Langmuir 2002, 18, 9771-9777
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Impact of Line Tension on the Equilibrium Shape of Liquid Droplets on Patterned Substrates Juergen Buehrle, Stephan Herminghaus, and Frieder Mugele* Universita¨ t Ulm, Abteilung Angewandte Physik, D-89069 Ulm, Germany Received May 20, 2002. In Final Form: August 12, 2002 We studied the morphology of liquid droplets on substrates with a lateral wettability pattern using numerical calculations. We analyzed the influence of the wettability contrast, the sharpness of the transitions between adjacent hydrophilic and hydrophobic stripes, and the line tension of the three-phase contact line on the modulation amplitude of the latter and on the shape of the liquid-vapor interface. In the presence of lateral variations of the line tension, we found that the modulation of the contact angle along the contact line is controlled by the local wettability of the substrate, by the local curvature of the contact line, and by gradients of the line tension. This result confirms a recently published theoretical extension of the modified Young equation.1 Furthermore, the numerical calculations demonstrate that and show how the line tension can be extracted from atomic force microscopy measurements of liquid droplets on patterned substrates.
Introduction Recent years have seen an increasing interest in strategies to control and manipulate small quantities of liquids. This desire stems from the demands of combinatorial chemistry and biochemical analysis and synthesis, where it turned out that reducing the amount of liquid handled in a chemical reaction dramatically improves both throughput and efficiency.2,3 Hence, a variety of microfluidic concepts are being developed and explored.4 A very simple and promising concept uses flat solid substrates with suitable patterns of wettability to define the positions of liquid reservoirs, flow channels, and reaction chambers, etc.5 In addition to the already mentioned fundamental improvements in performance, scaling down these structures as much as possible also allows for massive parallelization. Similar to the microelectronics industry, there arises the question as to how far one can drive the miniaturization of microfluidic devices. In “open” microfluidic devices with wettability patterns, limits arise from the fidelity of the liquid to the imposed patterns and from the quality and sharpness of the patterns themselves. Furthermore, small lateral scales involve high local curvatures of the solid-liquid-vapor contact line. In this case, the line tension, i.e., the excess free energy of the contact line per unit length, comes into play as an additional factor that may deteriorate pattern fidelity. In the present work, we used numerical calculations to study these questions in a simple, generic geometry, namely, a liquid droplet on a flat substrate with alternating hydrophilic and hydrophobic stripes. In particular, we focus on the impact of line tension whose order of magnitude by itself has been the subject of a scientific debate for a long time.6,7 While theoretical calculations8,9 * Corresponding author. Phone: +49/(0) 731 502 2931. Fax: +49/(0) 731 502 2958. E-mail:
[email protected]. (1) Swain, P. S.; Lipowsky, R. Langmuir 1998, 14, 6772. (2) Microsystem Technology in Chemistry and Life Science; Manz, A., Becker, H., Eds.; Topics in Current Chemistry; Springer: Berlin, 1998; Vol. 194. (3) Jakeway, S. C.; de Mello, A. J.; Russell, E. L. Fresenius J. Anal. Chem. 2000, 366, 525. (4) Whitesides, G. M.; Stroock, A. D. Phys. Today 2001, 54, 42. (5) Gallardo, B. S.; Gupta, V. K.; Eagerton, F. D.; Jong, L. I.; Craig, V. S.; Shah, R. R.; Abbott, N. L. Science 1999, 283, 57. Gau, H.; Herminghaus, S.; Lenz, P.; Lipowsky, R. Science 1999, 283, 46.
point to an order of magnitude of ≈10-11 to 10-10 J/m, experiments with macroscopic liquid droplets seemed to indicate a value of 10-6 to 10-5 J/m. Only recently, this discrepancy was resolved experimentally.10,11 In ref 11, the liquid-vapor interface of liquid droplets on hydrophilic Si substrates, which were covered with parallel stripes of hydrophobic silane molecules, was imaged with nanometer resolution using tapping mode atomic force microscopy (AFM). Two independent methods were presented to derive an upper limit for the line tension from the AFM data, which was on the order of the theoretical predictions. In addition to the general aspects of wetting on patterned substrates, the numerical calculations presented in this work validate an implicit assumption of the data analysis procedure used in those experiments. The paper is organized as follows. In the first section, we give a theoretical background of wetting on a patterned substrate. In the second section we describe the details of the numerical calculations and the data analysis procedure. In the third section we present the results regarding the influence of line tension and of the finite pattern sharpness. In the Discussion, we relate these numerical results to recent experimental work. Theory The lateral dimensions of the liquid microstructures considered in this work are small compared to the capillary length κ ) (σlv/gF)1/2 (σlv is the surface tension of the liquid, F is its density, and g is the gravitational acceleration). Therefore, we can neglect gravity. In this case, the free liquid-vapor interface of any connected liquid microstructure in mechanical equilibrium has to be a surface of constant mean curvature. Deviations from this shape would give rise to gradients in Laplace pressure, which, (6) Amirfazli, A.; Ha¨nig, S.; Mu¨ller, A.; Neumann, A. W. Langmuir 2000, 16, 2024. Amirfazli, A.; Kwok, D. Y.; Gaydos, J.; Neumann, A. W. J. Colloid Interface Sci. 1998, 205, 1. (7) Lyklema, J. Fundamentals of Interface and Colloid Science; Academic Press: San Diego, 2000; Vol. III. (8) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Clarendon: Oxford, U.K., 1982. (9) Getta, T.; Dietrich, S. Phys. Rev. E 1998, 57, 655. (10) Wang, J. Y.; Betelu, S.; Law, B. M. Phys. Rev. E 2001, 63, 031601. Wang, J. Y.; Betelu, S.; Law, B. M. Phys. Rev. Lett. 1999, 83, 3677. (11) Pompe, T.; Herminghaus, S. Phys. Rev. Lett. 2000, 85, 1930.
10.1021/la0204693 CCC: $22.00 © 2002 American Chemical Society Published on Web 11/07/2002
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in turn, would drive a fluid flow to reestablish equilibrium. Within the constraint of constant mean curvature, one can control the shape of the surface by imposing boundary conditions in the form of wettability patterns on the substrate. Here, we are interested in nonvolatile liquids. Hence, we can find the equilibrium liquid configuration by minimizing the free energy functional F at constant volume.
depends on the details of the preparation process. To mimic this situation in our calculations, we defined the wettability pattern by specifying the equivalent Young’s contact angle ϑYoung as a function of the lateral coordinate x in the following convenient form
F [A ] )
Here the parameter s controls the sharpness of the profile such that for small values of s we obtain a sinusoidal profile, whereas for large values of s it is rectangular. To avoid any ambiguity, we will henceforth reserve the expression “contact angle” for the actual contact angle θ, as determined from the numerical data for a given geometry, and use the expression “wettability” ϑ to describe the imposed pattern as defined in eq 4. In this sense, ϑ0 is the average wettability, ∆ϑ is the amplitude of the wettability contrast. q ) 2π/λ is the wavenumber of the pattern and λ its periodicity. Since the line tension is closely related to the wettability of the substrate, it will in general also depend on the position. For simplicity, we imposed the same functional form for the variation for Γ(x) as in eq 4,
σlvA lv +
∫dA sl{σsl(rb) - σsv(rb)} + ∫∂A
sl
dsΓ(r b) (1)
Here, A i is the area of the ith interface with interfacial energy σi (i ) lv (liquid-vapor), sv (solid-vapor), and sl (solid-liquid)). The first two terms on the right-hand side represent the interfacial energies, and the last one is the contribution due to the line tension Γ of the three-phase r on the contact line. Γ, σsl, and σsv depend on the position b substrate due to the wettability pattern. The line tension Γ includes contributions from both short-range and longrange (van der Waals) interactions in the vicinity of the contact line. These interactions, which also give rise to the effective interface potential, influence the surface profile only within a few nanometers above the substrate.9,12,13 Since we were interested in the asymptotic behavior of the liquid surface, i.e., several nanometers and more above the substrate, we implemented the line tension simply as a phenomenological parameter associated only with the line elements of the contact line strictly at the substrate level, i.e., at the height z ) 0. In contrast to the real experimental situation, this means that the surface profile in the present calculations is not distorted in the vicinity of the contact line by the action of the interface potential. The relative importance of the line contribution increases upon reducing the size of the liquid microstructure. This becomes evident if we rewrite eq 1 in dimensionless form by scaling all length scale to a characteristic length L of the system, which could be the average radius of a deformed droplet, and if we measure F in units of σlv.
F˜ [A˜ ] ) A˜ lv +
{
∫dA˜ sl
}
σsl(r b) - σsv(r b) 1 + σlv Lσlv
∫∂A˜ ds˜ Γ(rb) sl
(2)
Here, A˜ i ) A i/L2, s˜ ) s/L, and F˜ ) F /L2σlv. Obviously, the importance of the line energy term relative to the interfacial energy terms is determined by the dimensionless ratio Γ/(Lσlv). This fact becomes also apparent when we consider a liquid droplet on a homogeneous substrate. In this case, the liquid-vapor interface is known to have a spherical cap shape, and the functional in eq 2 reduces to a simple function of the contact angle θ and base radius R of the droplet. Minimizing this function leads to the well-known modified Young equation
cos(θ) )
σsv - σsl Γ Γ ≡ cos(ϑYoung) σlv σlvR σlvR
(3)
However, we were interested in the case of a substrate with a one-dimensional wettability pattern which consists of alternating hydrophilic and hydrophobic stripes. In experiments, the transition from one stripe to the next is never perfectly sharp. The wettability rather crosses over continuously within a transition region whose width (12) Indekeu, J. O. Physica A 1992, 183, 439. (13) Bauer, C.; Dietrich, S. Eur. Phys. J. B 1999, 10, 767.
ϑYoung(x) ) ϑ0 + ∆ϑ tanh(-s‚cos(qx))
Γ(x) ) Γ0 + ∆Γ tanh(-s‚cos(qx))
(4)
(5)
with the average line tension Γ0 and the modulation amplitude ∆Γ. Swain and Lipowsky investigated theoretically the general case with lateral variations in ϑ, in Γ, and in substrate topography.1 They found that minimization of eq 1 leads to a generalized form of the modified Young eq 3. Adapted to the case of a planar substrate with a one-dimensional wettability pattern, their result reads
cos(θ(x)) ) cos(ϑYoung(x)) -
κ(x) Γ(x) 1 dΓ b (6) b e ‚e σlv σlv dx x ⊥
e⊥ is the Here, b ex is the unit vector in the x direction and b unit vector locally perpendicular to the contact line. κ(x) is the local curvature of the contact line. The equation relates the local contact angle θ(x) to the local wettability ϑYoung(x). While the first two terms on the right-hand side are a natural extension of eq 3, the term proportional to dΓ/dx had not been recognized for an arbitrary geometry before. It is maximal when the contact line is parallel to the stripes and goes to zero when the contact line is perpendicular to the stripes. If we want to know the course of θ(x) for a given geometry, we need to calculate the actual surface profile by minimizing F in order to obtain the contact line γ(x) and its local curvature κ(x). Analytical solutions to this problem can only be obtained for highly symmetric situations14,15 or in the limit of small contact angles and weak wettability contrast.16 In most practical cases, it has to be solved numerically. Numerical Details The numerical calculations were performed using the SURFACE EVOLVER,17 version 2.14. This public-domain program package allows the minimization of the total energy of a given amount of liquid subject to various boundary conditions and constraints. In our case, the total (14) Rusanov, A. I. Colloid J. USSR (Engl. Transl.) 1977, 39, 618. (15) Boruvka, L.; Neumann, A. W. J. Colloid Interface Sci. 1977, 65, 315. (16) Joanny, J. F.; Gennes, P. G. d. J. Chem. Phys. 1984, 81, 552. (17) Brakke, K. Exp. Math. 1992, 1, 141.
Line Tension on the Equilibrium Shape of Droplets
Figure 1. Typical triangulation pattern used by the surface evolver (a). (b) Wettability pattern and course of the contact line γ (x) for two different average radii R. The dotted and the dashed lines indicate the directions along which the height profiles in c were extracted. (average wettability, ϑ0 ) 40°; wettability contrast, ∆ϑ ) 10°; transition sharpness, s ) 10).
energy includes the surface energies of the liquid-vapor, the solid-liquid, and of the solid-vapor interfaces and the line energy of the three-phase contact line, as specified in eq 1. To minimize the total energy, the surface evolver subdivides the liquid-vapor interface into triangular facets. Facets are bounded by oriented edges with vertexes at their ends. The total energy is evaluated by summing up first the areas of all facets multiplied by the liquid-vapor interfacial energy. The solid-liquid interfacial area is computed using Stokes’ theorem as a line integral along the three-phase contact line. Similarly, the line energy is calculated by multiplying the length of each segment of the contact line with the local value of Γ. In each minimization step, each vertex is moved in proportion to the gradient of the total energy with respect to the motion of this specific vertex. We used a script program to define the initial configuration and the constraints and to control the minimization procedure. In each run, we first equilibrated the surface on a homogeneous substrate and then gradually built up the wettability pattern. Since we were interested in the details of the surface profile near the contact line, we reduced the average length of the edges in several steps upon approaching the substrate (see Figure 1a). The typical edge length at the contact line was λ/500. We chose to work at a fixed volume corresponding to an average droplet radius of approximately 3λ. This size is comparable to the typical experimental conditions in ref
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11. Guided by those experiments, we referred our numerical calculations to a lateral periodicity of λ ) 1 µm, a modulation of the wettability of ∆ϑ ) 10° with ϑ0 ) 40°, and a liquid-vapor interfacial tension of 45 mJ/m2 whenever calculating absolute numbers. If the equilibrium shapes exhibit certain symmetries, computational time can be saved by calculating only appropriate segments of the actual droplet surface. In our case, the droplets have one symmetry plane perpendicular to the wettability pattern. Furthermore, there is a second symmetry plane parallel to the stripes. In principle, the latter can be centered either on a hydrophilic or on a hydrophobic stripe, depending on the droplet volume. However, for sufficiently large droplets, one can show that this does not affect the results. Following the results of Brandon et al.18 we centered the second symmetry plane on a hydrophilic stripe. Using these two symmetries, we could reduce the computational effort to a quarter of a droplet. With this geometry, the total number of vertices in each calculation run ranged between 10 000 and 50 000. Figure 1a shows a typical droplet configuration as generated by the SURFACE EVOLVER. To analyze these data, we used a custom written Matlab program.19 First, we extracted the contact line γ(x) (Figure 1b), i.e., all the vertexes with z ) 0, and computed the unit vectors b e⊥(x), which are locally perpendicular to γ(x), and the local contact line curvature κ(x). Then we interpolated the remaining vertices to a grid of 512 × 512 points. Next, we calculated interpolated height profiles along b e⊥(x) (Figure 1c). Finally, the local contact angles θ(x) were determined from the slopes of fit parabolas to height profiles at z ) 0.20 To check the accuracy of both the numerical calculations and the data analysis, we first performed calculation runs on homogeneous substrates for a series of values of Γ and for a series of radii. The surface shapes were found to be perfect spherical caps. The contact angles determined from the calculation data exceeded prescribed values (chosen between 30 and 50°) by only approximately 0.04°. By comparing the contact angles of droplets with different radii, we determined the line tension Γ from the calculation data using eq 3. Again, the preset values were confirmed, in this case to within 0.1%. Results Figure 2a shows the course of the contact line for a series of values of ∆ϑ in the absence of line tension. As expected, the amplitude of the modulation increases with increasing ∆ϑ. There is a slight asymmetry between transitions from hydrophilic to hydrophobic and vice versa, which is related to the finite size of the droplet.21 A comparison of the two curves γ(x) in Figure 1b shows that (18) Brandon, S.; Wachs, A.; Marmur, A. J. Colloid Interface Sci. 1997, 191, 110. (19) Matlab, Version 6.0 Release 12.; The MathWorks Inc.: Natick, MA (USA). (20) Note that the situation here is somewhat simpler than in the experiments. In the experiments the profiles are disturbed in the vicinity of the contact line by the effective interface potential. Therefore the local contact angle can only be obtained by extrapolating the profiles from outside the range of the effective interface potential down to the substrate. In the present numerical calculations, the line tension enters only as a phenomenological parameter that is not related to any interface potential of finite range. Therefore, the contact angle can be determined exactly at z ) 0. (21) For infinitely large droplets, the angles between e⊥ and ey at the boundaries from hydrophilic to hydrophobic and vice versa have opposite sign but the same magnitude. For droplets of finite size, however, their magnitude differs the more the farther we move to the right in Figure 1b, i.e., the farther we move away from the center of the droplet. This simple geometric necessity is also responsible for the asymmetry in the curvature peaks in Figure 2b.
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Figure 2. Contact line (a), local curvature (b), and local contact angle (c) versus x for ∆ϑ ) 5, 10, and 15°. For clarity, in c only every sixth point is plotted. Solid lines in c represent the imposed wettability profile (ϑ0 ) 40°; s ) 10).
the most left-hand transition (i.e. the one with the smallest value of x) is only weakly affected by the finite size. Simultaneously, with the increasing contact angle modulation, the maxima of the local curvature κ(x) become the more pronounced the larger ∆ϑ (Figure 2b). At each transition from one stripe to the next, there is a characteristic signature in κ(x) with a maximum and a minimum close to the edges of the transition region and a change in sign at the center. On the homogeneous regions, the curvature is comparably small and approximately constant. Its finite size is responsible for the shape of the individual height profiles in Figure 1c: In the center of the hydrophilic (hydrophobic) stripe, κ is positive (negative). To fulfill the requirement of constant mean curvature everywhere on the liquid-vapor interface, the curvature of the contact line must be balanced by an opposite curvature of the height profiles, as apparent in Figure 1c. The absolute value of κ in the homogeneous regions depends of course on the size of the droplet and tends to zero for larger volumes if the wettability profile is sufficiently sharp. Hence the bending of the height profiles is also reduced for larger droplets. In Figure 2c, we plot the cosine of the local contact angle along the contact line. The symbols, which represent the numerical results, and the solid lines, which show the imposed wettability profile according to eq 3 with ϑYoung) ϑYoung(x), agree perfectly. This is exactly what we expect in the absence of line tension. Thus, this plot substantiates the accuracy of the numerical calculations in the presence of a wettability pattern. Throughout the rest of this work, we will focus on the specific case of ∆ϑ ) 10°. In Figure 3, we show the same quantities as in Figure 2 but now for a series of different, spatially homogeneous
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Figure 3. Contact line (a), local curvature (b), and local contact angle (c) versus x for the innermost transition region (θ0 ) 40°; ∆ϑ ) 10°; s ) 10) with Γ (J/m) ) 0, 10-10, 10-9, and10-8, as indicated. The thin vertical solid lines indicate the edges of the transition region (see text). For clarity, in c only every sixth point is plotted. The thick solid lines in c show theoretical expectations based on eq 6 and the local curvature shown in b.
values of line tension, i.e., Γ(x)) Γ. Here, we concentrate on the transition region from hydrophilic to hydrophobic around x ) 0.25, where the angle between the average orientation of γ(x) and the x-axis is still small. In agreement with intuition, the presence of a positive excess energy per unit length of the contact line reduces the amplitude of the contact line modulation. Simultaneously, the peaks in κ(x) at the edges of the transition regions are less pronounced and the more rounded off the higher Γ becomes. For finite values of Γ, the local contact angle θ(x) (Figure 3c) does not simply follow the imposed wettability ϑYoung(x) (solid line for Γ ) 0) anymore. For each value of Γ, θ(x) deviates the more from ϑYoung(x) the larger the absolute value of κ(x). However, the solid lines, which include the correction due to the second term on the righthand side of eq 6, match nicely with the numerical results. (Remember that dΓ/dx ) 0, in the present case.) If we look at the quantitative impact of the line tension on the droplet shape, we notice that it is only weakly affected for Γ ) 10-10 J/m or less. For Γ ) 10-8 J/m (or more), however, the modulation of the contact line and the peaks in the local curvature are almost completely suppressed. For further analysis and in particular with regard to experiments, it is important to compare the width of the transition region with the width of the peaks in κ(x). We define the edges of the transition region (thin vertical lines in Figure 3) as the positions where the wettability lies within 1% of the saturation value in the center of each homogeneous stripe. Whereas the wettability is thus
Line Tension on the Equilibrium Shape of Droplets
Figure 4. Modified Young plot: cos(θ) versus κ(x) compiled from Figure 3b,c. The dashed lines represent asymptotic behavior on the homogeneous regions based on eq 3. The solid lines represent the full theory (eq 6) using the local curvature from Figure 3b.
Figure 5. Modified Young plot for variable wettability (ϑ0 ) 40°; ∆ϑ ) 10°; s ) 10) and variable line tension (Γ0 ) 2 × 10-10 J/m and ∆Γ ) 10-10 J/m; s ) 10).
virtually constant, the local curvature still varies significantly. In fact, at the edges of the transition region κ(x) is about 50% of the peak value. Hence, there are sizable parts of the contact line, where the local curvature varies while the wettability is constant. The existence of these regions is the basis for a strategy to determine Γ from experimental measurements of κ(x) and θ(x). If the wettability is constant, any variation in θ(x) must be due to the influence of Γ. From eq 6 it is clear that cos(θ(x)) varies linearly with κ(x). Thus, Γ can be obtained easily from the constant slope in a plot of the cosine of the local contact angle versus the local curvature. We stress that this method, which was also used in ref 11, is only valid for locations on the substrate where the wettability is constant. The existence of these regions is nontrivial, and in deed we will see below that their lateral extent depends crucially on the sharpness s of the wettability pattern. Figure 4 shows a plot of cos(θ(x)) versus κ(x) for the same data as presented in Figure 3. Variations of the wettability dominate the central part between the maximum and the minimum of κ(x) in each curve. Data points outside correspond to areas of homogeneous substrate wettability. As expected on the basis of the above discussion, the slopes for these data points reach a constant value and agree with the dashed lines, which represent plots of eq 3 for saturated wettabilities and with the appropriate values of Γ. The solid lines represent the full generalized Young eq 6 (with the gradient term equal to zero, in this case). Similar results can be obtained when local variations of Γ ) Γ(x) are also included. In Figure 5, we plot cos(θ(x)) versus κ(x) with a line tension varying between 1 × 10-10 J/m on the hydrophilic stripes and 3 × 10-10 J/m on the hydrophobic ones. (These specific values were motivated by the results obtained in ref 9.) Again, there are sections between the ends of the curve and the extrema of the curvature displaying a more or less linear relation between cos(θ) and κ from which the line tension on each homo-
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Figure 6. Modified Young plot. Dependence on the sharpness s of the wettability pattern. ∆w is the “1% to 99%”swidth of the transition regions (ϑ0 ) 40°; ∆ϑ ) 10°; Γ ) 0).
geneous stripe can be determined. It is interesting to note the importance of the line tension gradient term in eq 6. The solid line is calculated using the full theoretical formula. The dashed line shows a theoretical curve using only the first two terms on the right-hand side of eq 6. Obviously, the gradient term is crucial for a good agreement between theory and numerical calculations. The numerical results presented thus far were obtained for a rather sharp transition (s ) 10) between the hydrophilic and the hydrophobic stripes. We saw that it is possible to determine the line tension of a given solidliquid-vapor system under these conditions by measuring the local curvature and the local contact angle along the perimeter of a liquid droplet on a smooth solid substrate with a wettability pattern. Experimentally, the preparation and the characterization of a sharp wettability profile is a difficult task. Obviously, it is important to know how an experimental measurement is affected by the finite width of the transition regions. Therefore, we simulated droplet profiles for different values of the sharpness s and analyzed them as above (Figure 6). In the absence of line tension, we expect to see horizontal slopes close to the ends of each curve, i.e., on the homogeneous areas. While all the curves in Figure 6 do asymptotically meet this requirement, the range where the slope is horizontal decreases dramatically as the transition region becomes wider. Therefore, the resolution in the measurement of Γ will be compromised. In particular, the finite average slopes between the ends of each curve in Figure 6 and the adjacent maximum or minimum of the local curvature are easily mistaken as a sign of a finite line tension. If the transition regions are too wide, it is thus easy to overestimate Γ. Discussion From an experimental point of view, the most critical issue is to distinguish between the effects of line tension and the effects smeared transition regions. To avoid topographic roughness, wettability patterns are usually produced using monolayers of self-assembled molecules. One can either create a complete monolayer, which is patterned subsequently, for instance by means of UVphotolithography or by electron beam lithography. Or one can deposit the molecules in a patterned fashion, e.g. by microcontact printing. In both cases, the patterns can be characterized by AFM. In the simplest case, AFM height profiles give a measure of the surface coverage with nanometer resolution in the lateral and angstrom resolution in the vertical direction. For fluorinated silane molecules on Si, Pompe et al.22 obtained a typical width (22) Pompe, T.; Fery, A.; Herminghaus, S.; Kriele, A.; Lorenz, H.; Kotthaus, J. P. Langmuir 1999, 15, 2398.
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Figure 7. Modified Young plot. Experimental data from AFM measurements. Hexaethylene glycol on Si with hydrophobic stripes of phenyltrichlorosilane (θ0 ≈ 15°; ∆θ ≈ 4°).
between 150 and 200 nm using the microcontact printing technique with a pattern periodicity of 870 nm. However, there is no simple conversion for a height profile of silane molecules into a wettability profile because the wettability is generally an unknown nonlinear function of the surface coverage. A somewhat more sophisticated measure of the chemical composition of the substrate can be obtained by means of adhesion mapping or chemical force microscopy. In contrast to the height profiles, this technique should produce a more realistic map of the surface wettability. In the absence of a detailed substrate characterization one is left with the problem to determine the Γ directly from modified Young plots such as in Figure 5 or Figure 6 without knowing the wettability profile ϑ(x). Figure 7 shows a modified Young plot of experimental data24 obtained with the same techniques as in ref 11. Qualitatively, the shape of these AFM data resembles the numerical results. In particular on the hydrophobic part of the substrate, we find a region with cos(θ) ∝ κ, which is consistent with a line tension of Γ ≈ 5 × 10-11 J/m (solid line). This value should be considered as an upper limit, because the slope of the data points may in part result from the unknown finite width of the transition region. Inside the latter, where κ changes sign, the curve does not follow the expectations based on the numerical calculations. The contact angle decreases more quickly than expected. This behavior is probably related to the poorly defined wettability profile. On the hydrophilic region, however, the local contact angle is essentially independent of κ. Admitting generous errors bars, the experimental data are consistent with a line tension of at most several times 10-11 J/m, possibly with negative sign. This value is in agreement with theoretical estimates and with our earlier AFM11,25 and micro-interferometry26 measurements, but in sharp contrast to optical measurements.6,25 A comparison between the numerical calculations and experimental images of liquid droplets on patterned substrates also shows that a line tension of the order of 10-6 J/m or more is not realistic for molecular systems.11,25,27 First of all, with such a large value of Γ, the (23) A somewhat more sophisticated measure of the chemical composition of the substrate can be obtained by means of adhesion mapping or chemical force microscopy. In contrast to the height profiles, this technique should produce a more realistic map of the surface wettability. (24) Becker, T. Untersuchung fluider Nanostrukturen mit dem Kraftmikroskop; Applied Physics Department, University of Ulm: Ulm, Germany, 2000. (25) Mugele, F.; Becker, T.; Nikopoulos, R.; Kohonen, M.; Herminghaus, S. J. Adhesion Sci. Technol. 2002, 16, 951. (26) Kohonen, M.; Mugele, F.; Herminghaus, S. Manuscript in preparation. (27) Drelich, J.; Wilbur, J. L.; Miller, J. D.; Whitesides, G. M. Langmuir 1996, 12, 1913.
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contact angle for a micrometer-sized droplet can only be 0 or 180°, depending on the sign of Γ. Second, Figure 3 shows that the modulation of the contact line is essentially gone for Γ ) 10-8 J/m. If the line tension were even larger than 10-8 J/m, wettability patterns should not affect the droplet shapes on the micrometer scale at all. This is in marked contrast to many experimental observations.11,24,27,28 In principle, Γ can have either positive or negative sign.12 In particular, it is expected to change sign (from negative to positive) upon approaching a first-order wetting transition. We did not succeed to simulate droplet profiles for negative values of Γ. In this case, the contact line always became unstable. Individual points of the triangulation grid performed large excursions from the average position of the contact line. This observation is not entirely unexpected, because the gain in line energy for a small perturbation of the contact line with wave vector q is proportional to q2, whereas the cost in surface energy due to the deformation of the adjacent liquid-vapor interface16 is proportional to q. Hence the contact line is expected to become unstable on short length scales if Γ is negative. This conclusion is only true, however, if the line tension is implemented as a phenomenological parameter, which applies to the contact line as a singular object. In reality, as mentioned above, the line tension arises from the shortand long-range interactions. Physically, it is smeared out over the whole region where the film thickness is within the range of the effective interface potential, i.e., typically below a few nanometers. It was shown theoretically that the contact line instability for Γ < 0 is removed if the line tension is correctly taken into account via the effective interface potential.29 We are currently implementing the line tension in this way into our numerical calculations in order to be able to deal with negative values of Γ as well. With regard to microfluidic applications, these calculations indicate that line tension, of any realistic order of magnitude, does not compromise the pattern fidelity as long as the local curvature of the contact line is not required to exceed, say, 10 µm-1. Since the latter is not the case for the present state of the art open microfluidic systems, the parameter, which determines how precisely fluid microstructures can be designed, is currently the quality and sharpness of the wettability patterns. The situation may be different, however, if large electrostatic fields are applied, as for instance when the electrowetting effect is used to manipulate the liquid on the substrate. In this case, the electrostatic stray fields at the edge of the liquid structure give rise to an effective negative line tension, which can be substantially larger than 10-10 J/m, depending on the applied voltage. For sufficiently high voltage, it can render the contact line unstable and it can ultimately lead to the emission of small satellite droplets from the latter.30,31 A detailed analysis of this effect, which depends on long-range electric fields, is the subject of ongoing experimental and numerical work in our group. Conclusions Our numerical calculations showed that for liquid droplets on substrates with a striped wettability pattern the modulation of both the contact line and the local contact angle decreases both with increasing contact line tension (28) Wiegand, G.; Jaworek, T.; Wegner, G.; Sackmann, E. Langmuir 1997, 13, 3563. (29) Dobbs, H. Physica A 1999, 271, 36. (30) Vallet, M.; Vallade, M.; Berge, B. Eur. Phys. J. B 1999, 11, 583. (31) Mugele, F.; Herminghaus, S. Appl. Phys. Lett. 2002, 81, 2303.
Line Tension on the Equilibrium Shape of Droplets
and with decreasing sharpness of the wettability pattern. The contact angle is found to follow the predictions of a generalized form of the Young equation. If the line tension varies as a function of position, its gradient contributes significantly to the local contact angle. We demonstrated that the line tension can be extracted from the surface profile of a droplet. This can be done without knowing the detailed shape of the wettability pattern provided that it is sufficiently sharp. Comparison to experimental results shows that the line tension for the system under inves-
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tigation is of the order of 10-11 J/m, in agreement with theoretical predictions. A full and detailed agreement between experiments and numerical calculations, however, requires a higher quality of the experimental wettability patterns. Acknowledgment. This work was supported by the German Science Foundation within the priority program “Wetting and Structure Formation at Interfaces”. LA0204693
