 5760. (24) Koch, W.; Dietrich, S.; Napiorkowski, M. Phys. Rev. E. 1995, 51, 3300. (25) del Rio, O. I.; Neumann, A. W. J. Colloid Interface Sci. 1997, 196, 136. (26) Neumann, A. W.; Spelt, J. K., Eds. Applied Surface Thermodynamics; Marcel Dekker: New York, 1996; Chapts. 3 and 4. (27) Boruvka, L.; Neumann, A. W. J. Chem. Phys. 1977, 66, 5464.

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Second, it stipulates that there should be a linear relationship between the cosine of contact angle and the reciprocal of the drop base radius, providing that both line tension and liquid-vapor interfacial tension are constant. In a plot of cos θ vs 1/R the slope of a straight line fitted to the data points is set equal to the term -σ/ γlv. The liquid-vapor interfacial tension (γlv) can be determined from a separate experiment or its literature value can be used. The sign of line tension will also be determined by the slope of the fitted line; a negative slope will result in a positive line tension and a positive slope implies a negative line tension (γlv is always positive). The measurement of line tension through drop size dependence of contact angle may appear to be simple conceptually; however, as discussed in the literature (e.g., refs 10, 15, 22, and 28) there are intricate matters that need attention. A previous article15 showed that unsuitable contact angle methodologies or failure to satisfy underlying assumptions of the modified Young equation may be responsible for inconsistent line tension results reported for similar systems, e.g., in refs 3,7-10, and 29. In deriving the modified Young equation, in the form presented in eq 3, the following requirements for the selected solid surface apply: smoothness, homogeneity, and rigidity. Careful preparation of Self-Assembled Monolayers (SAM) of long-chain alkanethiols on gold produces a well-packed monolayer surface that is smooth, homogeneous, and stable.30-32 Thus, the above-mentioned conditions for the applicability of the modified Young equation are expected to be satisfied by using carefully prepared SAM surfaces. In our previous study,15 we used SAM surfaces which had been prepared of a single alkanethiol (i.e., 1-octadecanethiol, HS(CH2)17CH3); In the present study, SAM surfaces are composed of two thiol compounds coadsorbed onto gold (i.e., 1-hexadecanethiol, HS(CH2)15CH3, and 16-mercaptohexadecanoic acid, HS(CH2)15COOH). Two kinds of SAM surfaces were produced by changing the composition ratio of the two thiol compounds: (a) 12% HS(CH2)15COOH and 88% HS(CH2)15CH3 and (b) 20% HS(CH2)15COOH and 80% HS(CH2)15CH3. Coadsorption of two thiol compounds with different tail groups enables one to vary the solid surface free energy in a predictable and controlled manner.33 Hence one can expand on and cross-check the observed correlation between line tension and the solid-liquid interfacial tension in our previous study.15 The methodology for measuring drop size dependence of contact angle is important in arriving at meaningful results. This point has been discussed extensively elsewhere.34 Low-rate dynamic contact angle measurement (measurements are conducted as the drop is slowly grown) is used in this study in conjunction with the ADSA-P protocol. ADSA-P is a proven contact angle measurement technique with high objectivity and an accuracy of (0.1° (see the Experimental Section for more details).15,25,35-37 Moreover, (28) Yekta-Fard, M.; Ponter, A. B. J. Colloid Interface Sci. 1988, 126, 134. (29) Drelich, J.; Wilbur, J. L.; Miller, J. D.; Whitesides, G. M. Langmuir 1996, 12, 1913. (30) Bain, C. D.; Troughton, E. B.; Tao, Y.; Evall, J.; Whitesides, G. M.; Nuzzo, R. G. J. Am. Chem. Soc. 1989, 111, 321. (31) Porter, M. D.; Bright, T. B.; Allara, D. L., Chidsey, C. E. D. J. Am. Chem. Soc. 1987, 109, 3559. (32) Finklea, H. O.; Robinson, L. R.; Blackburn, A.; Richter, B.; Allara, D.; Bright, T. Langmuir 1986, 2, 239. (33) Bain, C. D.; Whitesides, G. M. Science 1988, 240, 62. (34) Kwok, D. Y.; Gietzelt, T., Grundke, K.; Jacobasch, H. J.; Neumann, A. W. Langmuir 1997, 13, 2880. (35) Kwok, D. Y.; Lin., R.; Mui, M.; Neumann, A. W. Colloids Surf. A. 1996, 116, 63. (36) Cheng, P.; Li, D.; Boruvka, L.; Rotenburg, Y.; Neumann, A. W. Colloids Surf. 1990, 43, 151.

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Figure 1. Synthesis procedure for preparing 16-mercaptohexadecanoic from 16-oxyhexadenanoic acid.

ADSA-P not only measures contact angle, but also simultaneously drop radius, volume, and liquid surface tension. This additional information in a low-rate dynamic measurement setting was demonstrated15,34,35 to facilitate an evaluation of the quality of measured contact angles; it identifies any slip stick motion (through plots of radius and contact angle versus time) and liquid reaction with the solid surface (through plots of surface tension and contact angle versus time). Such measurements should be disregarded. Low-rate dynamic measurements of contact angles ascertain that advancing contact angles are measured at all times. Experimental Procedure Materials. Unless stated otherwise, all reagents and chemicals were obtained from commercial sources at the highest purity available and used without further purification. House doubly distilled water was used (pH ∼5.5). To produce the solid surfaces, ripple-free premium glass microscope slides were purchased from Fisher Scientific Co.; 16-mercaptohexadecanoic acid was synthesized following the procedure outlined in Figure 1. 16Oxyhexadenanoic acid (1) was reacted with 48% hydrogen bromine in glacial acetic acid to yield 16-bromohexadecanoic acid (2).38 The exchange of bromine was carried out in ethanolic solution with thiurea similar to the procedure used in ref 39. The resulting isothiuronium salt (3) was hydrolyzed to the 16mercaptohexadecanoic acid (4) by treatment with alcoholic sodium hydroxide. To purify the product, it was recrystallized several times from a mixture of ethanol and water; mp 64-65 °C, 1H NMR (CDCl3, APX 300 MHz spectrometer, Bruker) δ 1.21.35 (m, 22H), δ 1.56-1.65 (m, 4H), δ 2.31-2.36 (t, 2H), δ 2.482.56 (q, 2H). Solid Surface Preparation. SAM surfaces were produced as follows. The glass microscope slides were cut into pieces of about 2 × 2 cm. Then a small hole was drilled in the center of each piece (diameter ∼1 mm). This allows the formation of a sessile drop from below the solid surface using a syringe such that the drop volume can be increased in a controlled fashion (see contact angle measurement section). Before the gold film deposition, the glass slides were washed with a copious amount of acetone followed by mild Argon plasma cleaning. The gold films were formed by first sputtering an approximately 30-Åthick layer of titanium as an adhesion promoter between the glass and the gold32 followed by sputtering a 400-Å-thick gold film on top. The bell-jar pressure during sputtering was 12 mTorr. Finally the monolayer was formed by taking the gold substrates out of the sputtering machine and immediately immersing them into ethanol solution of the thiols. Two different ratios of the two thiol components were used to prepare the solutions. The composition ratio of the two thiol species at the monolayer on the gold surface is not necessarily equal to the concentration ratio of the thiols in the solution.40,41 According to Figure 1 of ref 40, to have a monolayer composition of 12% HS(CH2)15COOH-88% (37) Li, D.; Cheng, P.; Neumann, A. W. Adv. Colloid Interface Sci. 1992, 39, 347. (38) Coyle, L. C.; Danilov, Y. N.; Juliano, R. L.; Regen, S. L. Chem. Mater. 1989, 1, 606. (39) Agrawal, K. C.; Bears, K. B.; Sehgal, R. K.; Brown, J. N.; Rist, P. E.; Rupp, W. D. J. Med. Chem. 1979, 22, 583. (40) Bain, C. D.; Whitesides, G. M. J. Am. Chem. Soc. 1988, 110, 6560.

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Figure 2. Schematic of the experimental setup. HS(CH2)15CH3 and 20% HS(CH2)15COOH-80% HS(CH2)15CH3 at the surface, the ethanol solutions should have the following compositions: 20% HS(CH2)15COOH-80%HS(CH2)15CH3 and 40%HS(CH2)15COOH-60%HS(CH2)15CH3, respectively. The total concentration of both solutions was kept at 1 mM. Experience has shown that the faster the gold substrates are taken out of the sputtering machine and immersed into the thiol solutions, the higher the quality of the produced SAM surfaces, as seen by monitoring contact angle hysteresis. After 2 h, the surfaces were taken out of the solution and rinsed copiously with ethanol. The surfaces were stored in absolute ethanol for subsequent use. Before performing contact angle measurements, the samples were removed from the ethanol and dried by a vigorous stream of nitrogen. Contact Angle Measurements. Low-rate dynamic advancing contact angles were measured for sessile drops in vapor saturated air using ADSA-P. Details of the ADSA-P methodology can be found elsewhere (e.g., see refs 25, 36, 37), but the essential points are as follows: ADSA-P is a technique that determines the liquid-fluid interfacial tension, contact angle, drop volume, drop radius, and drop surface area all at once. All it requires as input are the density difference across the liquid-fluid interface and several randomly selected points (e.g., 20) along the profile of an axisymmetric meniscus (e.g., a sessile drop). The profile points are obtained by digital image analysis. To calculate the outputs, ADSA-P fits the Laplace equation of capillarity to the selected points of the experimental profile, using a multiparameter optimization process, which includes interfacial tension and drop apex curvature as parameters. The experimental setup is shown schematically in Figure 2. The drop is formed by pumping the liquid from below the surface through the small hole drilled in the sample using a motor-driven syringe. The drop is centered in the viewing field of the microscope-CCD-camera assembly. A quartz glass cover was used to contain the solid surface and the drop, and hence to produce a vapor-saturated air atmosphere. The drop was illuminated from behind through a frosted glass diffuser to produce a uniform background. Digital images acquired from the drop were stored in the computer for later analysis by ADSA-P. The liquid is pumped into the drop continuously during the experiment leading to an increase in the drop volume and hence steady advance of the three-phase contact line. In a typical experiment, a sequence of images was acquired of a drop growing from a drop radius (R) of about 1 mm up to approximately 5 mm. The lower 1-mm limit is dictated by the size of the small hole drilled in the sample. The upper 5-mm limit is set because there is no or very little change in the contact angle as the radius is increased beyond this limit; the line tension effects are believed to be negligible at such large radii. The growth of the drop will not be uniform because pumping a constant volume of liquid per unit time into a large drop has less effect pumping the same amount into a small drop. Therefore, the time interval for acquiring images was varied (i.e., smaller intervals at the beginning) to obtain an even number of experimental points throughout the range for R. The average rate of advancing was ∼0.45 mm/min. The fact that low-rate dynamic angles rather than statically measured contact angles are used in eq 3 is justified. Studies have shown that there is no difference between proper static and low-rate dynamic contact angle measurements in both capillary rise42 and sessile drop35 configurations. All the (41) Bain, C. D.; Evall, J.; Whitesides, G. M. J. Am. Chem. Soc. 1989, 111, 7155. (42) Kwok, D. Y.; Budziak, C. J.; Neumann, A. W. J. Colloid Interface Sci. 1995, 173, 143.

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Langmuir, Vol. 16, No. 4, 2000 2027

Table 1. Summary of the Experimental Results (22 ( 1 °C)a

liquid

liquid-vapor surface tension (mJ/m2)

line tension (µJ/m)

cis-decalin 1-bromonaphthalene 2,2′-thiodiethanol formamide glycerol water

32.2 44.3 53.8 59.1 65.1 72.7

1.13 1.37 1.45 1.72 2.76 3.81

a

r

significance level for r (%)

contact angle θ∞ (deg)

solid-liquid surface tension (mJ/m2)

no. of experimental measurements

0.91 0.86 0.84 0.79 0.74 0.76

99 99 99 99 99 99

46.4 61.1 74.3 78.2 82.2 96.6

1.2 6.2 11.7 15.4 20.2 33.1

470 573 785 696 591 850

Measurements were performed on mixed SAM surface [CH3 (88%) and COOH (12%)]. Table 2. Summary of the Experimental Results (22 ( 1 °C)a

liquid

liquid-vapor surface tension (mJ/m2)

line tension (µJ/m)

cis-decalin 1-bromonaphthalene 2,2′-thiodiethanol formamide glycerol water

32.3 44.3 53.8 59.1 65.1 72.7

1.06 1.99 1.54 1.56 2.53 3.30

a

r

significance level for r (%)

contact angle θ∞ (deg)

solid-liquid surface tension (mJ/m2)

no. of experimental measurements

0.74 0.79 0.71 0.70 0.73 0.76

99 99 99 99 99 99

38.6 54.1 59.1 62.6 69.8 80.5

0.7 3.5 6.4 8.8 13.7 23.1

335 573 474 788 691 850

Measurements were performed on mixed SAM surface [CH3 (80%) and COOH (20%)].

This strategy should yield a good estimate for the trend of the contact angle changes as the radius of the drop changes. The reasons are as follows: (a) The minute imperfections on each of the surfaces are distributed randomly across the entire surface, causing contact angles at approximately the same radius to be slightly different from run to run. Statistically, to reduce the degree of randomness associated with any measurement, the primary choice is to perform multiple measurements. (b) The degree of reliability and confidence of any measurement is increased as the sample size is increased. Therefore, because of the large number of contact angle data used for each system (see Tables 1 and 2), the outcome of this strategy should produce a high level of confidence (or significance).

Results

Figure 3. Three individual runs for 2,2′-thiodiethanol on three similar mixed SAM [CH3 (88%) and COOH (12%)] surface. measurements were performed at room temperature, i.e., 22 ( 1 °C. It was shown for various liquids on siliconized glass surfaces (which are similar to our surfaces) that dθ/dT is about 0.1 deg/ °C43; therefore, a temperature control of (1 °C is sufficient. Data Handling. Figure 3 shows three experimental runs for thiodiethanol on three similar mixed SAM surfaces. As can be seen in all runs, the contact angle decreases as the drop radius increases; all runs are tracking one another closely, but they are not identical. The reason is that even with careful preparation, producing identical solid surfaces is unattainable under normal laboratory conditions. If one takes the data in Figure 3 and utilizes eq 3 to interpret the observed trend, one would arrive at different estimates for line tension from one run to another. In our previous study,15 we suggested an approach to increase the reliability of the interpretation. Basically, we perform multiple runs for the same liquid on one and the same solid surface (at least six runs) and average the results according to the following scheme. The entire range for R is divided into several bins with the aid of a computer program. A bin is represented by a small interval of R that at least contains one data point from each run between the two vertical lines in Figure 3. The program sets up the bins and then averages the values of the data points from several runs within each bin. The width of bins will not necessarily be equal because our experimental setup is such that it acquires images at fixed times and not at fixed radii of drops. However, in practical terms, this is not a matter of significant consequence. (43) Neumann, A. W. Adv. Colloid Interface Sci. 1974, 4, 105.

Figure 4 shows typical results from averaging at least six runs for cis-decalin and formamide on SAM-1 [12% HS(CH2)15COOH-88% HS(CH2)15CH3] surfaces, and Figure 5 presents results of averaged runs for 1-bromonaphthalene and 2,2′-thiodiethanol on SAM-2 (20% HS(CH2)15COOH and 80% HS(CH2)15CH3) surfaces. Both these figures indicate that the average contact angle decreases with increasing contact radius. To interpret the observed trend, in Figures 6 and 7 the data from Figures 4 and 5 are plotted in terms of cos θ versus 1/R. The correlation coefficient |r| is a measure of the quality of the fit of data, in our case to a straight line. In addition, to determine whether the observed correlation is genuine or due to chance, the significance level for r should be calculated. The higher the significance level, the less the probability of the observed correlation being due to chance. The correlation coefficients and the significance levels of the fitted lines are given in Tables 1 and 2. Considering the number of data involved, the values for the correlation coefficient suggests that a linear correlation between cos θ and 1/R for all systems is appropriate. The significance levels of 99% for the r values indicate that the correlations are genuine and not due to chance. All this considered, the interpretation that the observed drop size dependence of contact angles is due to line tension (cf. eq 3) is statistically well founded. Tables 1 and 2 summarize the findings for all systems studied in this work. By looking at the line tension column in Tables 1 and 2, one can see immediately that all values are positive and in the order of 10-6 J/m. This confirms our previous findings for solidliquid-vapor systems,7,8,11,15 and is in good agreement with

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Figure 4. Averaged results for drop size dependence of contact angles of (a) cis-decalin and (b) formamide on mixed SAM [CH3 (88%) and COOH (12%)] surface.

Figure 5. Averaged results for drop size dependence of contact angles of (a) 1-bromonaphthalene and (b) 2,2′-thiodiethanol on mixed SAM [CH3 (80%) and COOH (20%)] surface.

the results obtained in refs 13 and 44 using techniques other than drop size dependence of contact angles. Finally, we turn our attention to a suggested functional relationship between line tension and solid-liquid interfacial tension. Refs 11 and 45, using the thermodynamic phase rule for moderately curved capillary systems, (44) Nguyen, A. V.; Stechemesser, H.; Zobel, G.; Schulze, H. J. J. Colloid Interface Sci. 1997, 187, 547. (45) Gaydos, J.; Neumann, A. W. Kolloid Z. 1994, 56, 624.

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Figure 6. Interpretation of drop size dependence of contact angles through the modified Young equation for (a) cis-decalin and (b) formamide on mixed SAM [CH3 (88%) and COOH (12%)] surface.

Figure 7. Interpretation of drop size dependence of contact angles through the modified Young equation for (a) 1-bromonaphthalene and (b) 2,2′-thiodiethanol on mixed SAM [CH3 (80%) and COOH (20%)] surface.

stipulate that line tension may be described as a function of only the solid-liquid interfacial tension. There was no suggestion of an explicit functional form, however. A plot of line tension versus solid-liquid interfacial tension for the results of this study is given in Figure 8. It is apparent that the larger the solid-liquid interfacial tension, the larger the line tension measured on both surfaces. The observed trend is in broad agreement with our previous

Measuring Line Tension for Solid-Liquid-Vapor Systems

Figure 8. Line tension versus solid-liquid interfacial tension for all of the studied systems.

reports.11,15,46 Also, Nguyen et al.44 have noticed the same positive correlation between the line tension and the solidliquid interfacial tension. A point to note is that solidliquid interfacial tensions for our systems are obtained by using an equation of state approach.47 Discussion Exclusion of Alternative Causes for Drop Size Dependence of Contact Angles. It has been argued that drop size dependence of contact angles can have causes other than line tension. In the literature it has been attributed to viscous effects, solid surface deformation, thin film effects, and heterogeneity of the surface. All or some of these causes might prove to be operative for some systems; nevertheless, with careful experimentation, it is unlikely that the above-mentioned effects will be the primary reason for the observed drop size dependence of contact angles. A discussion of the exclusion of suggested alternative causes for the drop size dependence of contact angles follows. Viscous Effects. Viscous friction effects are negligible in low-rate contact angle measurements. The measured contact angles at advancing rates between 0 and 1 mm/ min are essentially rate independent.35,42,48 Our average rate of advancing was ∼0.45 mm/min. Solid Surface Deformation. Solid surface deformation is not a likely explanation for the observed dependence of θ on R. This effect is cited to be important only for gel and rubber surfaces.49-51 The surfaces used in this study clearly do not fall into this category. Thin Film Effects. Li and Neumann,52 using a flat thin film model, showed that the effect of a putative thin liquid film on the drop size dependence of contact angles for the type of solids and liquids used in an experiment like ours is in the order of 10-4 deg; this is far less than the observed drop size dependence of contact angles in this study. As a result, even if a film was present, our observations cannot be explained by its effect. Heterogeneity Effects. Heterogeneity due to polar sites can cause three-phase line contortions. It is proposed (e.g., refs 3 and 53) that these contortions induce changes in (46) Amirfazli, A.; Chatain, D.; Neumann, A. W. Colloids Surf. A. 1998, 142, 183. (47) Li, D.; Neumann, A. W. J. Colloid Interface Sci. 1992, 148, 190. (48) Elliott, G. E. P.; Riddiford, A. C. J. Colloid Interface Sci. 1967, 23, 389. (49) Rusanov, A. I. Colloid J. (Transl. Kolloidn. Zh.) 1975, 4, 621. (50) Shanahan, M. E. R. J. Phys. D. 1987, 20, 947. (51) Extrand, C.; Kumagai, Y. J. Colloid Interface Sci. 1996, 184, 191. (52) Li, D.; Neumann, A. W. Adv. Colloid Interface Sci. 1991, 36, 125. (53) Drelich, J.; Miller, J. D.; Good, R. J. J. Colloid Interface Sci. 1996, 179, 37.

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the geometry of the drop that mimics an effect similar to the drop size dependence of contact angles comparable in magnitude to our observation. Arguments of this nature mainly predict a trend opposite to our observation for the dependence of contact angles on the drop size. Moreover, these arguments are chiefly applicable to drops with heights that are no more than 1 order of magnitude larger than the size of the heterogeneous patches.3 The size of drops used in our study is much larger than the typical size for heterogeneous patches, i.e., about 1 µm or less, for well-prepared solid surfaces.54 So even if heterogeneities were present they would unlikely be the source for our observations. Theoretical Support and Model Calculations. The findings of this work are not only corroborated by other experimental evidence, as mentioned earlier, but they also seem to be consistent with some theoretical predictions. Sagis and Slattery55 expanded the approach of Li and Slattery56 using a singular perturbation scheme to describe the long-range molecular forces, and concluded that line tension is responsible for the dependence of contact angle on drop size. The system that Sagis and Slattery examined was similar in size to ours (i.e., sessile drops of the order of millimeter). Their theoretical calculations for n-alkane on a Teflon surface also yielded positive line tension of in the order of 10-6 J/m. A wide range of experimental values are published for line tension,1-15 and there are arguments in the literature10,53,54 that line tension values near the upper range of the experimental data, e.g., 10-6 J/m, are not acceptable, because they are a manifestation of the contortions of the three-phase line due to surface heterogeneties. Such arguments mainly stand on the basis that microscopic contortions of the three-phase line would add to the total energy of the line phase. The matter that three-phase line contortions will add to the total energy of the line phase is not disputed here; however, the arguments largely lack theoretical or experimental quantification. Therefore, assertions that line tension values can be caused to have an apparent value larger by up to 4 orders of magnitude than a perceived value of, say, 10-10 J/m is mere speculation. With use of a model calculation, we will show that arguments such as the one above will not hold for systems with only a limited amount of heterogeneity. The limited heterogeneity is taken as the case where the total area covered by patches which make the surface heterogeneous is less than, for example, 3% of the total area of the surface and the patch sizes are about 1 µm or less. Figure 9 shows a typical image of the three-phase line for a drop of 1-bromonaphthalene on SAM-2 surface. The image shown in Figure 9B was magnified at 128× and its resolution is better than 1 µm; the three-phase line appears to be smooth, which supports the assumption of the limited heterogeneity. Our model is based on the one presented in ref 57, which describes the effect of corrugation of the three-phase line on the drop size dependence of contact angles. The main difference between the two models is that, unlike in ref 57, the line tension is assumed to have a unique value for both the main and the patch (representing “impurities”) material. This simplification is not expected to have any impact on the present consideration, which concerns the (54) Pompe, T.; Fery, A.; Herminghaus, S. Langmuir 1998, 14, 2585. (55) Sagis, L. M. C.; Slattery J. C. J. Colloid Interface Sci. 1995, 176, 173. (56) Slattery, J. C. Interfacial Transport Phenomena; SpringerVerlag: Berlin, 1990; p 150. (57) Li, D.; Lin, F. Y. H.; Neumann, A. W. J. Colloid Interface Sci. 1991, 142, 224.

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Amirfazli et al.

Figure 10. Schematic for the model heterogeneous surface. The contorted three-phase is represented by the solid line. Table 3. Model Calculations for the Effect of Three-Phase Line Contortions on the Drop Size Dependence of Contact Angles for Several Different Model Heterogeneous Surfaces

Figure 9. Typical segments of three-phase line for a drop of 1-bromonaphthalene on mixed SAM [CH3 (80%) and COOH (20%)] surface at two different magnifications. (a) 5×; (b) 128×. The resolution is better than 1 µm in the image shown in B.

order of magnitude rather than small differences. The other inconsequential difference is that our model heterogeneous surface (square patches distributed evenly on the surface such that they form a pattern of concentric rings, see Figure 10) is defined differently from the one in the ref 57. This difference is inconsequential because all the assumptions made in defining the model in ref 57 are also met by our model surface. The reason to opt for such a model surface is 2-fold. First, it simplifies calculating R1 values (see below); second, it may be perceived as being more realistic. Without discussing the details of arriving at the relationship below (see ref 57 for details), we only mention that the value of K (the ratio of line tensions on the main material to the impurity) is set equal to 1 in eq 9 of ref 57. As a result that equation will be simplified as follows: cos θ ) γlv[R1(R1 + R2 + R3) cos θ1∞ + R1R2 cos θ21∞] - σ(R1 + R2) γlv(R1 + R2)(R1 + R3)

(4) where θ is the apparent phenomenological contact angle; R1 and R2 are the radii of curvature of the three-phase line on the main material and the impurity patches, respectively; R3 is the apparent radius of the drop; θ1∞ and θ2∞ are the contact angles on the main and the patch material, respectively. The illustrative nature of the model in ref 57 should be kept in mind because it assumes that the contortions of the three-phase line are represented by circular arcs on each of the main and the patch portions of the surface. In using eq 4, the following assumptions were made. The radius of curvature for the three-phase line on the patches, R2, is assumed to be half the patch size. To calculate R1, the radii of curvature for the threephase line on the main material, the following approach is taken. Based on the assumed patch size, 0.5 µm, for example, and the percentage of ‘impurities’, 1%, the number of patches on the periphery/edge of a drop with

patch size (µm)

R2 (µm)

R1 (µm)

∆θ (σ ) 2 × 10-6 J/m)

∆θ (σ ) 2 × 10-8 J/m)

3.0

1.50

1.0

0.50

0.5

0.25

0.1

0.05

7.15 13.50 19.70 2.40 4.50 6.55 1.20 2.25 3.30 0.25 0.45 0.65

0.797 0.776 0.766 0.785 0.766 0.758 0.781 0.763 0.756 0.775 0.760 0.754

0.030 0.032 0.032 0.015 0.016 0.016 0.012 0.012 0.012 0.009 0.008 0.008

a particular contact radius, 1.5 mm, can be calculated. Using simple geometrical relations, the distance between the evenly distributed patches on the periphery of the drop can be calculated (remember that square patches are forming a pattern of concentric rings). R1 is then simply taken as the half of this distance. Table 3 summarizes the R1 values for each patch size corresponding to 3%, 1%, and 0.5% (from top to bottom in each row, respectively) “impurities” on the surface. If the three-phase line is not contorted, i.e., for a homogeneous solid surface, eq 3 can be used to calculate the difference between the contact angles of two drops with different radii (∆θ) as a function of line tension. The ∆θ for two drops with radii of 1.5 mm and 4.5 mm would be 0.75° and 0.01°, if the line tension values are taken as 2 × 10-6 J/m and 2 × 10-8 J/m, respectively. The values above for ∆θ are calculated assuming a θ∞ value of 70° and a liquid surface tension (γlv) of 72.8 mJ/m2. In the next step, if there are high-energy patches (i.e., polar sites as often reported) with contact angles (θ2∞) of 10°, for example, the three-phase line will be contorted. To calculate ∆θ, the change between the macroscopic, apparent contact angles, for this case, eq 4 should be used. Table 3 shows the calculated ∆θ values from eq 4 for two cases where σ is of the order of 10-6 J/m and 10-8 J/m, respectively. All other parameters used in the calculations are the same as the ones used in the previous calculations involving eq 3, i.e., θ1∞ ) 70° and γlv ) 72.8 mJ/m2. The last column of Table 3, where σ is assumed to be in the order of 10-8 J/m, shows that the introduction of threephase line contortions did not have a notable impact on drop size dependence of contact angles (represented by

Measuring Line Tension for Solid-Liquid-Vapor Systems

∆θ). The ∆θ values are very similar to the ∆θ values calculated from eq 3, i.e., 0.01°. In other words, the threephase line contortions did not mimic the effect of a larger line tension than 10-8 J/m, which would have required a substantial increase in the value of ∆θ, i.e., 0.75°. The patch sizes in Table 3 represent typical values expected for a well-prepared surface. Similar calculations using various values for θ1∞, θ2∞, and σ yielded similar results for drops in the millimeter size range. Therefore, we conclude that the line tension values obtained in our present and previous studies, e.g., ref 15, possibly perceived to be relatively large, are not artifacts of commonly cited three-phase line contortions. In the second last column of Table 3, in which the line tension is assumed to be in the order of 10-6 J/m, the ∆θ values to some small degree depend on the size and the distribution of the heterogeneous patches. Noting that in real surfaces the minute imperfections will be randomly distributed, one can expect that there will be some scatter of the data from run to run depending on the distribution of the imperfections. Therefore, averaging of the results from several runs, as we argued earlier, should help reveal the general trend more clearly. According to this model calculation, it seems that averaging is not only warranted but also necessary. Contact Angle Hysteresis and Solid Surface Quality. The measured contact angle hysteresis for water on both of the mixed SAM surfaces was approximately 10°. One may argue that this level of contact angle hysteresis is indicative of roughness and/or heterogeneity of these surfaces, and conclude that these surfaces are not as suitable as single species SAM, which had a contact angle hysteresis of 6-8°. A possible explanation to counter such arguments is as follows. The polar tail group (e.g., carboxylic acid) of alkanethoils will restructure after being in contact with liquid.58,59 This restructuring means that (58) Evans, S. D.; Sharma, R.; Ulman, A. Langmuir 1991, 7, 156.

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the receding contact angle would be observed on a modified surface, i.e., the difference in advancing and receding contact angles would be at least partially a “true” hysteresis. There is also experimental evidence40,41 that at the present composition levels the two species in the monolayer do not phase segregate into macroscopic domains; hence, the surface can be considered homogeneous. Also, it is important to note that the molecules used for these surfaces are of the same chain length. Conclusions We established that a drop size dependence of contact angle for drops smaller than ∼5 mm in radius exists for several organic liquids on two different SAM surfaces. Viscous friction effects, solid surface deformation, thin film effects, or heterogeneity of the solid surface were ruled out as possible causes. Statistical analysis of the data substantiated that there is indeed a linear correlation between the cosine of the contact angle and the inverse of the drop radius. Therefore, through the modified Young equation, drop size dependence of contact angles was interpreted as a line tension effect. Line tension values for solid-liquid-vapor systems are positive and their magnitude is of the order of 10-6 J/m. A model calculation was used to show that moderate three-phase line contortions cannot cause the relatively large change of contact angle with drop size. The trend of an increase of line tension with increasing solid-liquid interfacial tension was reconfirmed. Acknowledgment. This investigation was supported by the Natural Sciences and Engineering Research Council of Canada under grants no. A8278 and no. EQP173469 (A.W.N.), and by a University of Toronto open fellowship (A.A.). LA990609H (59) Lee, T. R.; Carly, R. I.; Biebuyck, H. A.; Whitesides, G. M. Langmuir 1994, 10, 741.