Determination of line tension for systems near wetting

Journal of Colloid and Interface Science 265 (2003) 152–160 www.elsevier.com/locate/jcis

Determination of line tension for systems near wetting A. Amirfazli,a,∗ A. Keshavarz,b L. Zhang,b and A.W. Neumann b a Department of Mechanical Engineering, University of Alberta, Edmonton, AB T6G 2G8, Canada b Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, ON M5S 3G8, Canada

Received 19 September 2002; accepted 6 May 2003

Abstract Contact angle measurements for three n-alkanes, heptane, octane, and nonane, on two different self-assembled surfaces (SAM) are reported as a function of drop size. These liquids all formed low contact angles (below 20◦ ); the measurements were performed using an accurate method for systems with low contact angle, ADSA-D. The observed drop size dependence of the contact angles was interpreted using the modified Young equation. It was concluded that the observed drop size dependence of contact angles was due to line tension. The choice of systems also provided the opportunity to examine the behavior of the line tension for systems near wetting (i.e., low contact angles). It was determined that the line tension is positive and ranges from below 10−7 to just below 10−6 J/m for the systems studied; the observations suggested that the line tension decreases as the contact angle decreases and likely vanishes at complete wetting.  2003 Elsevier Inc. All rights reserved. Keywords: Line tension; Wetting; Contact angle; Drop size dependence; Drop shape analysis

1. Introduction Since line tension (σ ) influences contact angle (θ ) values, for systems near wetting, depending on its magnitude and sign, it may delay or accelerate the onset of complete wetting (i.e., free spreading of liquid on a substrate, where θ = 0). It is important to address the issue of line tension near wetting, as processes such as lubrication are closely related to wetting. As complete wetting is approached the three-phase line begins to disappear and the question may arise of whether line tension vanishes as well, i.e., σ = 0. The answer is not necessarily obvious, because a priori it is not a thermodynamic requirement for line tension to vanish when the three-phase line ceases to exist. Nevertheless, this situation is similar to the disappearance of the interface between two bulk phases at the critical point, where interfacial tension (γ ) could have a limiting value; thermodynamics does not require γ to be zero (in other words, one could not automatically presume γ = 0 at the critical point). Both experiments and theoretical reasoning (statistical mechanics) had to be used to show that the interfacial tension indeed vanishes. The situation described has prompted considerable interest on the part of theoreticians. To our knowledge thus far very little experimental effort has been reported in this area. * Corresponding author.

E-mail address: [email protected] (A. Amirfazli). 0021-9797/$ – see front matter  2003 Elsevier Inc. All rights reserved. doi:10.1016/S0021-9797(03)00521-6

Before entering into the discussion of line tension near wetting, it is important to note that generally in the transition from a partial wetting regime where the contact angle is nonzero to a wetting regime when θ is zero, two main paths are possible. (1) First-order wetting: if the spreading coefficient, S (where S = γsv − [γsl − γlv ]), vanishes with a discontinuous first derivative at the wetting transition, the transition is called first-order. In other words, in the case of a firstorder wetting transition from the partial wetting configuration, where a drop is in contact with an adsorbed film, a sudden transformation occurs to a step configuration where a thin film is connected with a much thicker film. (2) Critical or continuous wetting: this is a common mode of wetting in which the transition from partial to complete wetting is continuous; in other words, in a critical wetting transition there are no coexisting thin and thick films, but only a single uniform wetting layer with θ = 0. The critical wetting is qualitatively different from the first-order wetting and, therefore, the behavior of line tension in these wetting modes is not necessarily similar. Churaev, Starov, and Derjaguin were among the first researchers who dealt with the question of the behavior of line

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tension when the contact angle approached zero, i.e., in the vicinity of the wetting transition [1,2]. They argued that line tension is negative near the wetting transition and vanishes as the first power of the contact angle at wetting; i.e., σ ∼ θ . The membrane method of Derjaguin was used to model the system, which was composed of a drop resting on a solid surface covered with an adsorbed thin film. To simplify the calculations, it was assumed that in the three-phase zone the meniscus profile sloped gently from the drop to the flat thin film. This assumption could be fairly accurate for a particular mode of wetting transition, i.e., critical wetting, but it may not be a valid assumption for a more common mode of first-order wetting [3]. They also chose to represent the disjoining pressure by a simple linear function of film thickness. The result of simplified calculations by the above three authors is in direct conflict with the findings of Joanny and de Gennes [4]. Joanny and de Gennes predicted that line tension is positive near wetting and its value diverges to infinity as wetting approaches, i.e., σ ∝ ln 1/θ . The theory employed by de Gennes was essentially equivalent to that of Churaev et al., but a more refined disjoining pressure isotherm was used. Joanny and de Gennes used the disjoining pressure isotherm suggested by Scheludko [5] for simple van der Waals fluids, where long-range forces are dominant. The Hamaker constant appearing in the disjoining pressure isotherm was assumed to be positive at all times. Perhaps it should not be entirely surprising that de Gennes and Churaev et al. had arrived at dissimilar answers when one considers the comment in Ref. [1]: “one should hesitate to ascribe too great a significance to one form or the other of the analytical relationships, since the results are obtained using special shapes for disjoining isotherms, and simplifications and/or limitations are invoked during the derivation.” It is also worth noting that the outcome of de Gennes’s model calculation would to a certain extent depend on the selected cutoff distance. The cutoff is defined as the distance from the three-phase confluence zone where the contribution of long range forces to line excess properties will be significant; the cutoff distance is usually selected based on a judgment call. Consensus about vanishing of line tension at wetting seemed to take form when Widom et al. [6,7], in agreement with Churaev et al., argued that indeed line tension is negative near wetting and it vanishes at wetting. The independent work of Widom et al. was based on mean-field theory and they used a local approach where short-range forces were operative. In the so-called local approach, the intermolecular forces are modeled only for the molecules present at the region where all phases meet (in other words, the zone created as a result of truncation of three interfaces at the three-phase confluence zone). The assumption is that interfaces outside the contact region are effectively in their isolated states (not influenced by the imbalance of intermolecular forces in the three-phase zone). Modeling of the three-phase line using a local approach has been questioned by a number of researches, e.g., [1,3,8–11], because the excess free energy contribution from outside of the region where three phases

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meet and mingle is neglected. However, the work of Widom et al. was significant in that for the first time the issue of the mode of the wetting transition was put forward [3]. In particular, they interpreted their results in the context of firstorder wetting. Interestingly, however, after a short period, Szleifer and Widom [12] asserted that line tension changes from a negative to a positive value near wetting and its value perhaps diverges toward infinity at wetting (closely in agreement with similar predictions by Joanny and de Gennes). The basic approach of Szleifer and Widom did not differ fundamentally from that of Widom et al. [6,7] except in that in Ref. [12], compared to Refs. [6,7], a more refined free energy function was used (free energy functions describe the molecular interactions within the three-phase confluence zone). It is important to note that Szleifer and Widom had to argue qualitatively the divergence of line tension at wetting since application of the more refined free energy function posed computational difficulties (the smallest value for contact angle which could be reached was ∼13◦ ). They explained the discrepancy between their two different findings as being due to the mode of wetting transition. Szleifer and Widom [12] stated that the wetting transition in their work is the “normal” first-order wetting, where a thin film is in contact with a macroscopically thick film at wetting; they argued that in earlier studies [6,7], the system experienced a “special” case of first-order wetting, where a thin film was in contact with another film of microscopically finite thickness. In simpler terms, what this meant may be explained as follows: depending on the postulated form of the free energy density, Ψ , and the geometry of the space over which Ψ is defined, different results are possible. In 1992 Varea and Robledo [13], using the lattice mean field theory of the nearest neighbor 3D Ising model, investigated the behavior of line tension near wetting for systems with dominant short-range forces. The model calculations turned out to be complicated and a numerical scheme had to be used. Nevertheless, computational difficulties were encountered and their calculations could not be completed down to the actual wetting transition (the lowest contact angle was ∼13◦ ). Verea and Robledo, on the basis that their calculated line tension values increased sufficiently rapidly with decreasing contact angles, qualitatively concluded that the value of line tension would diverge to infinity at wetting. Also in 1992, Indekeu [11], on the basis of an interface displacement model, argued that line tension displays singular behavior at wetting. His calculations showed that the behavior of line tension near wetting depends on (1) the mode of wetting transition and (2) the range of intermolecular forces. For instance, in first-order wetting, depending on the intermolecular forces, line tension does not vanish, but either diverges or reaches a maximum positive value in the form of a cusp-like singularity. According to Indekeu, in this case, the three-phase line changes in nature and converts to a region of macroscopic inhomogeneity at wetting. At critical wetting, however, line tension vanishes as wetting approaches; in this case it is believed that the three-

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phase line also vanishes. Later in a review paper Indekeu [3] discussed how his findings can be used to interpret and reconcile certain aspects of the conflicting results obtained by different groups regarding the behavior of line tension near wetting. Essentially his arguments are similar to those of Dobbs [14] and center on the fact that the behavior of line tension near wetting is influenced by a range of parameters. Using several examples, both Indekeu and Dobbs have shown that each of the previous findings could be justified, if it was viewed and compared with other models using a similar set of parameters. For instance, Indekeu [3] showed that his interface displacement model would result in behavior for line tension near wetting, i.e., σ (0) ∝ −θ , similar to that predicted by Churaev and co-workers [1,2] and early papers by Widom et al. [6,7], if the special case of critical wetting, where short range forces are dominant, were considered. In another example, it was shown that in the case of first-order wetting with short-range intermolecular forces, the interface displacement model predicts that line tension increases from a negative value to a positive value near wetting and reaches a maximum finite value at wetting. Indekeu asserts that this is qualitatively consistent with the findings by Szleifer and Widom [12] and Varea and Robledo [13]; he states although line tension plots (σ vs θ ) in Refs. [12,13] seem to indicate that σ is diverging at wetting (note that in both studies, due to computational problems, σ values were not available near wetting), but the data can also be considered consistent with the finding of the interface displacement model which predicts a finite limiting value for σ . The overall picture suggests that a few changes in the construction of a model have a profound effect on the outcome of the line tension behavior near wetting. There seem to be credible reasons behind the approaches and the assumptions made; however, the subject is far from being clear. While the interface displacement model appears to have brought about consensus with respect to certain aspects of line tension behavior near wetting, Indekeu acknowledges that his model is still too phenomenological to be satisfactory from a more fundamental point of view. His description for line tension, based on an assumed profile of the meniscus, is only a simplistic representation of a more realistic free-energyfunction-type analysis. Also, gravity and curvature effects are left out in almost all of the above analysis, which according to Refs. [9,12,15] might alter the results in a significant way. Determination of line tension for systems with low contact angles can perhaps provide the much-needed experimental evidence to sort through and verify the extent to which the above predictions are valid. This paper, therefore, presents some exploratory experiments and results. The scope of this paper will be the study of solid–liquid– vapor systems. Our method of determining line tension for solid–liquid–vapor systems takes advantage of the dependence of contact angles on the drop size according to the modified Young equation. This approach allows the line tension to be measured directly using available sophisticated contact angle and drop size measurement techniques such

as axisymmetric drop shape analysis—diameter (ADSA-D). Whereas in some other methods, e.g., Ref. [16], the presence of surfactants is necessary, there is no need for them in our method; hence the complications (i.e., nonequilibrium) introduced by their presence are avoided.

2. Theoretical background To describe thermodynamically a multiphase, multicomponent system, e.g., a sessile drop on a solid surface surrounded by vapor, the fundamental equations for all phases, i.e., bulk, surface, and line, should be considered collectively. In other words, any particular configuration (state) of the aforementioned composite system should be in compliance with the set of fundamental equations for bulk, surface, and line phases. For instance, in determining the equilibrium state for the above composite system, the equilibrium principle should be applied to the collection of the fundamental equations, i.e., bulk, surface, and line. Since determination of line tension at equilibrium is of primary interest, we focus on presenting the working relationships for the equilibrium state. Gibbs’ approach, the minimum energy principle, is the most widely used method to determine the equilibrium state for capillary systems. The minimum energy principle characterizes the equilibrium state as having minimum total energy for a given total entropy of a system. It can be shown that minimizing the total energy for the aforementioned system results in the following equation for the mechanical equilibrium condition at the three-phase line (interested readers can find the mathematical details of the derivation in Ref. [17] and Chap. 1 of [18]), γlv cos θ = γsv − γsl − σ κgs ,

(1)

where γ refers to the interfacial tension and subscripts l, s, and v refer to liquid, solid, and vapor phases, respectively; κgs is the geodesic curvature of the three-phase line, which in the present case is the curvature in the plane of the solid surface. It is important to note that in deriving the modified Young equation (i.e., Eq. (1)) the solid phase is assumed to be ideal; i.e., it is considered to be rigid, smooth, and homogeneous. Consequently, the sessile drop would assume an axisymmetric shape, which means the three-phase line is circular. As a result, the geodesic curvature will be equal to the reciprocal of the radius of the three-phase circle (i.e., κgs = 1/R). It is also assumed that the system possesses moderately curved boundaries, in addition to the assumption that the only external body force present is gravity. In the above equation there are still too many unknowns to make it suitable as a working relationship for determining line tension in an experiment. To develop Eq. (1) into a working relationship, one can use the classical Young equation, i.e., Eq. (2), which describes the relation between contact angle and interfacial tensions of a large drop, i.e., when

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R → ∞: cos θ∞ =

γsv − γsl . γlv

(2)

Combining Eqs. (1) and (2) will yield the sought after working relationship as 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. Second, there should be a linear relationship between the cosine of the contact angle and the reciprocal of the drop base radius, provided that both line tension and liquid–vapor interfacial tension are constant. Therefore, an experiment to determine line tension can be conceived, as follows. The contact angle (θ ) for drops of various size (R), placed on a specific solid surface, can be measured and the results may be plotted in terms of cos θ versus 1/R. Then to determine line tension, the term −σ/γlv is equal to the slope of a straight line fitted to the data points. 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, as γlv is always positive. 3. Experimental procedure 3.1. Contact angle measurements For capillary systems with low contact angles (below 20◦ ) the profile of the sessile drop becomes progressively flatter, and the precision of methods relying on the meridian profile of the drop decreases rather significantly (e.g., goniometer, ADSA-P). The decrease in precision is primarily due to the difficulty in acquiring accurate coordinate points along the drop profile (or increased subjectivity in aligning a tangent at the point of contact with the solid surface in the case of a goniometer). In response to experimental demands in measuring low contact angles, ADSA-CD (contact-diameter) was developed by Skinner et al. [19]. In this technique the sessile drop is viewed from the top rather than the side, and hence the flatness of the drop does not reduce the precision of contact angle measurement. Moreover, since the entire contact line can be viewed when the drop is imaged from the top, the technique lends itself readily to averaging procedures and therefore it can be used to determine the average contact angle, if the drop shape is not perfectly axisymmetric. ADSA-CD or, more precisely, the newer version of this program, ADSA-D (diameter) (Chap. 10 of [18]), was used in this study to investigate the drop size dependence of contact angles for low-contact-angle systems. The required input data for ADSA-D are the liquid–fluid surface tension, the density difference across the liquid–fluid

Fig. 1. Schematic of the experimental setup for ADSA-D.

interface, the drop volume, the gravitational acceleration constant, and the contact or equatorial diameter of the drop, the latter in the case of θ > 90◦ . The contact diameter, however, is measured automatically from the image of the drop (by using image processing schemes); i.e., one does not need to enter a value for contact radius (unlike the other parameters above). The contact angle is calculated by fitting a series of Laplacian profiles to the experimentally observed drop contact diameter. An optimization procedure yields the contact angle which, for a given drop volume and liquid surface tension, provides the best fit to the experimental circumference of the contact circle. The accuracy of contact angle measurement is as high as 0.1◦ . A schematic of the experimental setup for ADSA-D is shown in Fig. 1. Before any measurements are taken, the stage where the solid surface is placed was leveled using a sensitive Starett bubble level and adjusting the set screws of the platform. A freshly prepared solid surface is then placed on the stage. Using a Chempette digital micropipettor a sessile drop was formed on the surface. The volume of the drop was adjusted to the desired value (0–5 µl), by means of the adjusting screw, before drop deposition. The micropipettor was calibrated in a gravimetric experiment. Using the micropipettor at a set volume, a drop was formed and weighed; knowing the density of the liquid, the drop volume was calculated from its weight. The procedure was repeated for several drops of different sizes to obtain a calibration curve for the micropipettor. A precision scale (Mettler H20) with a precision of 10−5 g was used. Using this scale the typical error for calculated drop volume would be less than 1.5%; it was shown [21] that a 1.5% error in the volume value used for ADSA-D input would cause an error significantly below 0.1◦ , which is the accuracy limit for ADSA-D methodology. The calibrated volume reading from the digital micropipettor was used as the input volume for ADSA-D. An image was acquired immediately after drop deposition. Image acquisition was usually a team effort (one person de-

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positing the drop while the other, seated at the computer, controlled the image acquisition timing) to minimize the effect of any evaporation of the drop, which may be important for accurate imaging of tiny drops with low contact angles. The elapsed time between drop deposition and image acquisition was normally under 3 s. It should be noted that evaporation of a drop over a period of time means that the system is not necessarily in static equilibrium; however, application of ADSA-D will allow us to capture the equilibrium contact angle at the moment the drop was placed on the surface. As mentioned above, ADSA-D uses the contact diameter to calculate the contact angle. Contact diameter of a drop would not change over some time even if conditions were such that evaporation occurs. The drop initially hinges on contact points/line; i.e., no change in the contact diameter will be observed, which is the geometrical input for calculation of contact angle by ADSA-D. Only when evaporation is allowed for extended period of time (well over 3 s), the contact diameter will shrink (receding contact line), which will result in inappropriate contact angle measurement. Droplet images were acquired from above using a vertically mounted Wild Heerburg M7S Zoom stereomicroscope fitted with a Cohu 4800 CCD camera and a coaxial incident illuminator (Volpi Intralux 5000 fiber optic system). Magnification of the microscope was selected so that the droplet would cover as large an area of the viewing field as possible (see Fig. 2). This would ensure a consistent accuracy level for the image processing module of ADSA-D. Therefore, it was necessary to change the microscope magnification from 6 to 32 times during the experiment depending on the drop size. At each magnification, of course, the experimental setup should be calibrated. The calibration was performed using a test slide having circular reference disks of known diameter. At each magnification an image was acquired from one of the cir-

Fig. 2. The top view of a sessile drop in an ADSA-D experiment. The white crosses at the edge of the drop are points which are manually selected to be fitted to a circle representing the contact circle for the drop. This information is used by ADSA-D for contact angle calculations. The liquid used is decane and the solid surface is FC 732. The contact angle is approximately 70◦ .

Table 1 Physical properties of test liquids (all data provided by supplier, except surface tension) Name Heptane Octane Nonane

Supplier

Purity (%)

bp (◦ C)

Surface tensiona (mJ/m2 )

Sigma–Aldrich Aldrich Aldrich

99+ 99+ 99+

98 125–127 151

19.9 21.4 22.7

a Surface tension value at room temperature (22 ± 1 ◦ C).

cular disks that occupied the largest possible area in the viewing field of the monitor. The file containing the coordinates of the points along the edges of the circular disk, as well as the exact disk size (in centimeters), was passed to ADSA-D, where it is used to calibrate the experimental setup and determine the physical size of droplets. The experimental setup as a whole is placed on a heavy granite table to minimize vibration. The platform where the solid surface is placed allows translation in the X and Y directions in the plane of the surface so that another drop can be placed on the surface and imaged. At least four drops of the same size for each liquid were used to measure contact angles for each type of solid surface. The measurements were performed at room temperature (21–23 ◦ C); the effect of temperature on surface tension is on the order of 0.1 mJ/m2 for every 1 ◦ C for n-alkanes (the liquids used in this study) [20]. This means that a 2◦ change in temperature would induce an ∼1% change in the surface tension of n-alkanes in general. It was asserted in [21] that when using ADSA-D, surface tension variations (or errors) of up to 5% cause deviations of the order of only 0.1◦ in contact angle. Therefore, the level of temperature control was satisfactory in view of the size of the observed effects; see below. 3.2. Materials Three different liquids were used in the contact angle measurements. They were obtained from commercial sources with high purity and used without further purification. Table 1 summarizes the relevant physical properties of the test liquids; except for the surface tension values, which were obtained experimentally in our laboratory using ADSA-P, the data are those provided by the manufacturers. The materials used to produce self-assembled monolayer surfaces were as follows: gold 99.99% pure (Target Materials Inc.); ripple-free premium glass microscope slides (Fisher Scientific Co.); 99.99% pure titanium (Target Materials Inc.); absolute ethanol (Commercial Alcohols Inc.); 1-hexadecane thiol, CH3 (CH2 )15 HS (Aldrich, 98%, mp 18–20 ◦ C, bp 184–191 ◦ C). 16-Mercaptohexadecanoic acid COOH(CH2 )15 HS was synthesized following the procedure outlined in [22]. 3.3. Solid surface requirements and preparation The quality of the solid surface plays a key role in obtaining meaningful contact angle data in the context of the

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Young equation, i.e., Eq. (2). Excessive roughness and/or chemical heterogeneity of the solid surface could mask or distort the contact angle effects sought in this study. It is therefore important to produce solid surfaces of sufficient quality to ensure that the contact angles accurately reflect the interaction between bulk, surface, and line phases as described by the modified Young equation, i.e., Eq. (3). Aside from smoothness and homogeneity requirements, solid surfaces should be stable and chemically inert with respect to the test liquids; as well, they should be of sufficiently low surface energy to result in contact angles of the desired magnitude. Self-assembled monolayers of long-chain alkanethiols (e.g., 1-hexadecane thiol) on a gold film placed on a silicon or glass substrate were used (e.g., [23,24]). Gold and sulfur bond strongly, which results in stable monolayer surfaces. Moreover, by changing the tail group of the molecule forming a monolayer, the surface free energy (wettability) can be manipulated without a fundamental change in surface structure (smoothness and/or homogeneity) [25,26]. Two kinds of SAM surfaces were produced: a methylterminated SAM surface composed of 100% 1-hexadecane thiol (HS(CH2 )15 CH3 ) and a mixed SAM surface made of 12% HS(CH2 )15 COOH, and 88% HS(CH2 )15CH3 . The preparation procedure for these surfaces was described in detail elsewhere [22,27]. Atomic force microscopy (AFM) was used to determine the roughness of the surfaces produced. The tested surfaces were generally very smooth with a mean roughness under 2 nm over an area of 5 by 5 µm. AFM phase images were produced for the mixed SAM surfaces. These images indicated that at the composition ratios used, methyl and carboxylic acid thiols do not phase segregate into different domains; i.e., the surface is a homogeneous mixture of methyl and carboxylic acid terminated thiols.

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Table 2 Examples of the reproducibility of contact angle data for 1.4-µl drops of heptane on methyl-terminated SAM surfaces and 8.2-µl drops of nonane on mixed SAM surface Drop No.

1

2

3

4

Contact angle (◦ ) for 1.42-µl drops of heptane Contact angle (◦ ) for 8.22-µl drops of nonane

5.8

5.6

5.7

5.5

14.5

14.7

13.5

14.6

Note. Contact angles were measured using ADSA-D.

Fig. 3. Averaged results for drop size dependence of contact angles from four individual runs for heptane on methyl-terminated SAM surface.

4. Results and discussion The findings of two separate sets of experiments each using a different solid surface are reported. Three alkane liquids were used on mixed SAM and methyl-terminated SAM surfaces; the alkanes formed drops with low contact angles (generally under 20◦ and as low as 4◦ ). Replicate experiments were performed for each system. Table 2 shows the measured contact angles for 1.4-µl drops of heptane placed on methyl-terminated SAM surfaces along with 8.2-µl drops of nonane on mixed SAM surfaces. As can be seen, the reproducibility of data is generally satisfactory given the size of the effect observed (e.g., see Fig. 3). Table 3 summarizes all averaged contact angle data and the corresponding averaged drop radii. Figures 3 and 4 show typical results for heptane on a methyl-terminated SAM surface and octane on the mixed SAM surface, respectively. Each data point in these figures is the average value from the experimental runs. The trend of the data in the top frames of Figs. 3 and 4 indicates clearly that the larger the drop

Fig. 4. Averaged results for drop size dependence of contact angles from four individual runs for octane on mixed SAM (12% COOH and 88% CH3 ) surface.

radius, the smaller the contact angle. To interpret the data from the top frames of Figs. 3 and 4 in terms of the modified Young equation, Eq. (3), these data (i.e., θ vs R) are replotted in terms of cos θ versus 1/R in the bottom frame. Equation (3) stipulates that, if the drop size dependence of contact angles is caused by a line tension effect, there should

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Table 3 Summary of average contact angle (θ ) and drop radius (R) from all runs along with their 95% confidence intervals for all systems studied System

Contact angle Contact angle Radius Radius 95% 95% conf. level (cm) conf. level θ (◦ )

Heptane on methyl-terminated SAM

7.56 6.81 6.20 5.65 5.41

0.19 0.19 0.16 0.18 0.14

0.119 0.163 0.231 0.284 0.385

0.010 0.016 0.021 0.011 0.015

Octane on methyl-terminated SAM

13.41 10.85 10.46 10.26 9.57

0.32 0.28 0.43 0.26 0.30

0.104 0.153 0.195 0.232 0.321

0.004 0.013 0.052 0.001 0.014

19.41 18.40 17.74 16.69 16.28

0.61 0.53 0.53 0.43 0.48

0.128 0.184 0.209 0.249 0.312

0.005 0.015 0.024 0.025 0.011

6.35 5.65 5.45 5.17 4.56

0.20 0.18 0.19 0.21 0.14

0.109 0.145 0.199 0.246 0.348

0.012 0.008 0.007 0.021 0.011

Octane on mixed SAM surface

10.24 10.19 8.73 8.11 7.71

0.27 0.28 0.17 0.18 0.16

0.119 0.143 0.208 0.262 0.356

0.012 0.018 0.006 0.012 0.009

Nonane on mixed SAM surface

14.36 13.54 12.29 11.47 10.56

0.76 0.41 0.45 0.29 0.35

0.157 0.182 0.218 0.276 0.374

0.014 0.008 0.011 0.006 0.002

Nonane on methyl-terminated SAM

Heptane on mixed SAM surface

be a linear relationship between cos θ and 1/R (provided that both line tension and liquid–vapor interfacial tension are constant). In Figs. 3 and 4 the straight line represents the best fit to the experimental data points using a least-squares method. To quantify the appropriateness of our linear fit and the strength of the linear relation between cos θ and 1/R, the correlation coefficient (r) is calculated. As can be seen from Tables 3 and 4 the correlation coefficients are all equal or better than 0.97; this verifies the appropriateness of a linear fit to the data. Moreover, in order to determine whether the observed correlation is genuine or due to chance, the significance level for r was calculated. The higher the significance level, the less the probability of the observed correlation being due to chance; the calculated significance levels for the r values were 99% or better. Therefore, interpretation of the observed drop size dependence of contact angles as being due to line tension (cf. Eq. (3)) is statistically well founded. The slope and the intercept of the fitted lines can be compared with the appropriate terms in Eq. (3) to find the line tension (σ ) and the contact angle (θ∞ ). These data along

Table 4 Summary of experimental results: line tension, θ∞ , and correlation coefficient, r, for pure methyl-terminated SAM surfaces (experiments were performed at 23 ◦ C) Liquid

Heptane Octane Nonane

Liquid–vapor Contact angle Line tension (µJ/m) surface tension θ∞ (◦ ) γlv (mJ/m2 ) 19.8 21.1 22.6

4.0 6.7 13.7

0.148 0.424 0.851

r

Significance level for r

0.99 0.97 0.97

99% 99% 99%

Note. The values for contact angle (θ∞ ) and line tension (σ ) are calculated from intercept and slope of the averaged cos θ vs 1/R plots, respectively.

Table 5 Summary of experimental results: line tension, θ∞ , and correlation coefficient, r, for SAM-1 surfaces [CH3 (88%)–COOH (12%)] (experiments were performed at 21 ◦ C) Liquid

Heptane Octane Nonane

Liquid–vapor Contact angle Line tension r Significance (µJ/m) level for r surface tension θ∞ (◦ ) γlv (mJ/m2 ) 20.0 21.3 22.8

3.7 5.8 6.3

0.086 0.292 0.895

0.98 0.98 0.99

99% 99% 99%

Note. The values for contact angle (θ∞ ) and line tension (σ ) are calculated from intercept and slope of the averaged cos θ vs 1/R plots, respectively.

with contact angle data from the other systems have been analyzed as explained above. Tables 4 and 5 summarize the average line tension results and also provide the correlation coefficients and their significance level for all test liquids on each of the two types of surface. These findings confirm the line tension results presented in [22,28,32]. This is important from the perspective of the general controversy regarding the sign and magnitude of line tension [22,27,28]. For a given line tension value (say 1 µJ/m) the effect of line tension on drop size dependence of contact angle increases as θ∞ decreases (this is the expected behavior according to the modified Young equation; see Eq. (3)). Moreover, the change in cos θ as R is varied is smaller at low θ values compared to when θ is large (e.g., the change in cos θ is one order of magnitude less when θ∞ is approximately 50◦ than when θ∞ is 20◦ ). This situation provides an opportunity to determine line tension from drop size dependence of contact angle with a greater certainty at small contact angles. Furthermore, it should be noted that the line tension results found here are entirely independent of considerations of the drop profile, i.e., the use of side view to measure contact angle, as employed in Refs. [22,28] (see Section 3.1). It has been speculated in the literature that drop size dependence of contact angles can have causes other than line tension. Among other causes there has been mention of viscous effects, solid surface deformation, thin film effects, heterogeneity of the surface, and evaporation and hydrostatic pressure effects. All or some of these causes might prove to be operative for some systems; nevertheless, with careful experimentation, it is unlikely that the above effects will be

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Fig. 5. Line tension versus solid–liquid interfacial tension. The plot contains the results from this study and those obtained by Amirfazli et al. [22,28,29].

the primary reason for the observed drop size dependence of contact angles. In [27] all of the above possible artifacts which might have affected the observed drop size dependence of contact angles were carefully considered. It was concluded that none of the above effects had contributed significantly to the observed drop size dependence of contact angles and hence the line tension values as obtained must be accepted. It was mentioned earlier that one of the goals of this study is comparison of theoretical results with experimental ones at low contact angles. The results show that the line tension is positive and ranges from a minimum of +8.6 × 10−8 J/m to a maximum of +8.5 × 10−7 J/m. When this set of new data is compared with previous contact-anglebased findings from Refs. [22,28,29] (see Fig. 5), it becomes apparent that there is a positive correlation between line tension and solid–liquid interfacial tension (solid–liquid interfacial tension values are calculated using an equation of state approach (Chap. 5 of [18]) knowing the contact angle and liquid surface tension for each system). Such a trend was also predicted based on the thermodynamic phase rule for moderately curved capillary systems [30,31,33]. Gaydos et al. showed that there are two degrees of freedom for a two-component solid–liquid–vapor sessile drop system and justified the existence of the following relationship: σ = σ (γsl ). Considering Fig. 5 and the thermodynamic predictions in [33], this means that as complete wetting (θ = 0) is approached, γlv = γsv and γsl = 0 (Chap. 4 of [18]) and it is likely that line tension vanishes also. The observed trend that generally, as contact angle decreases, so does line tension (see Fig. 6), is in agreement with some of the theoretical findings [1,2,6]. In fact if the wetting transition is of the critical type, at complete wetting there would not be a threephase line to ascribe a tension value to it. However, as discussed in the Introduction, there are theoretical predictions that state that line tension will not tend to zero, but either

159

Fig. 6. Line tension versus contact angle. The plot contains the results from this study and those obtained by Amirfazli et al. [22,28].

increase without bound or assume a limiting value [12,13]. Such predictions are mostly valid for first-order wetting. The question remains, however, whether the nature of the common line and its tension in a first order wetting transition (the common line is characterized as the confluence zone between a thin and a thick film surrounded by vapor phase) can be compared readily to the common line and its tension for a classical three-phase line of a solid–liquid–vapor system like a sessile drop resting on a solid surface. Based on the results for the classical system described here, the conclusion would be to expect that the line tension vanishes at complete wetting. The fact that line tension is found to be positive in this study contradicts the theoretical predictions of Derjaguin and Churaev [1,2] and also the initial position of Widom [6]. However, the positive sign for line tension near wetting is in line with the prediction by de Gennes [4,15], and the later position taken by Szleifer and Widom [12]. The issue of theoretically predicting the sign for line tension is a complicated one. It is generally believed [36] that the interface displacement described in [3] has been able to reconcile different predictions by various investigators regarding the sign of the line tension near wetting (see the Introduction). However, the interface displacement model is a phenomenological model which outlines a framework. The choice of parameters, e.g., disjoining pressure isotherm or free energy density profiles, still plays a major role in the outcome of the theoretical predictions. In the absence of a verifiable choice for the above functionals, it might be difficult to ascribe significance to such findings. Perhaps experimental studies similar to ours or specially of wetting transition at various temperatures (e.g., see [36]) are needed to provide the insight for choosing appropriate forms of disjoining pressure isotherm and/or free energy density profiles used in theoretical calculations. The positive sign for line tension observed here is, however, in agreement with our earlier

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observation [22,28–30] and others [32,34,35] looking at systems removed from complete wetting.

5. Summary and conclusions Line tension behavior for systems with small contact angles, i.e., near the wetting transition, was determined. For this purpose six systems of low contact angle (i.e., θ < 20◦ ) were studied. It was determined that the line tension is positive and ranges from below 10−7 to just below 10−6 J/m for these systems. The observations suggest that line tension decreases as the contact angle decreases and likely vanishes at complete wetting. The observed trend fits well into our previous observations for systems away from wetting stipulating that there exists a positive correlation between line tension and solid–liquid interfacial tension (note that as solid–liquid interfacial tension tends to zero, complete wetting is approached). Consideration of the magnitude of line tension and the observed trend in this study provided yet another indication that the line tension values (in the order of 10−6 µJ/m) obtained from the systems studied earlier (far from wetting) [22,28] are reasonable. Our findings are not fully corroborated by any of the theoretical predictions for line tension near wetting, but the theoretical predictions are generally not internally consistent either. As discussed, predictions are based on phenomenological models or frameworks that depending on the choice for free energy density profiles or disjoining pressure isotherms, can have large uncertainties. From a physical perspective a vanishing line tension at the wetting as suggested by our data is more plausible than some theoretical predictions, which assert line tension will tend to infinity at wetting. Further experimental studies are needed to target development of realistic parameters for theoretical formulations, e.g., finding the free energy densities by conducting contact angle experiments at various temperatures near wetting. Acknowledgment This work was supported by a grant from the Natural Science and Engineering Research Council of Canada (NSERC).

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