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Simulating Anisotropic Droplet Shapes on Chemically Striped Patterned Surfaces H. Patrick Jansen, Olesya Bliznyuk, E. Stefan Kooij,* Bene Poelsema, and Harold J. W. Zandvliet Physics of Interfaces and Nanomaterials, MESA+ Institute for Nanotechnology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands ABSTRACT: The equilibrium shape of droplets on surfaces, functionalized with stripes of alternating wettability, have been investigated using simulations employing a finite element method. Experiments show that a droplet deposited on a surface with relatively narrow hydrophobic stripes compared to the hydrophilic stripes adopts a strongly elongated shape. The aspect ratio, the length of the droplet divided by the width, decreases toward unity when a droplet is deposited on a surface with relatively narrow hydrophilic stripes. The aspect ratio and the contact angle parallel to the stripes show unique scaling behavior as a function of the ratio between the widths of the hydrophobic and hydrophilic stripes. For a small ratio, the contact angle parallel to the stripes is low and the aspect ratio high, while for a large ratio, the contact angle parallel is high and the aspect ratio low. The simulations exhibit similar scaling behavior, both for the aspect ratio of the droplets and for the contact angles in the direction parallel to the stripes. Two liquids with different surface tensions have been investigated both experimentally and in simulations; similarities and differences between the findings are discussed. Generally, three parameters are needed to describe the droplet geometry: (i) the equilibrium contact angles on the hydrophilic and (ii) hydrophobic areas and (iii) the ratio of the widths of these chemically defined stripes. Furthermore, we derive a simple analytical expression that proves to be a good approximation in the quantitative description of the droplet aspect ratio.
’ INTRODUCTION In recent years, anisotropic wetting has attracted a lot of scientific interest. It can be used for a wide range of applications including lubrication, printing, waterproofing, and microfluidics.15 There are different methods to pattern a surface so that it exhibits anisotropic wetting properties. Two possible ways to create a surface that exhibits an anisotropy in wetting behavior involve chemical or geometrical patterns, leading to what are generally referred to as two- and three-dimensional surfaces, respectively. Well-defined patterns arising from these methods are ideal for comparing simulations with experiments. A number of simulations and experiments on surfaces with spatially varying wettability have already been performed. For example, experiments as well as simulations have been done on grooved surfaces.615 Droplets deposited on such a surface typically have two distinct contact angles, one perpendicular to the grooves Θ^ and the other parallel to the grooves Θ . In all studies, it was found that Θ^ > Θ due to pinning of the droplet in the perpendicular direction. Owing to the more favorable spreading in the direction parallel to the grooves, the droplets are generally elongated along the grooves. Chen et al.7 used the freely available software package Surface Evolver (SE) to model a droplet on a surface consisting of parallel grooves, where the droplet was deposited on six ridges. They showed that such a droplet is no longer spherical, but adopts an elongated shape. In another study, Semprebon et al.13 used experiments as well as Lattice Boltzmann simulations to study a droplet on top of a single ridge. By increasing the volume of the droplet, the perpendicular contact angle Θ^ is increased while Θ remains constant )
)
resulting in a more pronounced elongation of the droplet. The difference in contact angles is directly related to the elongation and to the volume of the droplet, normalized to the ridge size. Not only were the equilibrium contact angles and the static shape of droplets on two- or three-dimensional striped surfaces investigated, but the contact angle hysteresis for these patterns has also been considered. Brendon et al.1618 as well as various other groups used a variation in contact angles in order to simulate the effect of different volumes on the contact angle hysteresis.1925 Generally, the contact angle hysteresis parallel to the grooves or stripes is much lower than that perpendicular to the stripes. Finally, another topic which has been addressed is the modeling of the corrugation of the contact line when a liquid droplet spans a region with a change in wettability. Numerous studies have shown that the contact line is not straight but has a wave-like behavior.9,26,27 In this paper, we perform simulations on the anisotropic equilibrium droplet shape and we will restrict ourselves to twodimensional chemically patterned surfaces. There are a number of ways to create a chemical pattern on a surface, which include chemical vapor deposition (CVD) with different exposure times or diffusion-controlled silanization,4,2831 destruction of a monolayer with UV-light after a CVD procedure,3234 and microcontact printing.35,36 In our case, we have used lithographic tools to pattern silicon wafers with a photoresist, which is subsequently
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Received: October 10, 2011 Revised: November 9, 2011 Published: November 10, 2011
r 2011 American Chemical Society
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used to protect the wafer against chemical vapor deposition of 1H,1H,2H,2H-perfluorodecyltrichlorosilane (PFDTS). The photoresist is then removed leaving a patterned surface consisting of alternating stripes of bare SiO2 (which were covered by the photoresist) and of PFDTS.15,3739 The focus of our work will be on chemically defined parallel stripes of alternating wettability that have different stripe widths with respect to each other. Most simulations done on these patterns were either performed to study contact angle hysteresis or were focused on the shape of the contact line. A few studies consider the shape of the entire droplet,4,7,12,18,21 but are limited to investigating the shape as a function of volume. Unfortunately, the effect of stripe width was not addressed. Furthermore, simulations have also been done to study dynamics on chemically patterned as well as on superhydrophobic surfaces,4042 where the emphasis was on droplet spreading behavior. Our present simulation approach does not provide realistic dynamics of droplet formation and behavior. For this purpose, Lattice Boltzmann simulations are ideally suited, which is part of our present research focus but lies outside the scope of this paper. We describe an investigation into the macroscopic properties of a droplet on a chemically striped patterned surface, and will therefore focus on the macroscopic contact angles and the aspect ratio ξ of the droplet. These simulations will determine the relevant parameters for the entire shape of a droplet, while it also allows us to investigate more liquids with different surface tensions. Additionally, our results described here provide a benchmark to assess the accuracy and reliability of the aforementioned Lattice Boltzmann simulations, since the final result of both approaches for an identical situation should be the same. The simulations will then be compared to experimental data measured on chemically striped patterned surfaces. Finally, a simple model is presented yielding an analytical expression for the droplet aspect ratio which agrees remarkably well with both the experiments and the simulations.
’ STRIPED PATTERNS As mentioned in the Introduction, we will focus on a specific type of chemically heterogeneous surface, namely, those consisting of stripes of different wettability. The surface patterns consist of alternating hydrophobic and hydrophilic stripes that are parallel to each other. Typically, a droplet placed on such a surface will adopt a nonspherical equilibrium shape due to anisotropic spreading.34,37 A schematic representation of this shape is shown in Figure 1a. Spreading parallel to the stripes is easier than spreading perpendicular to the stripes. When the droplet spreads perpendicular to the stripes, it has to overcome relatively high energy barriers formed by the hydrophobic stripes resulting in limited wetting in the perpendicular direction. In contrast, parallel to the stripes such discrete energy barriers do not exist. In previous work,38 we have shown that the initial stage of spreading, when the droplet comes into contact with the surface, is isotropic. After this first inertial stage, surface properties take over and control the subsequent spreading of the droplet. It was shown that in this capillary regime spreading perpendicular to the stripes is markedly slower and eventually stops, while spreading parallel to the stripes still continues. This eventually results in a droplet having an elongated shape on a chemically striped surface. The aspect ratio ξ = L/W of the droplet, defined by the length L at the baseline (major axis) divided by the width W at the baseline (minor axis), depends on the ratio of hydrophobic to
)
Figure 1. (a) Schematic representation of a droplet on a chemically striped patterned surface, consisting of hydrophobic and hydrophilic stripes, referred to as dry and wet areas, respectively. The droplet typically exhibits an elongated shape due to preferential spreading along the y-axis. (b) Contact angle Θ^ of the droplet perpendicular to the direction of the stripes, i.e., normal to the y-axis. (c) Parallel contact angle Θ , i.e., normal to the x-axis.
hydrophilic area. Bliznyuk et al.37 introduced a dimensionless parameter α α¼
wPFDTS wSiO2
ð1Þ
)
)
in which wPFDTS and wSiO2 represent the widths of the hydrophobic and hydrophilic stripes, respectively. The parameter α is directly related to the macroscopic surface energy of a specific surface design: a small α corresponds to a high surface energy, i.e., a hydrophilic surface, while a large α with a low surface energy corresponds to a hydrophobic surface. In our previous work, we concluded that α is an important parameter, since the aspect ratio ξ of the droplets as well as the contact angles in the parallel direction Θ exhibit scaling behavior as a function of α. The definition of the contact angles perpendicular Θ^ and parallel Θ to the stripes is depicted in Figure 1b and c, respectively. The experimental studies were done with droplets of fixed volume of 3 μL and stripe widths in the micrometer range; the minimum width of the stripes used is 2 μm and the maximum width amounts to 30 μm. For our experiments, the droplet volumes and the experimental procedures are identical to those described in a recent paper by Bliznyuk et al.37 Briefly summarizing, using standard cleanroom facilities silicon wafers with a thin layer of natural oxide are coated with positive photoresist, enabling pattern creation via optical lithography and providing surface protection during vapor deposition of 1H,1H,2H,2Hperfluorodecyltrichlorosilane (PFDTS, ABCR, Germany). After formation of the PFDTS self-assembled monolayer, the photoresist is washed off, leaving a chemically patterned surface.
’ MODEL CALCULATIONS The shapes of three-dimensional sessile droplets are computed using a finite element method as implemented in Surface Evolver, which is freely available.43 Surface Evolver (SE) is an interactive program for the study of surfaces shaped by energies which are a combination of surface tension, gravitational energy, squared mean curvature, user-defined surface integrals, and/or knot energies.43 In SE, a surface is implemented as a union of triangles formed by connecting vertices. Generally, the number of vertices can be increased for better accuracy of the shape. Using SE, the surface is evolved toward its minimal energy by a gradient descent method in a number of iteration steps. In our case, we only consider the surface tension of the liquid and the interaction with the substrate surface. The droplet is in contact with a substrate and it will reach a certain equilibrium size and shape due to energy minimization. In the simulations, gravity is ignored, since the droplets that were typically used for our experiments have a sufficiently small volume to neglect the 500
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influence of gravity. The equilibrium drop shape is obtained by a numerical procedure which minimizes the total free energy of the droplet. The free energy G of the system is given by7,17,18 ð2Þ
where σlv is the interfacial tension between the liquid and the vapor. Slv and Ssl are the liquidvapor and the solidliquid interfacial areas of contact. θi represents the intrinsic contact angle that generally depends on the position x and y on the substrate. A typical substrate surface is divided into alternating stripes with high (hydrophobic) and low (hydrophilic) contact angles; from now on, these stripes will be referred to as dry and wet stripes, respectively. The transition region is assumed to be sharp, i.e., we consider the modeled boundary between stripes as a stepfunction in surface energy and therewith a sharp discontinuity in contact angle. The intrinsic contact angle θi is defined by Young’s equation.
)
ð3Þ
where σlv, σsv, and σsl represent the interfacial tension between the liquid and the vapor, the solid and the vapor, and the solid and the liquid, respectively. Line tension is neglected in the simulations, since it is small compared to the size of the droplets that were used in our experiments. Pompe et al.44 have experimentally shown that the magnitude of the line tension amounts to 1010 J/m using scanning force microscopy (SFM) on chemically patterned surfaces. Buehrle et al.26 report a similar value using simulations on chemically striped patterns. As mentioned above, the substrate surface is divided into alternating wet and dry stripes. The width of the stripes with respect to each other can be varied. From the previous experimental study,37 it was found that only the ratio of hydrophobic to hydrophilic area α (given in eq 1) is the relevant parameter and not the absolute width of the stripes provided a number of conditions are met. The radius of the droplet needs to be lower than the capillary length (∼2 mm) such that gravity does not affect the shape of the droplet. Also, the radius of the droplet must be sufficiently large such that the droplet spans a number of stripes. When the size of the droplet is comparable with the width of a stripe, the scaling also does, of course, not hold, since only the wet stripe will be covered with liquid. A volume of liquid is placed on the surface and it is confined to stay on top of a region of 30 stripes; if this restriction was not imposed, we encountered problems in evolving the droplet toward an equilibrium shape due to the fact that there is no gradient in energy between the different stripes. In that case, vertices, i.e., the points that form a triangle, do not have a specific direction to move along and started to move randomly resulting in a runaway of vertices. This produced an overall droplet shape that is not physical. In order to avoid this effect, vertices are placed at the boundary between the stripes and confined to stay there ensuring that a runaway situation does not occur. The initial shape of the liquid volume is that of a cube. Upon starting the iterations, the vertices of the droplet are moved in the direction prescribed by (i) the surface forces acting on them, (ii) a fixed volume constraint, and (iii) the constraint that the droplet has to be on the surface. This will continue until the total energy of the droplet no longer changes significantly, i.e., no
change on the order of 109. The surface is then refined for a more precise shape determination by adding additional vertices, and the iterations are recommenced. This procedure is repeated a number of times until the final refined equilibrium shape is reached. For a more detailed description of the technical procedure, we refer to the SE manual.43 The contact angles Θwet and Θdry are input parameters for the simulations, as well as the ratio α in eq 1. During a simulation run, the liquid volume is kept fixed at a predefined chosen value. For a constant α value, different droplet volumes have been simulated. When the volume of the droplet becomes too high, the contact angle perpendicular to the stripes exceeds the intrinsic contact angle on the dry stripe due to the confinement of the droplet. In actual experiments, the droplet would then move to the next stripe, but in the simulation, this is not allowed; the droplet is forced to stay on the last wet stripe. In fact, this approximates the metastable free energy minimum state the experimental droplets are in preventing them from reaching the global minimum due to pinning of the contact line on the more hydrophobic stripes. As a consequence, the shape of the liquid mass is adapted in such a way that the contact angle perpendicular to the stripes is not larger than the equilibrium value on a dry stripe, in agreement with the experiments.
’ RESULTS In Figure 2, the equilibrium shape of a droplet is shown after the evolution on a surface with α = 1 at different viewing angles. It can be seen that Θ and Θ^ are very different. Θ is always lower than Θ^, since Θ is determined by the combined effect of the wet and dry stripes, while Θ^ is mainly determined by the contact angle on the dry stripe; in this particular example, we have taken Θ^ = 104 and Θ = 70. The elongated shape is also apparent from the side views of the droplet; the top view gives a good overview of the entire droplet shape. Furthermore, the corrugated contact line is nicely displayed in Figure 2d. The droplet spreads on both the wet and dry stripes, but it spreads further on the wet stripes, resulting in a wave-like contact line. An enlarged image of the corrugated contact line is also shown. In the remaining part of this work, we will not consider the )
σsv σsl σlv
) )
cosðθi Þ ¼
Figure 2. Equilibrium shape of a droplet on a striped surface with α = 1. (a) Side view of the droplet shape in the direction parallel to the stripes; the contact angle Θ is approximately 70. (b) Side view of the droplet shape perpendicular to the stripes showing the contact angle Θ^ of approximately 104. (c) Top-view of the droplet with the patterned surface underneath. The dark and bright areas are the wet (hydrophilic) and the dry (hydrophobic) stripes, respectively. Note the elongated shape of the droplet as well as the corrugated contact line. (d) The corrugated contact line as seen from a non-normal perspective.
)
ZZ G ¼ Slv cosðθi ðx, yÞÞ dx dy Ssl σlv
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)
)
Figure 3. (a) Aspect ratio ξ of the droplet plotted as a function of α. For low α values, ξ is large due to strong preferential spreading; for large α values, the aspect ratio approached 1, corresponding to almost spherical droplets. Results from experiments and simulations are shown by triangles and circles, respectively. The unmarked solid line represents a model calculation as described in the text. The contact angles 41 (dry) and 106 (wet) used for these simulations correspond to those of glycerol. (b) Θ^ and Θ plotted as a function of α from experiments (diamonds, triangles) and simulations (squares, circles). Θ^ is close the equilibrium angle on the dry stripe and Θ follows the CassieBaxter equation, represented by the unmarked solid line.
corrugated contact line, since we are only interested in the macroscopic properties of the droplet. Simulations were done considering patterns with Θwet = 41 and Θdry = 106. These values are identical to the contact angles as described in previous work by Bliznyuk et al.37 The liquid used during this experiment is glycerol, which has a surface tension of about 65 mN/m, i.e., very similar to water. Liquids with different values for the surface tension will exhibit other contact angles on pristine dry and wet surfaces. In the simulations, the initial angles on the two chemical species are adjusted to account for the change in surface tension. During simulations, an elongated shape is typically obtained similar to the experimental ones. The aspect ratio ξ of the simulated droplet is plotted as a function of α in Figure 3a. A pronounced decrease of the aspect ratio from values exceeding 3 toward unity is observed with larger wet stripe widths, i.e., with increasing macroscopic hydrophobicity, very similar to the experimental data. For a surface with low α, it is much more favorable for the droplet to elongate, so droplets on such a surface will exhibit large ξ values. For high α surfaces, the droplet is almost spherical since there is only little preferential spreading leading to an aspect ratio close to unity. In Figure 3b, the dependence of the directional contact angles Θ^ and Θ on the relative hydrophobicity α as determined from the experiment is compared to the simulation results. In the experiments, the contact angles are determined using a goniometer, while for the simulations, the edge of the droplet, i.e., the liquid vapor interface, is fitted with a circle. Owing to the corrugation of the contact line, it was not possible to take the tangent of the drop profile to determine the contact angle in the direction parallel to the stripes. Instead, we used the height of the droplet as obtained from the fitted circular profile. The contact angle is then determined using h ð4Þ Θ|| ¼ arccos 1 R )
)
)
)
)
Overall, there is good agreement between experimental and simulated data. In the direction perpendicular to the stripes, Θ^ is equal to the contact angle on the dry stripe and amounts to approximately 106. There does not appear to be much dependence on α. The contact line is pinned at the edge between a wet and a dry stripe and is not able to advance over the next dry stripe. It will adopt the equilibrium contact angle of that stripe for almost the entire range of α values. Only for very small α values, i.e., below approximately 0.2, does the contact angle Θ^ decrease due to volume effects. Also plotted in Figure 3b are the measured and simulated contact angles Θ in the direction parallel to the stripes as a function of α. In contrast to the almost unchanged value of Θ^ with varying hydrophobicity, Θ behaves remarkedly different, exhibiting a pronounced increase with increasing α. As the contact line probes the effect of many alternating wet and dry stripes, the value of Θ is determined by both wet and dry surface areas. Again, it is clear that the experiments are in good agreement with the simulations. From the experiments, it was found that Θ is adequately described by the CassieBaxter equation. Bliznyuk et al.37 rewrote the CassieBaxter (CB) such that the fractions f1 and f2 are expressed in terms of α. This yields " # α cosðΘdry Þ þ cosðΘwet Þ Θ|| ¼ arccos ð5Þ 1 þ α The CB relation as shown in Figure 3b by the solid line was obtained using Θwet = 41 and Θdry = 106 as parameters. These values correspond to those measured for the respective pristine surfaces. Our simulations are in good agreement with the experiments. As expected, the CB expression with these limiting contact angle values provides a good description of the simulation results. In addition to glycerol, we performed similar experiments using another liquid. To assess the effect of surface tension on the various aforementioned parameters, we have chosen decanol. Decanol was purchased from Merck (Germany); it has a surface tension of 28 mN/m, i.e., considerably lower than that of glycerol, which amounts to 65 mN/m. Figure 4a shows the droplet aspect ratio ξ as a function of α for decanol. The contact angles of decanol on pristine surfaces were determined to be Θwet = 30 and Θdry = 61; we used these values as input parameters for our simulations. The experimental
in which h is the height of the droplet and R represents the radius of curvature. Θ^ was determined by taking the tangent of the drop profile, which proved to be more accurate than fitting it with a circle. The profile of the droplet was, however, not perfectly fitted at the baseline in the perpendicular direction using the circular fit. Overall, the uncertainty in the contact angles amount to 2 at most for both the experimental data and the simulation results. 502
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Figure 4. (a) Aspect ratio ξ as a function of α for decanol, obtained from experiments (triangles) and simulations (circles) using Θwet = 30 and Θdry = 61. The unmarked solid line corresponds to the analytical model described in the text. (b) Directional contact angles Θ^ and Θ as a function of α for decanol, from experiments (diamonds, triangles) and simulations (squares, circles). The unmarked solid line represents the CassieBaxter relation as given in eq 5.
data and the simulation results are plotted in the same graph to allow an easy comparison. The simulations show good agreement with the experiments. As with the glycerol results in Figure 3a, the aspect ratio of the decanol droplets decreases with increasing α; the most pronounced difference between the two liquids pertains to the magnitude of the aspect ratio. For glycerol, values above 3 were observed, while for decanol, the highest value is approximately 2.4. Apparently, the lower surface tension gives rise to less elongation of the droplets. The directional contact angles Θ^ and Θ obtained with a decanol droplet are plotted as a function of α in Figure 4b. The Θ^ values exhibit a bit more scattering in the experimental set as the glycerol droplets; as with glycerol, a pronounced decrease of Θ^ occurs for small α. The values obtained by the simulations do not seem to depend strongly on α, not even at small α values. Nevertheless, the simulated perpendicular contact angles are of the same order of magnitude as the experimental ones. Also shown in Figure 4b is the dependence of Θ on the relative hydrophobicity α for both the experiments and the simulations. The agreement between both sets is good; only for small α do the simulated Θ values become lower than the experimental data. In addition, the results also start to deviate from the CB equation, represented by the solid line in Figure 4b. We ascribe this small discrepancy to an error in fitting the droplet profiles obtained with Surface Evolver by a circle.
)
)
Figure 5. Schematic representation of the droplet shape by a spherical cap: In the directions perpendicular (a) and parallel (b) to the stripes, the contact angles of the droplet are Θ^ and Θ , respectively.
)
)
agreement. This once again confirms that the ratio of hydrophobic-to-hydropholic stripe widths α is the quantity that determines the aspect ratio ξ of the droplets, and not the absolute stripe widths. Obviously, there is a limit to the minimum size of the droplets for which this scaling behavior holds. This is presently under investigation, but lies outside the scope of this work. The similarities between the experimental data and the simulations demonstrate that simulations can be used to predict the equilibrium properties of droplets on stripepatterned surfaces. Unfortunately, the simulations are very time-consuming and require significant computational effort. Moreover, they need to be repeated for every design. Considering the droplet shape in both orthogonal directions, we now derive a model which provides a simple analytical expression for the aspect ratio ξ as a function of α. From the experimental and simulation results, we found that Θ can be described by the CB equation (eq 5) while Θ^ is almost constant over a large α range and equal to Θdry. Moreover, considering the small droplet volumes allows us to neglect the effect of gravity on the droplet shape. Thus, we can model the circumference of the droplet in the high-symmetry directions as a hemispherical cap with height h. In Figure 5, the droplet profiles perpendicular (left) and parallel (right) to the stripes are schematically shown. To derive the aspect ratio ξ = L/W, we need to know the width W and the length L of the droplet as a function of the aforementioned contact angles. Using Figure 5 and taking into account that the height on both sides is the same )
’ DISCUSSION As described in the previous section, there is good agreement between the simulations and the experimental results as shown in Figure 3 and Figure 4, both for the aspect ratio as well as the directional contact angles as a function of relative hydrophobicity. The number of stripes that is spanned by the droplet in the experiments is in the range of 80700 stripes. In the simulations, that we have performed using Surface Evolver, the number of stripes under the droplet is limited to 30, which implies that effectively droplets with a significantly smaller volume are simulated compared to that were used in the experiments. The effective volume of the simulated droplets corresponds to 0.3 nL up to 0.15 μL, depending on the number of stripes spanned by the experimental droplets. In comparison, a volume of 3 μL has been used for the experiments. Nevertheless, despite this difference, the simulations and the experimental data show good
h ¼ R^ þ r^ ¼ R|| r|| 503
ð6Þ
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Using simple trigonometry, we can state the following: R^ ¼
W 1 2 sinðΘ^ Þ
ð7Þ
r^ ¼ R^ cosðΘ^ Þ
ð8Þ
L 1 2 sinðΘ|| Þ
ð9Þ
R|| ¼
results using various liquids with different properties are in good agreement with the experimentally measured parameters. Additionally, we present a simple model, which provides analytical expressions for the droplet aspect ratio in terms of the directional contact angles. Despite its simplicity, this model enables us to calculate the aspect ratio as a function of relative hydrophobicity in a good approximation. Simulations as described in this work, as well as the analytical model, enable us to predict the contact angles and elongation of a droplet on a chemically defined anisotropic patterned surface, for different liquids and substrate wettability. This allows us to effectively design and optimize chemically striped patterned surfaces in general, and more specifically for applications where control over droplet motion is required.
r|| ¼ R|| cosðΘ|| Þ
ð10Þ
Using these equations, we can derive the aspect ratio ξ. # " L 1 cosðΘ^ Þ sinðΘ|| Þ ξ¼ ¼ ð11Þ W sinðΘ^ Þ 1 cosðΘ|| Þ
Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT The authors thank Prof. Frieder Mugele, University of Twente, for helpful discussions. We gratefully acknowledge the support by MicroNed, a consortium that nurtures microsystems technology in The Netherlands. ’ REFERENCES (1) Coninck, J. D.; de Ruijter, M. J.; Voue, M. Curr. Opin. Colloid Interface Sci. 2001, 6, 49–53. (2) Stone, H. A.; Kim, S. AIChE J. 2001, 47, 1250–1254. (3) Stone, H. A.; Stroock, A. D.; Ajdari, A. Annu. Rev. Fluid Mech. 2004, 36, 381–411. (4) Liao, Q.; Wang, H.; Zhu, X.; Li, M. Science in China Series E: Technological Sciences 2006, 49, 733–741. (5) Moradi, N.; Varnik, F.; Steinbach, I. Europhys. Lett. 2010, 89, 26006–+. (6) Gau, H.; Herminghaus, S.; Lenz, P.; Lipowsky, R. Science 1999, 283, 46–49. (7) Chen, Y.; He, B.; Lee, J.; Patankar, N. A. J. Colloid Interface Sci. 2005, 281, 458–464. (8) Chung, J. Y.; Youngblood, J. P.; Stafford, C. M. Soft Matter 2007, 3, 1163–1169. (9) Dorrer, C.; R€uhe, J. Langmuir 2007, 23, 3179–3183. (10) Khare, K.; Herminghaus, S.; Baret, J.-C.; Law, B. M.; Brinkmann, M.; Seemann, R. Langmuir 2007, 23, 12997–13006. (11) Zhao, Y.; Lu, Q.; Li, M.; Li, X. Langmuir 2007, 23, 6212–6217. (12) Kusumaatmaja, H.; Vrancken, R. J.; Bastiaansen, C. W. M.; Yeomans, J. M. Langmuir 2008, 24, 7299–7308PMID: 18547090. (13) Semprebon, C.; Mistura, G.; Orlandini, E.; Bissacco, G.; Segato, A.; Yeomans, J. M. Langmuir 2009, 25, 5619–5625PMID: 19379004. (14) Yang, J.; Rose, F. R. A. J.; Gadegaard, N.; Alexander, M. R. Langmuir 2009, 25, 2567–2571. (15) Bliznyuk, O.; Veligura, V.; Kooij, E. S.; Zandvliet, H. J. W.; Poelsema, B. Phys. Rev. E 2011, 83, 041607. (16) Brandon, S.; Marmur, A. J. Colloid Interface Sci. 1996, 183, 351–355. (17) Brandon, S.; Wachs, A.; Marmur, A. J. Colloid Interface Sci. 1997, 191, 110–116. (18) Brandon, S.; Haimovich, N.; Yeger, E.; Marmur, A. J. Colloid Interface Sci. 2003, 263, 237–243. (19) Gleiche, M.; Chi, L.; Gedig, E.; Fuchs, H. ChemPhysChem 2001, 2, 187–191. (20) Long, J.; Hyder, M. N.; Huang, R. Y. M.; Chen, P. Adv. Colloid Interface Sci. 2005, 118, 173–190.
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)
)
)
Finally, by using the CB relation in eq 5 for Θ and taking Θ^ = Θdry, we can now plot ξ as function of α. The solid lines in Figure 3a and Figure 4a show the relation between droplet aspect ratio and α for both liquids. For glycerol, the comparison between the aforementioned model and the data sets in Figure 3a reveals that, despite its simplicity, this model is a very acceptable approximation. The experiments and the simulations exhibit the same trend, but the model appears to overestimate ξ slightly. Considering the contact angles for glycerol in Figure 3b, for small α the simulated Θ values are lower than the ones corresponding to the CB equation, while for higher α, this curve is a good approximation. For small α, there were more difficulties in fitting the droplet profile with a circle, resulting in lower contact angles. Θ in the model is given by eq 5 and the experiments are slightly higher than this curve leading to a smaller difference between the contact angles and therefore a smaller ξ in comparison to the model. A comparison of the experiments, the simulations, and the model in the case of decanol in Figure 4a reveals some discrepancies which are most pronounced for the lower α range in the graph; ξ in the experiments and simulations is much higher than ξ as obtained from the model. This is primarily due to the approximation that Θ^ is constant over the entire α range, while this is obviously not the case for the experimental results at low α. The Θ^ values are considerably lower at small α, leading to a higher ξ as predicted on the basis of the simplified model. Since decanol has a lower surface tension than glycerol, the contact angles on the wet and dry stripes are also smaller. Consequently, the elongation of the droplet and the differences between Θ^ and Θ are smaller. The difference between Θwet and Θdry is lower for decanol, which leads to less preferential spreading. The energy barriers on the dry stripes are lower for smaller contact angles giving rise to wider droplets and therewith smaller ξ values.
’ AUTHOR INFORMATION
’ CONCLUSIONS Simulations with a finite element method using Surface Evolver were done to investigate the behavior of liquid droplets on chemically striped patterned surfaces. By changing the relative widths of the hydrophobic and hydrophilic stripes, previous experimental studies have shown that the shape of the droplets, i.e., aspect ratio and contact angles, exhibit scaling behavior. This scaling is only dependent on the ratio of the widths of the hydrophobic to hydrophilic stripes. In general, our simulation 504
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dx.doi.org/10.1021/la2039625 |Langmuir 2012, 28, 499–505
