Langmuir 2007, 23, 11999-12002
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Empirical Equation to Account for the Length Dependence of Line Tension Robert David* and A. Wilhelm Neumann Department of Mechanical & Industrial Engineering, UniVersity of Toronto, 5 King’s College Road, MB56 Toronto, Ontario, Canada M5S 3G8 ReceiVed August 17, 2007. In Final Form: October 5, 2007 Measured values of the three-phase line tension in the literature are correlated with the spreading parameter and with the radii of the drops or bubbles under investigation. The latter dependence contradicts an assumption of the modified Young equation. We suggest an alternative empirical formulation that describes the data consistently, and we discuss its possible physical significance.
1. Introduction In the Gibbsian model for interfaces, the actually gradual transition from one phase to another is represented by uniform phases separated by an infinitesimally thin surface to which excess energy (the surface tension) is assigned. At a three-phase line, line tension is defined analogously as the excess energy that is not already associated with any of the adjacent surface or volume phases. Equivalently, line tension is the work of formation of a unit length of line phase. These definitions lead to the following modified form of Young’s equation (written for an axisymmetric sessile drop of liquid on a solid surface)
γlv cos θ + γsl +
σ ) γsv r
(1)
where γlv is the liquid-vapor interfacial tension, γsl is the solidliquid interfacial tension, γsv is the solid-vapor interfacial tension, θ is the contact angle, σ is the line tension, and r is the radius of the three-phase line.1 Counting the thermodynamic degrees of freedom in such a system indicates that only two of the {γlv, γsv, γsl, σ, θ} parameters are independent.2 Thus, for example, σ ) f(γlv, γsv) for a particular liquid-solid pair. If we hope for a simpler situation in which σ is a universal function of only one other variable, then the logical choice is either γsl or θ because each of them depends on both the solid and liquid properties (as σ is expected to). Duncan et al.3 observed a positive correlation between experimentally measured values of σ and γsl. Amirfazli et al.4 noted a similar correlation between σ and θ, and there has been much discussion in the literature about the value taken by σ on approach to wetting (i.e., as θ f 0).4,5 In this letter, we will examine how the above thermodynamic picture of line tension is reflected in the aggregate of currently available experimental data.
2. Literature Data In eq 1, the line tension and the three interfacial tensions are, by definition, independent of r; only θ varies with r. Thus, the * Corresponding author. E-mail:
[email protected]. Tel: 416978-1270. Fax: 416-978-7753. (1) Boruvka, L.; Neumann, A. W. J. Chem. Phys. 1977, 66, 5464-5476. (2) Li, D.; Gaydos, J.; Neumann, A. W. Langmuir 1989, 5, 1133-1140. (3) Duncan, D.; Li, D.; Gaydos, J.; Neumann, A. W. J. Colloid Interface Sci. 1995, 169, 256-261. (4) Amirfazli, A.; Keshavarz, A.; Zhang, L.; Neumann, A. W. J. Colloid Interface Sci. 2003, 265, 152-160. (5) Indekeu, J. O. Int. J. Mod. Phys. B 1994, 8, 309-345.
most common experimental method for measuring σ is to equate it to -γlv times the slope of a plot of cos θ versus r-1. In some configurations, such as liquid lenses, the forces balanced in eq 1 act in different directions, giving rise to the more general Neumann quadrilateral relation6
σ γlvilv + γslisl + iσ + γsvisv ) 0 r where, in a cross section, the i’s are unit vectors tangent to the corresponding interfaces at the three-phase line. The direction of σ is in the plane of the contact line and perpendicular to it, as has been experimentally verified by Gu et al.7 In these configurations, σ can still be measured in an analogous way using the size dependence of angles. We will continue to use the notation of eq 1 for simplicity, but our reasoning will apply to other configurations as well. Measurements of σ reported in the literature vary both in sign and over several orders of magnitude. Others have noted that the largest magnitudes for σ (∼1 µJ/m) tend to be measured for the largest drops and the smallest (∼10 pJ/m) tend to be measured for the smallest drops.8-10 Researchers investigating microscopic drops have also noticed that in straight-line fits of cos θ to r-1 the extrapolation to r-1 ) 0 did not match the contact angle of macroscopic drops.10-12 Figure 1a shows a plot of measured absolute values of σ versus r for all experiments that we are aware of within the last 20 years (about the period over which reliable results have appeared).3,4,8-30 (6) Chen, P.; Gaydos, J.; Neumann, A. W. Langmuir 1996, 12, 5956-5962. (7) Gu, Y.; Li, D.; Cheng, P. J. Colloid Interface Sci. 1996, 180, 212-217. (8) Drelich, J.; Miller, J. D.; Hupka, J. J. Colloid Interface Sci. 1993, 155, 379-385. (9) Sto¨ckelhuber, K. W.; Radoev, B.; Schulze, H. J. Colloids Surf., A 1999, 156, 323-333. (10) Mugele, F.; Becker, T.; Nikopoulos, R.; Kohonen, M.; Herminghaus, S. J. Adhes. Sci. Technol. 2002, 16, 951-964. (11) Yang, J.; Duan, J.; Fornasiero, D.; Ralston, J. J. Phys. Chem. B 2003, 107, 6139-6147. (12) Checco, A.; Schollmeyer, H.; Daillant, J.; Guenoun, P.; Boukherroub, R. Langmuir 2006, 22, 116-126. (13) Gaydos, J.; Neumann, A. W. J. Colloid Interface Sci. 1987, 120, 76-86. (14) Wallace, J. A.; Schu¨rch, S. J. Colloid Interface Sci. 1988, 124, 452-461. (15) Wallace, J. A.; Schu¨rch, S. Colloids Surf. 1990, 43, 207-221. (16) Li, D.; Neumann, A. W. Colloids Surf. 1990, 43, 195-206. (17) Drelich, J.; Miller, J. D. J. Colloid Interface Sci. 1994, 164, 252-259. (18) Drelich, J.; Miller, J. D.; Kumar, A.; Whitesides, G. M. Colloids Surf., A 1994, 93, 1-13. (19) Aveyard, R.; Clint, J. H. J. Chem. Soc., Faraday Trans. 1995, 91, 175176. (20) Chen, P.; Susnar, S. S.; Mak, C.; Amirfazli, A.; Neumann, A. W. Colloids Surf., A 1997, 129-130, 45-60. (21) Dussaud, A.; Vignes-Adler, M. Langmuir 1997, 13, 581-589.
10.1021/la702553h CCC: $37.00 © 2007 American Chemical Society Published on Web 10/19/2007
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Figure 2. Correlation of line tension with drop radius for the subset of data from Figure 1a with moderate S.
Figure 1. (a) Relationship between the magnitude of line tension and the length scale of the experiment. Each marker represents an individual experiment, with sometimes many in one publication. (b) The magnitude of line tension tends to decrease near wetting. Data for which the error bars encompassed σ ) 0 are not plotted.
Experiments on roughened, reactive, or photoactive surfaces and those for which only an upper limit for σ could be discerned are excluded. Experiments not explicitly based on the size dependence of the contact angle(s) are also excluded for the sake of our arguments, although they do follow the same pattern. Many of the systems studied were chemically similar, often consisting of hydrocarbon liquids and methylated or fluorinated solids. There is no apparent pattern distinguishing solid-liquidvapor from liquid-liquid-vapor data (not shown). However, other trends are visible. The absolute value of σ generally decreases near wetting (Figure 1b). We found that this trend was most apparent between σ and the spreading coefficient S ) γlv(cos θ - 1) rather than between σ and γsl or θ. In the former case, this may be because γsl is difficult to measure and its analog, the liquid-liquid interfacial tension, does not necessarily decrease to zero when a liquid lens approaches wetting; in the latter case, it may be because θ does not measure energy. S is also the characteristic magnitude of the interface potential from which theoretical estimates of σ are produced. However, from theory, σ is expected to reach a positive value or diverge as first-order wetting is approached, depending on the intermolecular potential.5,31 (22) Amirfazli, A.; Kwok, D. Y.; Gaydos, J.; Neumann, A. W. J. Colloid Interface Sci. 1998, 205, 1-11. (23) Amirfazli, A.; Chatain, D.; Neumann, A. W. Colloids Surf., A 1998, 142, 183-188. (24) Aveyard, R.; Clint, J. H.; Nees, D.; Paunov, V. Colloids Surf., A 1999, 146, 95-111. (25) Amirfazli, A.; Ha¨nig, S.; Mu¨ller, A.; Neumann, A. W. Langmuir 2000, 16, 2024-2031. (26) Yakubov, G. E.; Vinogradova, O. I.; Butt, H. J. J. Adhes. Sci. Technol. 2000, 14, 1783-1799. (27) Gu, Y. Colloids Surf., A 2001, 181, 215-224. (28) Seemann, R.; Jacobs, K.; Blossey, R. J. Phys.: Condens. Matter 2001, 13, 4915-4923. (29) Wang, J. Y.; Betelu, S.; Law, B. M. Phys. ReV. E 2001, 63, 031601. (30) Takata, Y.; Matsubara, H.; Kikuchi, Y.; Ikeda, N.; Matsuda, T.; Takiue, T.; Aratono, M. Langmuir 2005, 21, 8594-8596.
Figure 3. (a) Fit of eq 2 (c ) 0.74 mJ/m2) to the drop-size dependence of the contact angle measured by Amirfazli et al.25 for 1-bromonaphthalene on a self-assembled monolayer composed of 80% CH3 groups and 20% COOH groups. (b) Fit of eq 2 (c ) 2.23 mJ/m2) to the data of Drelich et al.8 for kerosene drops on a quartz surface immersed in water.
As for the sign of σ, most measurements of negative line tension are for microscopic drops near wetting (Figure 1b). Some line tensions, measured with bubbles rather than drops, were reported as negative but were actually positive because θ was measured outside the bubbles.8,18 They are shown as positive in Figure 1b. Finally, Figure 1a shows that σ depends on not only S but also r, contrary to the assumption made in formulating eq 1. This raises questions as to the applicability of the Gibbs model, on which eq 1 is based, to the quantity measured as σ.
3. Empirical Equation If we attempt to isolate the dependence of σ on the drop radius r by considering only experiments for which 1 e -S < 100, then a fit of the magnitude of σ to a power law in r gives an exponent close to 1 (Figure 2). Thus, we hypothesize that (31) Bausch, R.; Blossey, R. Phys. ReV. E 1993, 48, 1131-1135.
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σ(r, S) ) c(S)r for some function c. From the fit in Figure 2, c ) 0.62 mJ/m2 for moderate values of S. If we now put aside eq 1 and consider σ only phenomenologically, then we have
-γlv
∂ cos θ ) c(S)r ∂(r-1)
By integrating and assuming that γlv is independent of r, we get
()
γlv cos θ ) -c(S) ln
r0 r
(2)
where r0, the integration constant, stands in for γsv and γsl from eq 1. Equation 2 suggests fitting cos θ versus r-1 data to a logarithm rather than a straight line, as is normally done in accordance with eq 1. Figure 3a shows a typical example of such a fit. In most experiments in which the range of r is limited, logarithmic and straight line fits produce similar correlation coefficients. However, for experiments over wider ranges of r,8,18,27 logarithmic fits are more suitable (as already noted by Drelich and Miller32), with a typical example shown in Figure 3b. The values of c that have been measured from fits to eq 2 are plotted in Figure 4 as a function of S. The relationship between c and S, with r varying from the micrometer to millimeter range, is similar to the relationship that has been demonstrated between σ and γsl for r only in the millimeter range.4 At moderate S, the values of c are consistent with the value found above (0.62 mJ/m2) from the aggregate of the literature data. Therefore, eq 2 consistently describes the variation of contact angle with drop size both within and between experiments, unlike eq 1.
4. Discussion Neither eq 1 nor eq 2 fits well to data10,11,33 that cover even wider ranges of r (not shown). Thus, the correct relationship between cos θ and r-1 may be more complicated. Generally, over a narrow range of r (typically less than a decade), many functions appear as straight lines; therefore, eq 1 may be only a first-order approximation. A higher-order term resulting from the choice of dividing surface in the model system was considered by Rusanov et al., but it was found to vanish for large drops.34 The length dependence of σ is also evident in results from a second type of experiment, in which line tension is measured by analyzing the shape of the three-phase line formed by liquid on a striped solid surface of alternating wettability. A numerical solution showed that this wavy shape is a function of the dimensionless variable σ/2γlva, where a is the stripe width.35 With relatively constant σ and γlv, the three-phase line shape would change dramatically with a. However, observations for a ≈ 1 mm36 and 0.5 µm37 show similar line shapes, consistent with a line tension roughly proportional to a. Thus, the body of experimental data in the literature points to a length-dependent line tension; however, a physical explanation (32) Drelich, J.; Miller, J. D. Colloids Surf. 1992, 69, 35-43. (33) Checco, A.; Guenoun, P.; Daillant, J. Phys. ReV. Lett. 2003, 91, 186101. (34) Rusanov, A. I.; Shchekin, A. K.; Tatyanenko, D. V. Colloids Surf., A 2004, 250, 263-268. (35) Gaydos, J.; Neumann, A. W. AdV. Colloid Interface Sci. 1994, 49, 197248. (36) Hoorfar, M.; Amirfazli, A.; Gaydos, J. A.; Neumann, A. W. AdV. Colloid Interface Sci. 2005, 114, 103-118. (37) Herminghaus, S.; Pompe, T.; Fery, A. J. Adhes. Sci. Technol. 2000, 14, 1767-1782.
Figure 4. Magnitude of c, which is roughly proportional to the spreading parameter.
remains unclear. We will discuss a number of other researchers’ speculations before describing our own. The length dependence of σ has been attributed to the differences between experimental methods used on different length scales.10 However, different methods on the same length scale (e.g., sessile drops3 vs tapered cylinders dipped into liquid;7 optical15 vs interferometric9 imaging) have produced consistent results, and Figure 2 appears to indicate a single cause underlying the dependence of σ on r. One variable that changes with drop size is vapor pressure. However, for a constant drop size, measured line tension values are not correlated with vapor pressure (not shown). Also, in striped surface measurements in which a large quantity of liquid is used, the surface curvature approaches zero. Thermal equilibration times for submillimeter-sized drops are a few seconds at most, and even for larger drops, typical experimental durations of a few minutes ensure equilibrium. Larger values of σ have often been attributed to the roughness or heterogeneity of the solid surface.17 A nonlinear relationship between cos θ and r-1, based on this hypothesis, has been proposed.33 However, initial calculations of the effects of roughness and heterogeneity are too small to account for data measured with millimeter-sized drops.10,25 In addition, large values of line tension measured in liquid-liquid systems9,14,15,20 cannot be due to roughness or heterogeneity (though Wallace and Schu¨rch’s surface14,15 was somewhat solidlike). Often, the drop size dependence of θ is expressed in terms of the contact line curvature κ, which equals r-1 for an axisymmetric drop. However, in striped surface experiments, the contact line curvature changes along the profile while the size of the system (r, or drop height) remains constant. In one such measurement, no dependence of σ on κ was found.37 This suggests a long-range source for the length dependence of σ. One possibility, notwithstanding the reservations above, is large wavelength surface roughness;38 another is very long range intermolecular forces. The van der Waals forces usually considered in theoretical estimates of σ decay as d-6, with d being the distance between two molecules. The nearly linear dependence of σ on r from Figure 2 implies that the energy associated with the line scales not as its length L but as a power close to L2. Dobbs39 has shown that van der Waals forces could not produce such behavior in σ for a finite radius, axisymmetric sessile drop but that longerrange forces (decaying as d-4 to d-5) could. Although the most obvious longer-range force is gravity, its magnitude is negligible in the region near the contact line.40 (38) Decker, E. L.; Garoff, S. Langmuir 1997, 13, 6321-6332. (39) Dobbs, H. Int. J. Mod. Phys. B 1999, 13, 3255-3259. (40) De Gennes, P. G. ReV. Mod. Phys. 1985, 57, 827-863.
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Among the well-known intermolecular forces, decay slower than d-6 implies the presence of charges or dipoles.41 There is some recent evidence of such forces playing a role at interfaces between nominally dispersive media. A long-range force decaying as d-4 was measured between polystyrene particles at an oil/ (41) Israelachvili, J. Intermolecular and Surface Forces, 2nd ed.; Academic Press: San Diego, CA, 1992; p 28. (42) Aveyard, R.; Binks, B. P.; Clint, J. H.; Fletcher, P. D. I.; Horozov, T. S.; Neumann, B.; Paunov, V. N.; Annesley, J.; Botchway, S. W.; Nees, D.; Parker, A. W.; Ward, A. D.; Burgess, A. N. Phys. ReV. Lett. 2002, 88, 246102.
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water interface, acting through the oil phase.42 Finally, polar and other specific interactions have been found to affect contact angles in all but rare cases.43 Acknowledgment. R.D. gratefully acknowledges helpful discussions with Stephanie M. Dobson and funding from an NSERC postdoctoral fellowship. LA702553H (43) Tavana, H.; Neumann, A. W. AdV. Colloid Interface Sci. 2007, 132, 1-32.
