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Partial wetting of chemically patterned surfaces: The effect of drop size
Journal of Colloid and Interface Science 263 (2003) 237–243 www.elsevier.com/locate/jcis
Partial wetting of chemically patterned surfaces: The effect of drop size Simon Brandon,∗ Nir Haimovich, Einat Yeger, and Abraham Marmur Department of Chemical Engineering, Technion—Israel Institute of Technology, 32000 Haifa, Israel Received 12 June 2002; accepted 13 March 2003
Abstract Partial wetting of chemically heterogeneous substrates is simulated. Three-dimensional sessile drops in equilibrium with smooth surfaces supporting ordered chemical patterns are considered. Significant features are observed as a result of changing the drop volume. The number of equilibrated drops is found either to remain constant or to increase with growing drop volume. The shape of larger drops appears to approach that of a spherical cap and their three-phase contact line seems, on a larger scale, more circular in shape than that of smaller drops. In addition, as the volume is increased, the average contact angle of drops whose free energy is lowest among all equilibrium-shaped drops of the same volume appears to approach the angle predicted by Cassie. Finally, contrary to results obtained with two-dimensional drops, contact angle hysteresis observed in this system is shown to exhibit a degree of volume dependence in the advancing and receding angles. Qualitative differences in the wetting behavior associated with the two different chemical patterns considered here, as well as differences between results obtained with two-dimensional and three-dimensional drops, can possibly be attributed to variations in the level of constraint imposed on the drop by the different patterns and by the dimensionality of the system. 2003 Elsevier Science (USA). All rights reserved. Keywords: Contact angle; Contact angle hysteresis; Cassie equation
1. Introduction Wetting of chemically heterogeneous and/or rough surfaces, a process commonly occurring in both natural and man-made environments, has been studied for several decades. The equilibrium contact angle on such surfaces has been measured (e.g., [1,2]) and calculated (e.g., [3,4]) by a number of authors. Most notable are the simple predictions by Cassie [5] and Wenzel [6] for the equilibrium contact angles on chemically heterogeneous and rough surfaces, respectively. These predictions were confirmed to be qualitatively correct for a wide range of conditions both in experiments and in calculations. The Cassie and Wenzel angles can be calculated using 1 cos(θC ) = (1) cos(θi )(x, y) dx dy Ssl Ssl
and cos(θW ) = ρ cos(θi ), * Corresponding author.
E-mail address: [email protected] (S. Brandon).
(2)
where, θC , θW , and θi are the Cassie, Wenzel, and intrinsic contact angles, respectively. The solid–liquid interfacial area is given by Ssl , x and y are the spatial coordinates spanning this interface, and ρ is the surface roughness, which is defined as the ratio of the solid surface area to its value when projected on to the tangent plane. Although Eqs. (1) and (2) yield a single-valued contact angle for a given solid surface and liquid–fluid system, it is well known (see, e.g., [7]) that rough and/or chemically heterogenous surfaces may exhibit contact angle hysteresis, which is a symptom of multiple local equilibrium states. Moreover, the multiplicity of local minima in the system’s free energy is strongly dependent on the drop volume and on parameters characterizing the surface (see, e.g., [8]). It has been indicated [9,10], and recently proven [11,12] that the Cassie and Wenzel angles can serve as approximations to the contact angle at the global minimum in the system’s free energy. According to [11,12] this approximation becomes better as the drop size increases relative to the scale of heterogeneity and/or roughness. Here we present a computational investigation of the effects of drop size and pattern type on local and global minima in the free energy of three-dimensional sessile drops partially wetting chemically patterned smooth surfaces. Results are discussed in relation to the relevant Cassie angle
0021-9797/03/$ – see front matter 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0021-9797(03)00285-6
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as well as, in some cases, to two-dimensional simulations of a similar system. The three-dimensional simulations were achieved, as in the case of our previous study [13], using the public-domain “Surface Evolver” software [14,15], while two-dimensional results were obtained analytically using a model system discussed in [3].
2. Model formulation and solution technique Three-dimensional sessile drops in equilibrium with chemically heterogeneous (patterned) substrates are computed using an approach similar to that employed in [13]. Both the liquid drop and the solid surface on which it is placed are immersed in an ambient fluid; the liquid and fluid are assumed to be mutually immiscible. Ignoring gravitational effects1 and three-phase interactions at the contact line, the dimensionless free energy of the system is given by G = Slf − (3) f (x, y) dx dy, Ssl
where G is the free energy normalized with respect to σlf l 2 , σlf is the liquid–fluid interfacial energy, l is a length-scale characteristic of the chemical pattern, and Slf is the area of contact along the liquid–fluid interface. The spatial coordinates (x, y) are normalized with respect to l, while Ssl and Slf are rendered dimensionless using the scaling factor l 2 . Finally, the solid surface energy function, f (x, y), is defined as f (x, y) = cos(θY ) =
σsf (x, y) − σsl (x, y) . σlf
(4)
Here, σsl (x, y) and σsf (x, y) are the local solid–liquid and solid–fluid interfacial energies, respectively, and θY is the local Young contact angle, which, as discussed below, is in our case equal to the intrinsic contact angle (θi (x, y)). In this study we consider two forms of the solid energy function, representing either a one-dimensional periodic variation in solid surface energy, f (x) = cos(θ0 ) + φ cos(2πx),
(5)
or a two-dimensional periodic solid surface energy pattern, f (x, y) = cos(θ0 ) + φ cos(2πx) cos(2πy) − cos(2πx) − cos(2πy) , (6) where the chemical pattern given by Eq. (5) was already considered in our previous three-dimensional [13] as well as two-dimensional [3] analyses of contact angle hysteresis. 1 Strictly speaking this limits the analysis to drops with a radius ap proximately smaller than 2σlf /(ρl g) where ρl , σlf , and g are the liquid density, liquid–fluid interfacial energy, and gravitational acceleration, respectively.
Minimization of the free energy G (Eq. (3)) with respect to the liquid–fluid interface shape, while the drop’s volume is constrained to a fixed value, yields an equilibrium-shaped drop. This, within the framework of the main assumptions of this work, must satisfy both the Young–Laplace and the Young equations. As a direct consequence, the liquid–fluid interface is required to exhibit uniform curvature while (as mentioned above) the local contact angle at the three-phase boundary is constrained to an intrinsic value (θi ) which is uniquely defined for every position of this boundary via Eq. (4). Therefore, calculation of the liquid–fluid interfacial mean curvature and local contact angles from the numerically obtained equilibrium drop shapes serves as a test of their accuracy. Although local contact angles of three-dimensional drops are well defined, average contact angles of these systems are, in general, not unique. In this study we use two possible definitions of the average contact angle. The first (θ1 ), used in [13], is given by θ dl , θ1 = C (7) C dl where C refers to the closed integration path along the threephase contact line; the second average angle (θ2 ) is given by 3V H 3 = 0, (8) π where V is the volume of the drop and H is its liquid–fluid interfacial mean curvature. Notice that θ2 is the angle corresponding to that of a spherical cap whose mean curvature and volume are equal to the three-dimensional drop’s mean curvature and volume, respectively. The choice of θ1 stems from its simplicity while that of θ2 is motivated by Ref. [11] in which it was proven that, for drops partially wetting rough surfaces, θ2 approaches the Wenzel contact angle when the drop is sufficiently large. As mentioned above, the public-domain software package “Surface Evolver” was employed in the calculation of 3-D equilibrium drop shapes. This software, which in essence minimizes the system’s free energy (Eq. (3)), is described in detail in [14,15]. Additional information on our application of Surface Evolver to the study of 3-D wetting of heterogenous surfaces is given in [13]. Two-dimensional simulations of cylindrical drops were obtained with surfaces described by Eq. (5). In this case an analytical solution for the dimensionless free energy (per unit length of the drop) of the system is given by [3,8] cos3 θ2 − 3 cos θ2 + 2 −
G2D =
2rθ φ − 2r cos θ0 − sin 2π(r + x0 ) sin θ 2π − sin −2π(r − x0 ) ,
(9)
where G2D is the free energy per unit length of the drop normalized by σlf l, r is the “radius” of the base of the drop (normalized by l), and x0 is the x position (also normalized using l) of the center of the drop. An additional equation relating the volume (per unit length) of the cylindrical drops
S. Brandon et al. / Journal of Colloid and Interface Science 263 (2003) 237–243
to r is given by V 2D = r 2
θ − sin θ cos θ sin2 θ
,
(10)
where V 2D is the volume (per unit length) normalized with respect to l 2 . As implied in [8], an unconstrained 2-D drop (i.e., one which is free to slide along the surface) will achieve equilibrium only when x0 = n/2, where n = 0, ±1, ±2, ±3, . . . , in which case θ = θi (r + x0 ) = θi (−r + x0 ). In addition, note that the surface properties are periodic with a (dimensionless) period of 1, and it is therefore enough to consider n = 0 and n = 1 (i.e., x0 = 0 and x0 = 0.5).
3. Results and discussion In this manuscript we investigate the effect of drop size on the behavior of three-dimensional sessile drops partially wetting chemically patterned solid surfaces. The first system to be analyzed involves the 1-D patterned surface given by Eq. (5) with parameters that were considered in [3,13] (φ = 0.4 and θ0 = 90◦ ). In Figs. 1–3 we present calculated drop shapes corresponding to local minima in this system’s free energy for drops whose unitless volumes (rendered dimensionless using the scaling factor l 3 ) range between 1 and 100 in magnitude; note the system free energies, which are given in the figure captions.2 All drops were obtained using the same procedure (described in [13]) except for the shape of the initial volume placed on the solid surface. Here, this volume was chosen to be shaped as a parallelepiped whose 2 The values of G, for three-dimensional drops shown in this manuscript, were determined (using a number of mesh refinement tests) to be accurate to within four significant digits.
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aspect ratio (in the xy plane) was varied from case to case. Choosing different initial shapes resulted in the achievement of more than one equilibrium shaped drop for a given volume. An extensive selection of initial shapes (aspect ratios) suggests that most possible minima were obtained. A number of interesting observations can be made. First, in the cases V = 1 and V = 10 it appears that a lower free energy value is achieved when the drop is centered in a region of low-energy (large-contact-angle) substrate. Another observation involves the number of calculated minima which, consistent with results from two-dimensional calculations (see, e.g., [3]), either remains constant or grows with the volume of the drop. However, this observation should be regarded with caution, since there is no guarantee that all possible minima were calculated for each drop volume. Finally, note that, as expected, the shape of the three-phase contact line and the shape of the liquid–fluid interface become (on a large scale) increasingly circular and hemispherical, respectively, as the drop volume increases. The existence of multiple local minima in the free energy of a sessile drop partially wetting a solid surface is often reflected in its ability to exhibit contact angle hysteresis. The appearance of this phenomenon in the system considered above (Figs. 1–3) was studied in [13]. However, due to limitations of software and hardware, the hysteresis loop described in [13] was restricted to relatively small-scale drops. Here we reexamine contact angle hysteresis in this system. Specifically, we investigate the effect of drop size on this phenomenon by considering volumes ranging from the relatively small values discussed in [13] up to those almost three orders of magnitude larger. In Fig. 4 we present a hysteresis loop obtained in a manner similar to that described in [13]. Starting with a small equilibrium-shaped drop, its volume was increased in several finite steps; at the end of each step a new and larger
Fig. 1. The three-phase contact line (top row) and the three-dimensional shape observed at an angle from above the surface (bottom row) of V = 1 drops in equilibrium with a surface described by Eq. (5) (with φ = 0.4, θ0 = 90◦ ). The shading of the surfaces in the top row describes the distribution of f (x) = cos θi (x) along the solid substrate, where light regions correspond to relatively large f (x) (small θi ) values and dark regions represent relatively small f (x) (large θi ) values. (a) G = 3.888, (b) G = 3.706, (c) G = 3.905.
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Fig. 2. The three-phase contact line (top row) and the three-dimensional shape observed at an angle from above the surface (bottom row) of V = 10 drops in equilibrium with a surface described by Eq. (5) (with φ = 0.4, θ0 = 90◦ ). The shading of the surfaces in the top row describes the distribution of f (x) = cos θi (x) along the solid substrate, where light regions correspond to relatively large f (x) (small θi ) values and dark regions represent relatively small f (x) (large θi ) values. (a) G = 17.66, (b) G = 17.63, (c) G = 17.98.
Fig. 3. The three-phase contact line (top row) and the three-dimensional shape observed at an angle from above the surface (bottom row) of V = 100 drops in equilibrium with a surface described by Eq. (5) (with φ = 0.4, θ0 = 90◦ ). The shading of the surfaces in the top row describes the distribution of f (x) = cos θi (x) along the solid substrate, where light regions correspond to relatively large f (x, y) (small θi ) values and dark regions represent relatively small f (x) (large θi ) values. (a) G = 82.89, (b) G = 82.29, (c) G = 82.35, (d) G = 82.73.
equilibrium-shaped drop was calculated.3 This procedure was followed until relatively large drops were obtained. Next, the drop volume was decreased using a similar procedure involving several finite steps. Thus a set of “volumeincrease” and “volume-decrease” average contact angles were obtained for a wide range of drop volumes; interestingly, both measures of the average contact angle (θ1 and θ2 ) appear to be almost indistinguishable in Fig. 4. Notice also the dashed line in this figure, which corresponds to the contact angle consistent with Cassie’s equation (θC = θ0 = 90◦ ).
3 Note that each of the many drops represented by the symbols in Fig. 4 was simulated with the procedure used to generate the shapes shown in Figs. 1–3.
In comparing our extended three-dimensional results with those obtained for similar three [13]- and two [3]dimensional systems, a number of observations can be made. First, note that the hysteresis loop shown here involves (for drops in the vicinity of V = 1) the two minima corresponding to Figs. 1a and 1b while in [13] the loop is associated with the minima shown in Figs. 1a and 1c. More importantly, the hysteresis loop shown here is qualitatively different from that calculated for two-dimensional systems [3]. Here, the angles at the maxima of the volume-increase and minima of the volume-decrease plots change with increasing volume, while in [3] these angles are the volume-independent advancing and receding contact angles. This is possibly a result of the additional spatial degree of freedom in the 3-D system, which may cause the advancing and receding contact
S. Brandon et al. / Journal of Colloid and Interface Science 263 (2003) 237–243
Fig. 4. Contact angle hysteresis simulation for a surface described by Eq. (5) (with φ = 0.4, θ0 = 90◦ ). Empty squares correspond to volume-increase calculations and empty triangles correspond to volume-decrease calculations. (a) θ1 versus drop volume. (b) θ2 versus drop volume. The smooth lines in this figure are included to help guide the eye, while the dashed horizontal lines in the figure represent the average intrinsic angle of this system, which is also that predicted by Cassie’s equation. Error bars corresponding to the calculated angles are estimated to be smaller than the size of the symbols.
angles to be volume-dependent for the drop sizes considered in Fig. 4. It is expected [11,12] that the average contact angle associated with the global minimum in the free energy of a given system should, when the drop volume increases, approach
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the value predicted by Cassie (θC ). We next discuss this issue and, in particular, attempt to examine (and compare) the global equilibrium state as a function of drop volume for two different chemical patterns. These are the one-dimensional surface pattern considered in Figs. 1–3 (given by Eq. (5) with φ = 0.4 and θ0 = 90◦ ) and the two-dimensional surface pattern (given by Eq. (6) with φ = 0.2 and θ0 = cos−1 (−0.2)). The second surface pattern was chosen to support the same maximum and minimum values of the local intrinsic contact angle (cos−1 (−0.4) and cos−1 (0.4), respectively) as that exhibited by the first, one-dimensional, pattern. Plots of drop and contact line shapes for the lowest calculated energy drop at each volume are exhibited for the one-dimensional and two-dimensional patterned surfaces in Figs. 5 and 6, respectively. Notice that Fig. 5 includes a result for V = 1000, which is shown solely for its qualitative value. Due to software and hardware limitations it is the only V = 1000 drop to be calculated and can therefore not be safely considered to be the lowest energy drop for this value of V . Figures 5 and 6 reveal, as did Figs. 1–3, that the drop shapes approach that of a spherical cap as their volume increases. The twofold symmetry of the one-dimensional pattern and fourfold symmetry of the two-dimensional pattern are, to a certain extent, imposed on the drops; this effect does, however, significantly diminish with increasing drop volume. It is tempting to mentally extrapolate these results to even larger volumes where one would expect to see macroscopically spherical cap shaped drops. This extrapolation would also be useful as a test for the limit of the average contact angle θ2 (defined by Eq. (8)) to see if it approaches the angle predicted by Cassie’s equation. The closest we can come to this test is exhibited in Fig. 7, where θ2 /θC is plotted for the drops shown in Figs. 5 and 6. From these results it seems evident that the average angle (θ2 ) of global equilibrium-shaped drops does indeed approach the angle predicted by Cassie’s equation. However, this conclusion should be treated with caution since, as mentioned above,
Fig. 5. The three-phase contact line (top row) and the three-dimensional shape observed at an angle from above the surface (bottom row) of the lowest energy equilibrium drop (for each value of the volume) partially wetting a surface described by Eq. (5) (with φ = 0.4, θ0 = 90◦ ). The shading of the surfaces in the top row describes the distribution of f (x) = cos θi (x) along the solid substrate, where light regions correspond to relatively large f (x) (small θi ) values and dark regions represent relatively small f (x) (large θi ) values. (a) V = 1, (b) V = 10, (c) V = 100, (d) V = 1000.
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Fig. 6. The three-phase contact line (top row) as well as the three-dimensional shape (observed at an angle from above the surface; bottom row) of the lowest energy equilibrium drop (for each value of the volume) partially wetting a surface described by Eq. (6) (with φ = 0.2 and θ0 = cos−1 (−0.2)). The shading of the surfaces in the top row describes the distribution of f (x, y) = cos θi (x, y) along the solid substrate, where light regions correspond to relatively large f (x, y) (small θi ) values and dark regions represent relatively small f (x, y) (large θi ) values. (a) V = 1, (b) V = 10, (c) V = 100.
Fig. 7. The normalized average contact angle (θ2 /θC ) of the lowest energy equilibrium drop (for each value of the volume) partially wetting a surface described by Eqs. (5) and (6). Circles correspond to drops wetting surfaces with a one-dimensional pattern (Eq. (5) with φ = 0.4, θ0 = 90◦ ) and squares correspond to drops wetting surfaces with a two-dimensional pattern (Eq. (6) with φ = 0.2 and θ0 = cos−1 (−0.2)). The dashed line depicts the expected large volume limit. Error bars corresponding to the normalized angles are estimated to be smaller than the size of the symbols.
there is always the possibility that we did not locate the lowest energy state at each drop volume. The effect of drop volume on contact angle of the global equilibrium state was more rigorously investigated in the two-dimensional system involving cylindrical drops. In this case the surface is described by Eq. (5) with the same parameters as those considered in the relevant three-dimensional system (φ = 0.4, θ0 = 90 ◦ ). A plot of this angle versus the two-dimensional drop volume (per unit length) is shown in Fig. 8. Looking at this figure it appears that, as the drop volume is increased, the angle of the global equilibrium state does approach the angle predicted by Cassie. However,
Fig. 8. The normalized average contact angle (θ/θC ) of the lowest energy two-dimensional cylindrical equilibrium drop partially wetting a surface described by Eq. (5) (with φ = 0.4, θ0 = 90◦ ). The dashed line depicts the expected large-volume limit.
the approach is significantly nonmonotonic and relatively slow, therefore emphasizing an uncertainty associated with the three-dimensional results shown in Fig. 7. It is theoretically possible that the three-dimensional system also exhibits a significantly nonmonotonic and slow approach to the Cassie limit. In this case, the rate of approach cannot be safely deduced from Fig. 7 (which exhibits only three data points for each surface). At the same time, the fact that the three-dimensional systems are less constrained than the two-dimensional system may promote a faster approach to the Cassie limit. This conjecture is supported by the apparent slightly faster rate of approach (with increasing drop volume) exhibited by three-dimensional drops on the 2-D patterned surface compared to that shown by the more constrained three-dimensional drops partially wetting the 1-D patterned surface (see Fig. 7).
S. Brandon et al. / Journal of Colloid and Interface Science 263 (2003) 237–243
4. Summary and conclusions The following points summarize the main results presented in this manuscript. • The Surface Evolver is a useful tool for studying threedimensional wetting phenomena. Hardware and software limitations render the application of this package to systems larger and/or more complex than those exhibited here extremely difficult and time-consuming. Still, the rapidly advancing nature of computer technology does offer hope that this situation is temporary. Already the systems studied here are more complex than those analyzed in our previous, related publication [13]. • Contact angle hysteresis, associated with a heterogeneous surface, is obtained for a one-dimensional chemical pattern for a large range of three-dimensional drop volumes. The appearance of the hysteresis loop is reminiscent of that obtained in a relevant two-dimensional calculation though it does show qualitatively different features. These may be due to the more constrained nature of the 2-D as compared with the 3-D drops.4 It is possible that the differences, in particular the volume dependence of the advancing and receding angles, will disappear for drop volumes larger than the values considered here. The plots shown in Fig. 4 show a trend that is not inconsistent with this conjecture. Note that the advancing and receding angles are taken to be the contact angles at the maxima of the volume-increase and minima of the volume-decrease plots, respectively. • The number of local minima associated with the 1-D patterned surface appears to either remain constant or increase with growing drop volume. This observation, which is consistent with results from two-dimensional simulations, hinges on the assumption that all possible local equilibrium shapes were calculated for each value of the drop volume. 4 Note that continuously changing the volume of a 2-D drop, forces its three-phase contact points to periodically move through regions of maximum and minimum surface energy. This is in contrast to 3-D drops, whose contact line has more freedom to continuously rearrange itself on the twodimensional heterogenous surface and can never only occupy regions of maximum (or minimum) surface energy.
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• Increasing the drop volume appears to promote the appearance of almost spherical cap-shaped drops whose contact line is, on a large scale, circular in shape. • The average contact angle (θ2 ) of the lowest energy (“global”) equilibrium drop at each value of V , is found to approach (with increasing drop volume) the value predicted according to Cassie’s equation. This finding is consistent with [11,12] and with results from a twodimensional analytical calculation. The qualitative difference between the behavior of the 2-D and 3-D drops (Figs. 7 and 8) either may be due to the relatively unconstrained nature of the three-dimensional systems or could be an incomplete result requiring further investigation.
Acknowledgment This work was supported in part by a grant from the Technion’s VPR funds.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
J.T. Woodward, H. Gwin, D.K. Schwartz, Langmuir 16 (2000) 2957. H. Kamusewitz, W. Possart, D. Paul, Colloids Surf. A 156 (1999) 271. S. Brandon, A. Marmur, J. Colloid Interface Sci. 183 (1996) 351. F.Y.H. Lin, D. Li, A.W. Neumann, J. Colloid Interface Sci. 159 (1993) 86. A.B.D. Cassie, Discuss. Faraday Soc. 75 (1952) 5041. R.N. Wenzel, Ind. Eng. Chem. 28 (1936) 988. A. Marmur, Colloids Surf. A 116 (1996) 55. A. Marmur, J. Colloid Interface Sci. 168 (1994) 40. D. Urban, K. Topolski, J. De Coninck, Phys. Rev. Lett. 76 (23) (1996) 4388. K. Topolski, D. Urban, S. Brandon, J. De Coninck, Phys. Rev. E 56 (3) (1997) 3353. G. Wolansky, A. Marmur, Colloids Surf. A 156 (1999) 381. G. Wolansky, personal communication. S. Brandon, A. Wachs, A. Marmur, J. Colloid Interface Sci. 191 (1997) 110. K.A. Brakke, Exp. Math. 1 (2) (1992) 141. K.A. Brakke, Surface Evolver Manual, Version 2.14, 1999. [Software, manual, and more are available at http://www.geom.umn.edu/ software/evolver/ or http://www.susqu.edu/facstaff/b/brakke/evolver/ evolver.html].
Partial wetting of chemically patterned surfaces: The effect of drop size Simon Brandon,∗ Nir Haimovich, Einat Yeger, and Abraham Marmur Department of Chemical Engineering, Technion—Israel Institute of Technology, 32000 Haifa, Israel Received 12 June 2002; accepted 13 March 2003
Abstract Partial wetting of chemically heterogeneous substrates is simulated. Three-dimensional sessile drops in equilibrium with smooth surfaces supporting ordered chemical patterns are considered. Significant features are observed as a result of changing the drop volume. The number of equilibrated drops is found either to remain constant or to increase with growing drop volume. The shape of larger drops appears to approach that of a spherical cap and their three-phase contact line seems, on a larger scale, more circular in shape than that of smaller drops. In addition, as the volume is increased, the average contact angle of drops whose free energy is lowest among all equilibrium-shaped drops of the same volume appears to approach the angle predicted by Cassie. Finally, contrary to results obtained with two-dimensional drops, contact angle hysteresis observed in this system is shown to exhibit a degree of volume dependence in the advancing and receding angles. Qualitative differences in the wetting behavior associated with the two different chemical patterns considered here, as well as differences between results obtained with two-dimensional and three-dimensional drops, can possibly be attributed to variations in the level of constraint imposed on the drop by the different patterns and by the dimensionality of the system. 2003 Elsevier Science (USA). All rights reserved. Keywords: Contact angle; Contact angle hysteresis; Cassie equation
1. Introduction Wetting of chemically heterogeneous and/or rough surfaces, a process commonly occurring in both natural and man-made environments, has been studied for several decades. The equilibrium contact angle on such surfaces has been measured (e.g., [1,2]) and calculated (e.g., [3,4]) by a number of authors. Most notable are the simple predictions by Cassie [5] and Wenzel [6] for the equilibrium contact angles on chemically heterogeneous and rough surfaces, respectively. These predictions were confirmed to be qualitatively correct for a wide range of conditions both in experiments and in calculations. The Cassie and Wenzel angles can be calculated using 1 cos(θC ) = (1) cos(θi )(x, y) dx dy Ssl Ssl
and cos(θW ) = ρ cos(θi ), * Corresponding author.
E-mail address: [email protected] (S. Brandon).
(2)
where, θC , θW , and θi are the Cassie, Wenzel, and intrinsic contact angles, respectively. The solid–liquid interfacial area is given by Ssl , x and y are the spatial coordinates spanning this interface, and ρ is the surface roughness, which is defined as the ratio of the solid surface area to its value when projected on to the tangent plane. Although Eqs. (1) and (2) yield a single-valued contact angle for a given solid surface and liquid–fluid system, it is well known (see, e.g., [7]) that rough and/or chemically heterogenous surfaces may exhibit contact angle hysteresis, which is a symptom of multiple local equilibrium states. Moreover, the multiplicity of local minima in the system’s free energy is strongly dependent on the drop volume and on parameters characterizing the surface (see, e.g., [8]). It has been indicated [9,10], and recently proven [11,12] that the Cassie and Wenzel angles can serve as approximations to the contact angle at the global minimum in the system’s free energy. According to [11,12] this approximation becomes better as the drop size increases relative to the scale of heterogeneity and/or roughness. Here we present a computational investigation of the effects of drop size and pattern type on local and global minima in the free energy of three-dimensional sessile drops partially wetting chemically patterned smooth surfaces. Results are discussed in relation to the relevant Cassie angle
0021-9797/03/$ – see front matter 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0021-9797(03)00285-6
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as well as, in some cases, to two-dimensional simulations of a similar system. The three-dimensional simulations were achieved, as in the case of our previous study [13], using the public-domain “Surface Evolver” software [14,15], while two-dimensional results were obtained analytically using a model system discussed in [3].
2. Model formulation and solution technique Three-dimensional sessile drops in equilibrium with chemically heterogeneous (patterned) substrates are computed using an approach similar to that employed in [13]. Both the liquid drop and the solid surface on which it is placed are immersed in an ambient fluid; the liquid and fluid are assumed to be mutually immiscible. Ignoring gravitational effects1 and three-phase interactions at the contact line, the dimensionless free energy of the system is given by G = Slf − (3) f (x, y) dx dy, Ssl
where G is the free energy normalized with respect to σlf l 2 , σlf is the liquid–fluid interfacial energy, l is a length-scale characteristic of the chemical pattern, and Slf is the area of contact along the liquid–fluid interface. The spatial coordinates (x, y) are normalized with respect to l, while Ssl and Slf are rendered dimensionless using the scaling factor l 2 . Finally, the solid surface energy function, f (x, y), is defined as f (x, y) = cos(θY ) =
σsf (x, y) − σsl (x, y) . σlf
(4)
Here, σsl (x, y) and σsf (x, y) are the local solid–liquid and solid–fluid interfacial energies, respectively, and θY is the local Young contact angle, which, as discussed below, is in our case equal to the intrinsic contact angle (θi (x, y)). In this study we consider two forms of the solid energy function, representing either a one-dimensional periodic variation in solid surface energy, f (x) = cos(θ0 ) + φ cos(2πx),
(5)
or a two-dimensional periodic solid surface energy pattern, f (x, y) = cos(θ0 ) + φ cos(2πx) cos(2πy) − cos(2πx) − cos(2πy) , (6) where the chemical pattern given by Eq. (5) was already considered in our previous three-dimensional [13] as well as two-dimensional [3] analyses of contact angle hysteresis. 1 Strictly speaking this limits the analysis to drops with a radius ap proximately smaller than 2σlf /(ρl g) where ρl , σlf , and g are the liquid density, liquid–fluid interfacial energy, and gravitational acceleration, respectively.
Minimization of the free energy G (Eq. (3)) with respect to the liquid–fluid interface shape, while the drop’s volume is constrained to a fixed value, yields an equilibrium-shaped drop. This, within the framework of the main assumptions of this work, must satisfy both the Young–Laplace and the Young equations. As a direct consequence, the liquid–fluid interface is required to exhibit uniform curvature while (as mentioned above) the local contact angle at the three-phase boundary is constrained to an intrinsic value (θi ) which is uniquely defined for every position of this boundary via Eq. (4). Therefore, calculation of the liquid–fluid interfacial mean curvature and local contact angles from the numerically obtained equilibrium drop shapes serves as a test of their accuracy. Although local contact angles of three-dimensional drops are well defined, average contact angles of these systems are, in general, not unique. In this study we use two possible definitions of the average contact angle. The first (θ1 ), used in [13], is given by θ dl , θ1 = C (7) C dl where C refers to the closed integration path along the threephase contact line; the second average angle (θ2 ) is given by 3V H 3 = 0, (8) π where V is the volume of the drop and H is its liquid–fluid interfacial mean curvature. Notice that θ2 is the angle corresponding to that of a spherical cap whose mean curvature and volume are equal to the three-dimensional drop’s mean curvature and volume, respectively. The choice of θ1 stems from its simplicity while that of θ2 is motivated by Ref. [11] in which it was proven that, for drops partially wetting rough surfaces, θ2 approaches the Wenzel contact angle when the drop is sufficiently large. As mentioned above, the public-domain software package “Surface Evolver” was employed in the calculation of 3-D equilibrium drop shapes. This software, which in essence minimizes the system’s free energy (Eq. (3)), is described in detail in [14,15]. Additional information on our application of Surface Evolver to the study of 3-D wetting of heterogenous surfaces is given in [13]. Two-dimensional simulations of cylindrical drops were obtained with surfaces described by Eq. (5). In this case an analytical solution for the dimensionless free energy (per unit length of the drop) of the system is given by [3,8] cos3 θ2 − 3 cos θ2 + 2 −
G2D =
2rθ φ − 2r cos θ0 − sin 2π(r + x0 ) sin θ 2π − sin −2π(r − x0 ) ,
(9)
where G2D is the free energy per unit length of the drop normalized by σlf l, r is the “radius” of the base of the drop (normalized by l), and x0 is the x position (also normalized using l) of the center of the drop. An additional equation relating the volume (per unit length) of the cylindrical drops
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to r is given by V 2D = r 2
θ − sin θ cos θ sin2 θ
,
(10)
where V 2D is the volume (per unit length) normalized with respect to l 2 . As implied in [8], an unconstrained 2-D drop (i.e., one which is free to slide along the surface) will achieve equilibrium only when x0 = n/2, where n = 0, ±1, ±2, ±3, . . . , in which case θ = θi (r + x0 ) = θi (−r + x0 ). In addition, note that the surface properties are periodic with a (dimensionless) period of 1, and it is therefore enough to consider n = 0 and n = 1 (i.e., x0 = 0 and x0 = 0.5).
3. Results and discussion In this manuscript we investigate the effect of drop size on the behavior of three-dimensional sessile drops partially wetting chemically patterned solid surfaces. The first system to be analyzed involves the 1-D patterned surface given by Eq. (5) with parameters that were considered in [3,13] (φ = 0.4 and θ0 = 90◦ ). In Figs. 1–3 we present calculated drop shapes corresponding to local minima in this system’s free energy for drops whose unitless volumes (rendered dimensionless using the scaling factor l 3 ) range between 1 and 100 in magnitude; note the system free energies, which are given in the figure captions.2 All drops were obtained using the same procedure (described in [13]) except for the shape of the initial volume placed on the solid surface. Here, this volume was chosen to be shaped as a parallelepiped whose 2 The values of G, for three-dimensional drops shown in this manuscript, were determined (using a number of mesh refinement tests) to be accurate to within four significant digits.
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aspect ratio (in the xy plane) was varied from case to case. Choosing different initial shapes resulted in the achievement of more than one equilibrium shaped drop for a given volume. An extensive selection of initial shapes (aspect ratios) suggests that most possible minima were obtained. A number of interesting observations can be made. First, in the cases V = 1 and V = 10 it appears that a lower free energy value is achieved when the drop is centered in a region of low-energy (large-contact-angle) substrate. Another observation involves the number of calculated minima which, consistent with results from two-dimensional calculations (see, e.g., [3]), either remains constant or grows with the volume of the drop. However, this observation should be regarded with caution, since there is no guarantee that all possible minima were calculated for each drop volume. Finally, note that, as expected, the shape of the three-phase contact line and the shape of the liquid–fluid interface become (on a large scale) increasingly circular and hemispherical, respectively, as the drop volume increases. The existence of multiple local minima in the free energy of a sessile drop partially wetting a solid surface is often reflected in its ability to exhibit contact angle hysteresis. The appearance of this phenomenon in the system considered above (Figs. 1–3) was studied in [13]. However, due to limitations of software and hardware, the hysteresis loop described in [13] was restricted to relatively small-scale drops. Here we reexamine contact angle hysteresis in this system. Specifically, we investigate the effect of drop size on this phenomenon by considering volumes ranging from the relatively small values discussed in [13] up to those almost three orders of magnitude larger. In Fig. 4 we present a hysteresis loop obtained in a manner similar to that described in [13]. Starting with a small equilibrium-shaped drop, its volume was increased in several finite steps; at the end of each step a new and larger
Fig. 1. The three-phase contact line (top row) and the three-dimensional shape observed at an angle from above the surface (bottom row) of V = 1 drops in equilibrium with a surface described by Eq. (5) (with φ = 0.4, θ0 = 90◦ ). The shading of the surfaces in the top row describes the distribution of f (x) = cos θi (x) along the solid substrate, where light regions correspond to relatively large f (x) (small θi ) values and dark regions represent relatively small f (x) (large θi ) values. (a) G = 3.888, (b) G = 3.706, (c) G = 3.905.
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Fig. 2. The three-phase contact line (top row) and the three-dimensional shape observed at an angle from above the surface (bottom row) of V = 10 drops in equilibrium with a surface described by Eq. (5) (with φ = 0.4, θ0 = 90◦ ). The shading of the surfaces in the top row describes the distribution of f (x) = cos θi (x) along the solid substrate, where light regions correspond to relatively large f (x) (small θi ) values and dark regions represent relatively small f (x) (large θi ) values. (a) G = 17.66, (b) G = 17.63, (c) G = 17.98.
Fig. 3. The three-phase contact line (top row) and the three-dimensional shape observed at an angle from above the surface (bottom row) of V = 100 drops in equilibrium with a surface described by Eq. (5) (with φ = 0.4, θ0 = 90◦ ). The shading of the surfaces in the top row describes the distribution of f (x) = cos θi (x) along the solid substrate, where light regions correspond to relatively large f (x, y) (small θi ) values and dark regions represent relatively small f (x) (large θi ) values. (a) G = 82.89, (b) G = 82.29, (c) G = 82.35, (d) G = 82.73.
equilibrium-shaped drop was calculated.3 This procedure was followed until relatively large drops were obtained. Next, the drop volume was decreased using a similar procedure involving several finite steps. Thus a set of “volumeincrease” and “volume-decrease” average contact angles were obtained for a wide range of drop volumes; interestingly, both measures of the average contact angle (θ1 and θ2 ) appear to be almost indistinguishable in Fig. 4. Notice also the dashed line in this figure, which corresponds to the contact angle consistent with Cassie’s equation (θC = θ0 = 90◦ ).
3 Note that each of the many drops represented by the symbols in Fig. 4 was simulated with the procedure used to generate the shapes shown in Figs. 1–3.
In comparing our extended three-dimensional results with those obtained for similar three [13]- and two [3]dimensional systems, a number of observations can be made. First, note that the hysteresis loop shown here involves (for drops in the vicinity of V = 1) the two minima corresponding to Figs. 1a and 1b while in [13] the loop is associated with the minima shown in Figs. 1a and 1c. More importantly, the hysteresis loop shown here is qualitatively different from that calculated for two-dimensional systems [3]. Here, the angles at the maxima of the volume-increase and minima of the volume-decrease plots change with increasing volume, while in [3] these angles are the volume-independent advancing and receding contact angles. This is possibly a result of the additional spatial degree of freedom in the 3-D system, which may cause the advancing and receding contact
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Fig. 4. Contact angle hysteresis simulation for a surface described by Eq. (5) (with φ = 0.4, θ0 = 90◦ ). Empty squares correspond to volume-increase calculations and empty triangles correspond to volume-decrease calculations. (a) θ1 versus drop volume. (b) θ2 versus drop volume. The smooth lines in this figure are included to help guide the eye, while the dashed horizontal lines in the figure represent the average intrinsic angle of this system, which is also that predicted by Cassie’s equation. Error bars corresponding to the calculated angles are estimated to be smaller than the size of the symbols.
angles to be volume-dependent for the drop sizes considered in Fig. 4. It is expected [11,12] that the average contact angle associated with the global minimum in the free energy of a given system should, when the drop volume increases, approach
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the value predicted by Cassie (θC ). We next discuss this issue and, in particular, attempt to examine (and compare) the global equilibrium state as a function of drop volume for two different chemical patterns. These are the one-dimensional surface pattern considered in Figs. 1–3 (given by Eq. (5) with φ = 0.4 and θ0 = 90◦ ) and the two-dimensional surface pattern (given by Eq. (6) with φ = 0.2 and θ0 = cos−1 (−0.2)). The second surface pattern was chosen to support the same maximum and minimum values of the local intrinsic contact angle (cos−1 (−0.4) and cos−1 (0.4), respectively) as that exhibited by the first, one-dimensional, pattern. Plots of drop and contact line shapes for the lowest calculated energy drop at each volume are exhibited for the one-dimensional and two-dimensional patterned surfaces in Figs. 5 and 6, respectively. Notice that Fig. 5 includes a result for V = 1000, which is shown solely for its qualitative value. Due to software and hardware limitations it is the only V = 1000 drop to be calculated and can therefore not be safely considered to be the lowest energy drop for this value of V . Figures 5 and 6 reveal, as did Figs. 1–3, that the drop shapes approach that of a spherical cap as their volume increases. The twofold symmetry of the one-dimensional pattern and fourfold symmetry of the two-dimensional pattern are, to a certain extent, imposed on the drops; this effect does, however, significantly diminish with increasing drop volume. It is tempting to mentally extrapolate these results to even larger volumes where one would expect to see macroscopically spherical cap shaped drops. This extrapolation would also be useful as a test for the limit of the average contact angle θ2 (defined by Eq. (8)) to see if it approaches the angle predicted by Cassie’s equation. The closest we can come to this test is exhibited in Fig. 7, where θ2 /θC is plotted for the drops shown in Figs. 5 and 6. From these results it seems evident that the average angle (θ2 ) of global equilibrium-shaped drops does indeed approach the angle predicted by Cassie’s equation. However, this conclusion should be treated with caution since, as mentioned above,
Fig. 5. The three-phase contact line (top row) and the three-dimensional shape observed at an angle from above the surface (bottom row) of the lowest energy equilibrium drop (for each value of the volume) partially wetting a surface described by Eq. (5) (with φ = 0.4, θ0 = 90◦ ). The shading of the surfaces in the top row describes the distribution of f (x) = cos θi (x) along the solid substrate, where light regions correspond to relatively large f (x) (small θi ) values and dark regions represent relatively small f (x) (large θi ) values. (a) V = 1, (b) V = 10, (c) V = 100, (d) V = 1000.
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Fig. 6. The three-phase contact line (top row) as well as the three-dimensional shape (observed at an angle from above the surface; bottom row) of the lowest energy equilibrium drop (for each value of the volume) partially wetting a surface described by Eq. (6) (with φ = 0.2 and θ0 = cos−1 (−0.2)). The shading of the surfaces in the top row describes the distribution of f (x, y) = cos θi (x, y) along the solid substrate, where light regions correspond to relatively large f (x, y) (small θi ) values and dark regions represent relatively small f (x, y) (large θi ) values. (a) V = 1, (b) V = 10, (c) V = 100.
Fig. 7. The normalized average contact angle (θ2 /θC ) of the lowest energy equilibrium drop (for each value of the volume) partially wetting a surface described by Eqs. (5) and (6). Circles correspond to drops wetting surfaces with a one-dimensional pattern (Eq. (5) with φ = 0.4, θ0 = 90◦ ) and squares correspond to drops wetting surfaces with a two-dimensional pattern (Eq. (6) with φ = 0.2 and θ0 = cos−1 (−0.2)). The dashed line depicts the expected large volume limit. Error bars corresponding to the normalized angles are estimated to be smaller than the size of the symbols.
there is always the possibility that we did not locate the lowest energy state at each drop volume. The effect of drop volume on contact angle of the global equilibrium state was more rigorously investigated in the two-dimensional system involving cylindrical drops. In this case the surface is described by Eq. (5) with the same parameters as those considered in the relevant three-dimensional system (φ = 0.4, θ0 = 90 ◦ ). A plot of this angle versus the two-dimensional drop volume (per unit length) is shown in Fig. 8. Looking at this figure it appears that, as the drop volume is increased, the angle of the global equilibrium state does approach the angle predicted by Cassie. However,
Fig. 8. The normalized average contact angle (θ/θC ) of the lowest energy two-dimensional cylindrical equilibrium drop partially wetting a surface described by Eq. (5) (with φ = 0.4, θ0 = 90◦ ). The dashed line depicts the expected large-volume limit.
the approach is significantly nonmonotonic and relatively slow, therefore emphasizing an uncertainty associated with the three-dimensional results shown in Fig. 7. It is theoretically possible that the three-dimensional system also exhibits a significantly nonmonotonic and slow approach to the Cassie limit. In this case, the rate of approach cannot be safely deduced from Fig. 7 (which exhibits only three data points for each surface). At the same time, the fact that the three-dimensional systems are less constrained than the two-dimensional system may promote a faster approach to the Cassie limit. This conjecture is supported by the apparent slightly faster rate of approach (with increasing drop volume) exhibited by three-dimensional drops on the 2-D patterned surface compared to that shown by the more constrained three-dimensional drops partially wetting the 1-D patterned surface (see Fig. 7).
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4. Summary and conclusions The following points summarize the main results presented in this manuscript. • The Surface Evolver is a useful tool for studying threedimensional wetting phenomena. Hardware and software limitations render the application of this package to systems larger and/or more complex than those exhibited here extremely difficult and time-consuming. Still, the rapidly advancing nature of computer technology does offer hope that this situation is temporary. Already the systems studied here are more complex than those analyzed in our previous, related publication [13]. • Contact angle hysteresis, associated with a heterogeneous surface, is obtained for a one-dimensional chemical pattern for a large range of three-dimensional drop volumes. The appearance of the hysteresis loop is reminiscent of that obtained in a relevant two-dimensional calculation though it does show qualitatively different features. These may be due to the more constrained nature of the 2-D as compared with the 3-D drops.4 It is possible that the differences, in particular the volume dependence of the advancing and receding angles, will disappear for drop volumes larger than the values considered here. The plots shown in Fig. 4 show a trend that is not inconsistent with this conjecture. Note that the advancing and receding angles are taken to be the contact angles at the maxima of the volume-increase and minima of the volume-decrease plots, respectively. • The number of local minima associated with the 1-D patterned surface appears to either remain constant or increase with growing drop volume. This observation, which is consistent with results from two-dimensional simulations, hinges on the assumption that all possible local equilibrium shapes were calculated for each value of the drop volume. 4 Note that continuously changing the volume of a 2-D drop, forces its three-phase contact points to periodically move through regions of maximum and minimum surface energy. This is in contrast to 3-D drops, whose contact line has more freedom to continuously rearrange itself on the twodimensional heterogenous surface and can never only occupy regions of maximum (or minimum) surface energy.
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• Increasing the drop volume appears to promote the appearance of almost spherical cap-shaped drops whose contact line is, on a large scale, circular in shape. • The average contact angle (θ2 ) of the lowest energy (“global”) equilibrium drop at each value of V , is found to approach (with increasing drop volume) the value predicted according to Cassie’s equation. This finding is consistent with [11,12] and with results from a twodimensional analytical calculation. The qualitative difference between the behavior of the 2-D and 3-D drops (Figs. 7 and 8) either may be due to the relatively unconstrained nature of the three-dimensional systems or could be an incomplete result requiring further investigation.
Acknowledgment This work was supported in part by a grant from the Technion’s VPR funds.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
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