Anisotropic Wetting Behavior Arising from Superhydrophobic Surfaces

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J. Phys. Chem. B 2008, 112, 7234–7243

Anisotropic Wetting Behavior Arising from Superhydrophobic Surfaces: Parallel Grooved Structure Wen Li,†,‡ Guoping Fang,*,‡ Yongfeng Li,‡ and Guanjun Qiao*,‡ Key Laboratory of Low Dimensional Materials and Application Technology (Ministry of Education) and Faculty of Materials and Optoelectronic Physics, Xiangtan UniVersity, Xiangtan, Hunan 411105 P. R. China, and State Key Laboratory for Mechanical BehaVior of Materials, Xi’an Jiaotong UniVersity, Xi’an 710049, P. R. China ReceiVed: December 21, 2007; ReVised Manuscript ReceiVed: March 8, 2008

It has been found experimentally that superhydrophobic surfaces exhibit strong anisotropic wetting behavior. This study reports a simple but robust thermodynamic methodology to investigate the anisotropic superhydrophobic behavior for parallel grooved surfaces. Free energy and its barrier and the corresponding contact angle and its hysteresis for various orientations of the groove structure are calculated based on the proposed thermodynamic model. It is revealed that the strong anisotropy of equilibrium contact angle (ECA) and contact angle hysteresis (CAH) is shown in the noncomposite state but almost isotropic wetting properties are exhibited in the composite state. Furthermore, for the noncomposite state, decreasing groove width and spacing or increasing groove depth can amplify the anisotropy for ECA. Meanwhile, decreasing groove width and increasing depth can amplify the anisotropy for CAH, while varying groove spacing can barely influence CAH. For the composite state, however, the surface geometry hardly leads to the anisotropic behavior. In addition, using a fitting approximation, a simple quantitative correlation between wettability and orientation can be established well, which is consistent with the numerical calculations. 1. Introduction Inspired by the so-called “lotus effect”1,2 discovered in 1997, superhydrophobic surfaces have received strong interest in both fundamental research and practical applications over the past decade; this can be reflected clearly by the fact that an exponentially increasing number of publications3 in this field have appeared in recent years. Due to very large contact angle (CA) and small contact angle hysteresis (CAH), i.e., the difference between advancing or maximum and receding or minimum contact angles, superhydrophobic surfaces would have ideal liquid-shedding or droplet-sliding properties.4,5 Such unique wettability is critical for the development of the so-called selfcleaning materials and could be useful in many industrial applications such as glass coatings, microfluidics, and pesticides.6–8 Now it has been well recognized that surface roughness is necessary to create superhydrophobic surfaces9–12 although the role of hierarchical or multiscale roughness has not been revealed completely.13–15 In particular, correlations between roughness and CA were well established almost 60 years ago,16–18 e.g., the classical Wenzel’s and Cassie’s equations. However, for a surface with microstructures or microtextures, such correlations are inadequate to gain a complete understanding on the role of surface geometry in superhydrophobic behavior, since roughness only represents a composite measure of all surface texture parameters. Instead, it is necessary to investigate the effect of all geometrical parameters that can describe completely a surface microstructure. This is especially important to the surfaces with regular or ordered microtextures, which can be achieved accurately using the latest micro/ * To whom correspondence should be addressed. Email: gjqiao@ mail.xjtu.edu.cn (Guanjun Qiao); [email protected] (Guoping Fang). † Xiangtan University. ‡ Xi’an Jiaotong University.

nanofabrication technologies.19 In such surfaces, for the same roughness value, different surface textures can exhibit completely different superhydrophobicity. In particular, even for a microtextured surface, different superhydrophobic behavior, depending on the concerned orientation, can be observed because of the regular surface microstructures. This indicates that such microtextured surfaces can manifest themselves into a unique anisotropic wettability compared to normal surfaces with random roughness.20 To gain a better understanding on the effect of surface microstructure and geometry, the investigations of the anisotropic wetting behavior is very important for fundamental research on superhydrophobic surface. Furthermore, the anisotropic wettability is especially applicable for the design of tunable or controllable superhydrophobic surfaces, where a certain property merely in a particular orientation or direction is, in practice, needed.21 For instance, if superhydrophobic surfaces are used in microfluidics, a small CAH and hence a reduced friction is required in a fixed or desired orientation (e.g., along the microchannel direction) whereas for a smooth liquid delivery opposite requirements for other orientations are needed to avoid the liquid loss. In fact, such orientation or directiondependent wettability is also demanded in many other fields such as coating and printing etc., where an accurate control of movement and deposition of liquid droplets on a surface is desired.22,23 It is interesting to note that recently exploratory research in this context has started. Yoshimitsu et al.20 investigated experimentally the sliding behavior of water droplets for composite states (i.e., entrapment of air in the troughs between asperities) in both pillar and groove structures. They found a better water-shedding property in the parallel direction of the grooved surface than the pillar structure. Followed by the work of Yoshimitsu et al., Zhao et al.24 and Chen et al.25 also

10.1021/jp712019y CCC: $40.75  2008 American Chemical Society Published on Web 05/21/2008

Wetting Behavior of Superhydrophobic Surfaces investigated the grooved surface. The former focused on the effect of the microstructure for noncomposite states (i.e., complete liquid penetration into the troughs between asperities), whereas the latter focused on static wetting behavior for composite states. In addition, Chung et al.26 exploited wrinkling phenomena to investigate the noncomposite state on grooved surface and gained very similar results to Zhao’s. Very recently, Zhang et al.27 studied the wetting anisotropy on hierarchical structures, but the mechanism still remains unclear. On the other hand, few studies have considered the difference of anisotropic wetting between the composite and noncomposite states. In particular, systematical theoretical studies on the mechanism for the wetting anisotropy are very rare. Consequently, available theoretical results can hardly be found to be consistent with experimental observations and even sometimes contradict with each other. For instance, in the two different studies by Zhao et al.24 and Chung et al.,26 the experimental and theoretical results of CA in the direction parallel to the grooves are obviously different. It should be pointed out that studies demonstrate that surface topographic structure24–26 and chemical composition28–30 can result in the anisotropic wettability and further attribute this wettability to the difference of energy barrier between different directions due to the geometrical and chemical inhomogeneities. However, a quantitative correlation between surface geometrical parameters and contact angles and a thorough mechanism responsible for this effect of microtextures have been lacked. While further experimental observations on the above aspects are still important, theoretical investigations are necessary. Very recently, we proposed a simple and robust thermodynamic approach31,32 for successful analysis of surface free energy (FE) and free energy barrier (FEB) and calculations of various CAs and CAH for a surface pillar microtexture. In this approach, a 3-dimensional microstructure can be simplified into a 2-dimensional system by analyzing the system along specific planes. Based on this simplification, the analysis of thermodynamic status, e.g., FE and FEB, and the subsequent numerical calculations of CA and CAH associated with FE and FEB can be readily conducted. Apparently, the thermodynamic status and wettability depend strongly on the chosen plane, which exactly reflects the effect of surface geometry of a certain microtexture. This also indicates that the proposed approach is especially suitable for the study of anisotropic wetting behavior of microtextured superhydrophobic surfaces. In the present study, using a simple superhydrophobic surface with a parallel grooved microtexture, which has been experimentally studied well before, we systematically investigate the effects of all surface geometrical parameters, i.e., groove width, height, and spacing, with emphasis on different anisotropic wetting behavior between composite and noncomposite states. Here, it should be pointed out that with the recent rapid improvements of micro/nanofabrication techniques such a superhydrophobic surface with a grooved microtexture can be prepared readily. The present work mainly focuses on the theoretical aspect; further experimental investigation is needed and could provide adequate supports for the present theoretical analysis. Nevertheless, based on previous available experimental observations, it is believed that this analysis could be used to reveal the thermodynamic mechanism for the anisotropic wetting behavior. 2. Thermodynamic Analysis In the previous experimental studies, the wetting behavior only in parallel and perpendicular directions was investigated because of the limitations of experimental conditions. This is

J. Phys. Chem. B, Vol. 112, No. 24, 2008 7235 inadequate to establish a quantitative correlation between wettability and orientation, since Youngblood et al.33 have clearly indicated that instabilities at any point on the contact line can make a contribution to the momentum of a moving droplet. A main aim for the present study is to develop a simple thermodynamic model to analyze FE and FEB completely for any orientation or direction of a surface microstructure. To this end, a simple model based on our previously proposed methodology31 can be developed using a typical grooved structure as illustrated in Figure 1a. For the 3D parallel grooved structure, a 2D model can be established from the resultant cross section by cutting the 3D structure along a given orientation. This 2D model, consisting of a solid pillar texture, can show a different surface geometry, depending on the cutting orientation (see Figures 1b and 1c). Since such a 2D model represents a simple wetting system, CA and CAH of a resultant surface pillar structure with different geometrical parameters for various orientations can be readily calculated using the corresponding 2D model. For the analysis of a 2D wetting system, some assumptions should be made as follows: (1) It is generally accepted that gravity, chemical heterogeneity, and interactions between water and solid and between fluidic molecules within a droplet can be neglected. (2) It is reasonable to assume that the droplet size is millimeter scaled and is much larger than the dimension of surface asperities. As a result, the line tension, i.e., the excess free energy of a solid-liquid-vapor system per unit length of the threephase contact line,34,35 becomes extremely small and makes little contribution to the wettability for such a macroscopic droplet. In fact, Pompe et al.36 have determined the value of the line tension, which is in the range of 10-11 to 10-10 J/m; the line tension can therefore be neglected. (3) Based on the above assumptions, the 2D droplet profile can be considered as a spherical cap and have a constant area for different orientations (analogue of constant volume in 3D model). In our previous work, we use a set of geometrical parameters of pillar width, spacing, and height to describe a 2D model resulting from a 3D surface pillar structure.32 In the present study, for a given cutting angle (R) in the 3D groove structure, the corresponding pillar width (aR) and spacing (bR) in the resultant 2D model can be expressed as:

aR ) a ⁄ sinR, bR ) b ⁄ sinR

(1)

where a and b are groove spacing and width (Figure 1), respectively, and the cutting angle (R) is given in the X-Y reference frame indicated in Figure 1b. Therefore, using the same methodology as before, we can derive a set of FE and FEB equations for this cutting angle. Here we give a detailed derivation for a FE equation as a typical example. For a wetting composite state, when the three-phase contact line moves from one instantaneous position C (with a CA of θC and drop size of LC) to an arbitrary position such as D (with θD and LD) at the cross section of cutting angle R (see Figure 2), the magnitude of FE per unit length of drop contact line for each of the two states can be written as:

(

FCcom ) γla θC

LC a 2b + + γls +C sinθC sinR sinR

)

(2)

LD a + γsa +C sinθD sinR

(3)

FDcom ) γlaθD

where C is the FE of the unchanged portion for the system (e.g., the FE associated with solid-liquid, liquid-air, and solid-

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Figure 1. (a) An enlarged view of a groove structure surface microtexture. (b) Top view of the groove structure surface. (c) Schematic cross sections of microtextured surface along various cutting angles R1, R2, and R3 as indicated in frame b. The inset shows formation of possible noncomposite and composite wetting states as liquid or air fills the troughs of the microtexture.

air interfaces beneath the drop from O to D); for this case, LD ) LC - aR - 2bR. Young’s equation is locally valid:

γlacosθY ) γsa - γls

(4)

where θY is the intrinsic CA of the substrate. Thus, the system FE changes from C to D can be expressed as:

(

com ∆FCfD ⁄ γla ) θD

)

(acosθY - 2b) LC LD - θC + (5) sinθD sinθC sinR

According to the assumption, eq 6 is derived from geometrical analysis:

θC

LC2 sin 2θC

- LC2ctgθC ) θD

LD2 sin 2θD

- LD2ctgθD

(6) 3. Results and Discussion

Similarly, change in FE and the geometrical constraint for a noncomposite wetting state can also be given as:

(

non ∆FCfD ⁄ γla ) θD

)

LD LC a + 2b - θC + + 4h cosθY sinθD sinθC sinR

(

)

(7) θC

LC2 sin 2θC

- LC2ctgθC +

and numerical computations can be implemented by a set of coded instructions on computer. With the same method, the geometrical constraint and FE equations between other arbitrary instantaneous positions can be derived. By changing the cutting angle, the whole FE and FEB value in different directions on the grooved structure can also be obtained. It is worth pointing out that previous computational method,31 32 involves tedious numerical calculations for each FE. As such, the FE obtained is not adequate to construct complete continuous FE and FEB curves. To overcome the weakness of the previous methodology, an improved modeling way and program codes in present study are developed, which allows us to gain a set of FE and FEB values easily.

LD2 4bh ) θD 2 - LD2ctgθD (8) sinR sin θ D

A reference FE state is assigned as a value of zero by a random choice (e.g., the instantaneous position C) with the initial drop size and CA of L0 and θ0, respectively. Since it is not possible to obtain θD as an explicit function of θC, ∆FCfD is determined by solving eqs 5–8 via successive approximations

To make our results and discussion easily understandable, first we give a typical example to show how to get various CAs and CAH from the analysis and calculations of FE and FEB. Figure 3 illustrates two FE curves for a cutting angle (R ) 0.95°). Note that FE (J/m) as described, i.e., F per unit length of contact line, is normalized with respect to γ (J/m2), and the unit of FE will then be meter. One can see that the two bowlshaped curves contain multivalued local minimum and maximum FE for both noncomposite and composite states. Such local extremes represent metastable equilibrium states and are related to various apparent CAs in experiments. However, there is only one global FE minimum for each curve, which is associated with the equilibrium CA (ECA) and exactly corresponds to the Wenzel’s or Cassie’s CA for noncomposite or composite states.31,32 It is also noted that the two curves intersect at CA )

Wetting Behavior of Superhydrophobic Surfaces

Figure 2. Illustration of FE analysis for a 2D drop on the cross section of microtextured surface along the direction of the cutting angle R as indicated in Figure 1a.

Figure 3. Normalized free energy as a function of the instantaneous contact angle in the direction of R ) 0.95° on a groove structure surface for composite (com) and noncomposite (non) wetting states (L0 ) 10-2 m, a ) b ) 2 µm, h ) 1 µm; intrinsic CA, θY ) 120°). The inset shows an enlarged view of a segment of FE curve illustrating the FEB; positions A, B, and C correspond to those in Figure 2, and ∆AB and ∆AC represent the FEB for retreating and advancing contact line, respectively. Note that FE is per unit length of the contact line and it is normalized with respect to the surface tension of the liquid.

180°. This indicates that there is no difference in FE between noncomposite and composite states if a droplet touches a solid surface at one point, or in other words, the droplet forms a sphere on the solid surface to achieve ideally the maximum CA (180°). In addition, it is seen from Figure 3 that for this cutting angle, the FE curve of the composite system is associated with a larger ECA and located higher than that of the noncomposite system. Here, it is worth noting that, besides a higher FE, a larger ECA also indicate a less stable thermodynamic state. Actually, the FE curves for noncomposite and composite wetting states will overlap, implying a equivalent thermodynamic state if their ECAs are equal to each other (see below). Figure 4 shows two FEB curves for the composite state for the same surface geometry used in Figure 3. From the FEB curves, advancing (θa) and receding (θr) CAs and ECA as well as theoretical CAH defined as (θa - θr) can be determined, which is described in detail in our previous work. 31,32 3.1. Thermodynamic Analysis on Anisotropic Wetting Behavior. 3.1.1. FE Analysis on GrooWe Structure Surface. Figure 5 shows a comparison of FE between noncomposite and composite states for various cutting angles from 0° to 90°. One can see that, for the given textured geometry, the FE of composite state is higher than that of noncomposite state for

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Figure 4. Determining receding and advancing CAs as well as CAH from the typical curves of advancing and receding FE barriers in the direction of R ) 0.95° on a groove structure surface for a composite wetting state (L0 )10-2 m, a ) b ) 2 µm, h ) 1 µm; intrinsic CA, θY) 120°). The CAH shown is the maximum value associated with zero FEB on the advancing and receding branches of the FE curve (see Figure 3). Note that FEB is per unit length of the contact line and it is normalized with respect to the surface tension of the liquid.

Figure 5. Comparison of variations of normalized FE with apparent CA between noncomposite (non) and composite (com) wetting states in various directions. The inset shows an enlarged view of segments of FE curves illustrating the different configurations of FEB in various directions; Rdenotes the cutting angle shown in Figure 1. In the direction parallel to the grooves (R ) 0°), the FE curve is smooth and is meaningless for the composite state (L0 ) 10-2 m, a ) b ) 2 µm, h ) 1 µm; intrinsic CA, θY ) 120°).

all the orientations, implying that this wetting system prefers the noncomposite state. One can also see that the difference in FE between noncomposite and composite states decreases as cutting angle increases. For example, the maximum FE difference occurs at R ) 0.11°, i.e., the direction parallel to the grooves, whereas the two curves overlap, resulting in a zero difference at R ) 90°, i.e., the direction perpendicular to the grooves. The above theoretical result is interesting enough to indicate that a droplet in the perpendicular direction can exhibit a larger possibility for the transition from noncomposite to composite states than that in the parallel direction. This is consistent with the recent experimental observation by Sommers et al.,37 which showed noncomposite states appeared mainly in the parallel direction for both wetted and dry channels on the micropatterned aluminum surfaces. Such an anisotropic behavior in the transition of thermodynamic states may find potential applications in the liquid delivery or transfer. For example, the rolling off of droplet induced by external stimulus on a surface or channel or the sliding in a tiled surface in a fixed or designed direction is desired; it is therefore a practical strategy to promote


Figure 6. Variations of equilibrium CA (ECA) with respect to cutting angles for noncomposite (non) and composite (com) wetting states (L0 )10-2 m, a ) b ) 2 µm, h ) 1 µm; intrinsic CA, θY ) 120°). The inset schematically describes the interactive force between the adjacent liquid molecules which pulls the broken point back and forms a relatively continuous three-phase contact line; a and b schematically represent the three-phase contact line and the fluidic molecule, respectively.

a thermodynamic transition from noncomposite to composite in order to achieve this desired controllable wetting behavior. Note that in the direction parallel to the grooves (R ) 0°), it is meaningless for the composite state (no air entrapment happens), and the calculated results show a smooth FE curve with no metastable states. θa, θr, and ECA are, therefore, equivalent to θY as if a droplet were located on an ideal smooth surface. This is also the reason why we select a cutting angle 0.11° rather than 0°. 3.1.2. Anisotropy Analysis of Static CA. Based on the FE curves (Figure 5), variations of ECA with respect to cutting angles can be drawn, as illustrated in Figure 6. One can see that the ECA for the noncomposite state increases with increasing cutting angle, showing a strong dependence on the direction and hence indicating an obvious anisotropic wetting behavior. However, the ECA of the composite state appears constant with increasing cutting angle, indicating no anisotropic behavior. The above result demonstrates that, for a given surface and a concerned wetting parameter, the anisotropic wetting behavior is different, depending on thermodynamic status. It should be pointed out that the 2D model has not taken the continuity of the three-phase contact line into consideration. Consequently, a sharp drop of ECA value in the parallel direction for the composite state is observed due to the limitation, as marked in Figure 6. In fact, such a broken point in the three-phase contact line can hardly occur. This happens because the interactive force between the adjacent liquid molecules can pull the broken point back and form a relatively continuous three-phase contact line, as illustrated in the inset of Figure 6. For the given surface texture, this interaction leads to a ECA value in the parallel direction close to that in the neighboring 2D drops ()138.6°) rather than θY ()120°). This result is similar to the experimental observation by Chen et al.,25 who showed that for the composite state, the static contact angle in the parallel direction (127°) are much larger than θY (114°). In contrast, for the noncomposite state, ECA in the parallel direction will exhibit a value close to θY, owing to its gradual change without a sharp drop in ECA, as shown in Figure 6. The above theoretical result is also supported by the latest experimental observation by Zhao et al.,24 although the results of Chung et al.26 indicate that CA in the parallel direction corresponds to the Wenzel’s CA, implying a small value. In addition, it is noted that that the present theoretical analysis,

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Figure 7. Variations of normalized FEB with apparent CA in various directions of the groove structure surface for noncomposite (non) and composite (com) wetting states (L0 ) 10-2 m, a ) b ) 2 µm, h ) 1 µm; intrinsic CA, θY ) 120°). Notations ‘re’ and ‘ad’ denote receding and advancing, respectively.

demonstrating an isotropic wetting behavior for the composite state, is also consistent with the 3D simulation38 using a chemically patterned surface. 3.1.3. FEB Analysis on GrooWe Structure Surface. Figure 7 illustrates FEB variations with respect to (apparent) CA for both noncomposite and composite wetting states. One can see that for a given direction (cutting angle), the receding FEB curves for noncomposite and composite states overlap (e.g., curve 2 for the noncomposte state and curve 8 for the composite state for a R ) 6.34°). And all the receding FEB curves cross the x-axis at the intrinsic CA (θY) 120°), indicating an isotropic distribution for receding CA on this grooved surface. However, the advancing FEB curves for noncomposite and composite states show quite different. One can see that the advancing CAs for the composite state remain unchanged with cutting angle, but the advancing CAs for the noncomposite state vary significantly. It is important to note that for both noncomposite and composite states, the advancing FEB in the parallel direction (curves 1 and 7) changes more dramatically than that in the perpendicular direction (curves 5 and 11). Accordingly, a small external energy (e.g., drop vibration) will cause a large fluctuation of apparent CA in the perpendicular direction, implying the occurrence of multiple CA values for various metastable equilibrium states. A similar situation can be seen for the receding FEB. This different trend of change in FEB could be responsible for the relatively larger fluctuation of apparent CA in the perpendicular than the parallel direction in most experiments. For a detailed discussion, the effect of different number of FEB with respect to cutting angles will be considered in section 3.1.5. 3.1.4. Analysis of Anisotropy in Dynamic CA. Based on the FEB curves (Figure 7), variations of various CAs with respect to cutting angles can be obtained. As seen in Figure 8a, the curves of receding CAs for both noncomposite and composite wetting states overlap and the values remain constant with increasing cutting angle, implying an isotropy of receding wetting behavior. As for the advancing CAs, one can see that θa for the noncomposite state increases with increasing cutting angle, indicating a strong anisotropic wettability. However, θa of the composite state remains constant and shows no anisotropy. Based on the FEB curves (Figure 7), variations of CAH with respect to cutting angles can also be obtained, as illustrated in Figure 8b. One can see that, for the noncomposite state, CAH increases considerably with increasing cutting angle, indicating



Figure 9. Number of metastable states with respect to cutting angles for noncomposite (non) and composite (com) wetting systems (L0 ) 10-2 m, a ) b ) 2 µm, h ) 1 µm; intrinsic CA, θY ) 120°).

Figure 8. (a) Variations of advancing CA (θa) and receding CA (θr) with respect to cutting angle for noncomposite (non) and composite (com) wetting states. (b) Variations of contact angle hysteresis (CAH) with respect to cutting angles for noncomposite (non) and composite (com) wetting states (L0 )10-2 m, b ) 2 µm, h ) 1 µm; intrinsic CA, θY ) 120°). Note that when parameter of groove spacing a is variable, the same curves are gained.

an obvious anisotropic behavior. In contrast, for the composite state, CAH remains unchanged with increasing cutting angle, indicating an isotropic wetting behavior. The present theoretical results are consistent with the recent experimental observations. In particular, Zhao et al.24 have shown that, for noncomposite states, CAH in the parallel direction is very close to that of the flat surface (although it is not zero as considered in a real surface) and it is lower than that in the perpendicular direction. It is noted that the sharp drop of θa and CAH in the special direction such as R ) 0° for composite state can be attributed to the limitation of this 2D model, as discussed in section 3.1.2. For compensation of the neglected interaction from the neighboring 2D drops, the CAH value in the parallel direction for the composite state will be much larger than zero (close to CAH shown in the neighboring 2D drops). In contrast, CAH of the noncomposite state in the parallel direction will show a close value for an ideal smooth surface, i.e., zero. 3.1.5. The Effect of Energy Barriers on Anisotropic Wetting. As discussed above, a grooved surface in the noncomposite state can exhibit an obvious anisotropic wetting behavior in term of most wetting parameters. In contrast, it is expected that such a surface in the composite state should show almost an isotropic wetting behavior. However, it has been demonstrated in recent experimental studies20,25 that for the composite state, the isotropic wetting behavior is not always observed. This indicates that there should be other driving forces that cause the anisotropic wetting behavior. Several authors20,24,28 have attributed such anisotropic wettability to the effect of metastable thermodynamic state such as the distribution of energy barrier. Here, we further show that the number of energy barrier also plays an important role.

A closer look at the FE curves (see the inset in Figure 5) clearly shows different configurations of FEB, such as shape, size, and frequency of occurrence, for different directions. Figure 9 illustrates variations of the number for both noncomposite and composite states with respect to cutting angles. One can see that for both states, the number increase largely with cutting angle, leading to a great number of FEB for the perpendicular direction. This may therefore cause the hysteretic effect of the three-phase contact line around this direction because of numerous metastable states, when a droplet advances on the grooved surface. As a result, the wetting system could exhibit an apparent CA very close to the advancing CA. However, for the parallel direction, wetting metastable states with lower FE will easily be attained due to the less number of FEB, indicating an apparent CA close to the equilibrium CA. The hysteretic effect also makes a droplet hardly reach its final equilibrium state, and accordingly, the anisotropic wetting behavior is promoted in the composite state and is further amplified in the noncomposite state. Actually, recent studies24,26 have shown a similar result for the noncomposite wetting state. The apparent CA experimentally measured in the perpendicular direction is agreement with the maximum macroscopic angle (i.e., the advancing CA in the present study); while in the parallel direction, the apparent CA turns to be close to the equilibrium CA. Similarly, under a dynamic wetting condition, the advancing or receding of the three-phase contact line, situated in the perpendicular direction, seems more difficult than that in the parallel direction due to the anisotropic distribution of energy barriers. In addition, one can also see from Figure 9 that the number of energy barrier of the composite state is obviously larger than that of the noncomposite state except for R ) 0° and 90°, so the effect of energy barrier on static and dynamic wetting for the composite state will be more remarkable for the given surface texture. This essentially indicates that the wetting behavior of the composite state is more sensitive to change in FEB than that of the noncomposite state. 3.2. The Effects of Surface Texture Geometry on Anisotropic Wetting Behavior. In our previous work,31,32 we have indicated that ECA and CAH are two of the most important wetting parameters, which can characterize static and dynamic (sliding) behavior well, respectively. In this section, we focus on the discussion of the effect of surface texture geometry on anisotropic wetting behavior, with an emphasis on variations of ECA and CAH with respect to groove width, height, and spacing. 3.2.1. The Effect of GrooWe Spacing. To investigate the role of groove spacing, parameter a is variable whereas the other


Li et al.

Figure 10. Variations of ECA with cutting angles for various groove spacings (a) of the microtextured surfaces for noncomposite and composite wetting states (L0 ) 10-2 m, b ) 2 µm, h ) 1 µm; intrinsic CA, θY ) 120°; the unit for a is micrometers).

geometrical parameters remain unchanged. Figure 10 shows variations of ECA with respect to cutting angle for different groove spacings. One can see that for various groove spacings, ECA in the noncomposite state increases with increasing cutting angle, indicating an obvious anisotropic wettability. It is interesting to note that such an anisotropic behavior can be amplified with decreasing groove spacing, in particular, under a large cutting angle; for the maximum angle, i.e., the perpendicular direction, the system exhibits a maximum ECA. This significantly implies that the anisotropic behavior can be controlled or adjusted by the design of surface geometry, indicating a promising strategy for the development of practical superhydrophobic surfaces. However, for various groove spacings, ECA with respect to cutting angle in the composite state remains unchanged, i.e., isotropic wetting behavior, although ECA for a given cutting angle also increases with decreasing groove spacing. Completely different from the variability of ECA, the modeling results show that θa, θr, and CAH for both noncomposite and composite states in various directions are hardly dependent on groove spacing. When b ) 2 µm and h ) 1 µm, variations of θa, θr, as well as resultant CAH with respect to cutting angle for diverse groove spacings have the same results as shown in Figure 8. One can find that regardless of change in groove spacing the wetting system still shows an anisotropy of dynamic wetting behavior for the noncomposite state but an isotropy for the composite. It is noted here that the present study is based on hydrophobic surfaces (i.e., intrinsic CA is larger than 90°), but this conclusion can be extended to hydrophilic grooved surfaces. It is also noted that Zhao et al.24 have investigated the role of the groove wavelength (analogical to (a + b) in the present study, see Figure 1) on hydrophilic grooved surfaces for the noncomposite state. They changed groove spacing and width simultaneously perhaps due to the limitation in experimental conditions and found that CAH in the perpendicular direction increased with the decrease of the groove wavelength. However, we argue that it is not the groove spacing but the width that affects CAH (see the next section). On Comparison of Figure 10 to Figure 8a, one can find that ECA in various directions for noncomposite state will shift upward to the unchanged θa as the groove spacing decreases. Since ECA can not be higher than θa, it will be infinitely close to θa when the groove spacing decreases to a value small enough. In this case, taking b ) 2 µm and h ) 1 µm as an example, we find that if groove spacing decreases to about 1nm, ECA will almost approach to θa, as shown in Figure 8a. This

Figure 11. (a) Variations of ECA with cutting angles for various groove widths (b) of the microtextured surfaces for noncomposite and composite wetting states. (b) Variations of θa, θr with cutting angles for various groove widths (b) of the microtextured surfaces for noncomposite and composite wetting states (L0 ) 10-2 m, a ) 2 µm, h ) 1 µm; intrinsic CA, θY ) 120°; the unit for b is micrometers; θa and θr denote advancing CA and receding CA, respectively).

means that further decrease in groove spacing can hardly alter the anisotropy of the static wetting behavior any more. For the composite state, ECA in various cutting angles will also approach to θa as groove spacing decreases but will always keep wetting isotropy. 3.2.2. The Effect of GrooWe Width. Figure 11a shows variations of ECA in both noncomposite and composite states for various groove widths. It is of interest to see that the effect of groove width on ECA for the noncomposite state is entirely opposite to that for the composite state. For a given cutting angle, ECA in the noncomposite state decreases with increasing groove width. However, ECA in the composite state increases with increasing groove width. As a result, the ECA curves for the two states can intersect at a certain cutting angle Rr when groove width decreases. For instance, for the given surface texture, the curves for the two states will intersect at the cutting angle Rr ) 29.7° when the groove width b ) 1 µm. It should be noted that the wetting system with a larger ECA can show a larger FE; this has been shown based on further calculations of FE, as illustrated in Figure 12. One can see that the FE curves for both states overlap in the direction of Rr ) 29.7°. Moreover, in the direction between Rr and 90°, the FE curve for the noncomposite state is located at a higher position, implying that the composite state is more stable. Thus, the composite state would be energetically favorable at this range of cutting angle. If groove width is larger than 2 µm, the ECA curves for noncomposite and composite states will never intersect, implying that the noncomposite will be an energetically preferred state on the whole grooved surface. The above analysis provides another possible theoretical interpretation for the Sommers et al.’s experimental phenomenon that a droplet on a certain



Figure 12. Comparison of variations of normalized FE with apparent CA between noncomposite and composite wetting states with different cutting angles (L0 ) 10-2 m, a ) 2 µm, b ) 1 µm, h ) 1 µm; intrinsic CA, θY ) 120°).

microstructure surface can exhibit both noncompsoite and composite states at the same time, depending on the orientation,37 which has also been mentioned in section 3.1.1. This finding therefore offers a fundamental principle for the design of microfluidic channels. Figure 11b illustrates variations of θa and θr for different groove widths. Meanwhile, the resultant CAH for various groove widths can be easily determined from them (as discussed below).) For the composite state, the groove width can hardly affect θa and θr. For the noncomposite state, θr also remains constant. However, groove width does affect θa and hence CAH in the noncomposite state, which is different from the effect of groove spacing. As seen in Figure 11b, when groove width is larger than a critical value of 2 µm, θa in the perpendicular direction will increase much more largely than that in the parallel direction, resulting in an amplification of the anisotropy of CAH. When groove width is smaller than 2 µm, the largest values for θa and CAH in the perpendicular direction for the noncomposite state can be reached. As groove width continues to decrease, a turning point Rr occurs. Both θa and CAH in the directions between Rr and 90° can reach the largest value, while they change from the largest to the smallest in the directions between Rr and 0°. On comparison of Figure 11b to Figure 11a, it is noted that, when groove spacing is smaller than 2 µm, the same turning point of Rr occurs in both figures. For instance, Rr shows 29.7° in both figures when the groove width is 1 µm. Therefore, the droplet, in the directions between Rr and 0°, where the noncomposite state is energetically preferred (see Figure 12 mentioned above), will roll easily due to the small CAH; on the other hand, the droplet may be restricted to move along the directions between Rr and 90° due to the large CAH, where the composite state is energetically favored. It should also be pointed out that the critical turning point Rr will shift to cutting angle 0° with decreasing groove width. For instance, Rr changes from 29.7° to 3.2° as groove width decreases from 1 to 0.1 µm for the given surface texture. This indicates that, if groove width decreases largely, the dynamic wetting behavior for the noncomposite state is similar to that for the composite state and the anisotropy will possibly disappear. 3.2.3. The Effect of GrooWe Depth. Figure 13a shows variations of ECA with respect to cutting angle for various groove depths. One can see that ECA for the composite state is hardly dependent on groove depth. In contrast, for the noncomposite state, one can find a critical direction Rr′ when groove

Figure 13. (a) Variations of ECA with cutting angle for various groove depths (h) of the microtextured surfaces for noncomposite and composite wetting states. (b) Variations of θa, θr with cutting angles for various groove depths (h) of the microtextured surfaces for noncomposite and composite wetting states (L0 )10-2 m, a ) b ) 2 µm; intrinsic CA, θY)120°; the unit for h is micrometers; θa and θr denote advancing CA and receding CA, respectively).

depth is larger than 2 µm. For instance, when the groove depth is 4 µm, ECA in the directions between Rr′ ()31°) and 90° can reach the largest theoretical value ()180°), while it changes from the largest to the smallest in the directions between Rr′ and 0°. When groove depth is smaller than 2 µm, ECA in the perpendicular direction will increase much more largely than that in the parallel direction, i.e., the effect of groove depth in the perpendicular direction is much more sensitive than that in the parallel direction, causing the maximum anisotropy in ECA. This is consistent with a recent experimental observation by Zhao et al.24 They have shown that the ECA of the noncomposite state in the parallel direction is almost independent of groove depth, while it is influenced obviously in the perpendicular direction. It is particularly noted that when groove depth increases to a critical value larger than 1 µm, the ECA curves for noncomposite and composite states will intersect at a certain direction Rr, as also shown in Figure 13a. This effect is quite similar to that of groove width. Figure 13b illustrates variations of θa and θr for different groove depths. Comparing the effects of groove depth to groove width (also see Figure 11b), one can see a similar trend. The curves of θa (so as CAH) in the noncomposite state shift upward with increasing groove depth and finally become very close to the unchanged curve of the composite state. It is also noted that, when groove depth is larger than 1 µm, the same critical value Rr could be found, as shown in Figure 13a, and in the range of Rr to 90°, both θa and CAH can reach the theoretical largest value. 3.3. A Fitting Approximation for the CA CurWe. In order to better understand the static and dynamic wetting behavior, in particular, on its anisotropy, it is useful to establish theoretical


Li et al.

(

) (

)

θEsin(R) 2 θEcos(R) 2 + )1 A B

(9)

where R and θE represent cutting angle and equilibrium contact angle, respectively. Parameters A and B are constants, which can be determined by the use of the ECA values in the parallel and perpendicular directions, respectively. For the given surface texture (a ) b ) 2 µm, h ) 1 µm), A ) 122° and B ) 138.6° for the noncomposite state and A ) B ) 138.6° for the composite state. Inserting the values of A and B, ECA as a simple function of cutting angle can be finally derived. Figure 14a also shows the fitting curves using the above derived function. As seen, there is a fairly good agreement between the modeling results and the numerical calculations. Similarly, based on the shapes of the CAH curves (Figure 14b), we can also propose a modified elliptic equation to establish a function for the analysis of CAH as follows:

( Figure 14. (a) Fitting curves for variations of equilibrium CA (θE) with respect to cutting angles for noncomposite (non) and composite (com) states in polar coordinates. (b) Fitting curves for variations of contact angle hysteresis (θH) with respect to cutting angles for noncomposite (non) and composite (com) states in polar coordinates. Polar angle and polar axis represent cutting angle R and magnitude of CA, respectively. They are symmetrically reproduced in the range of polar angles between 90° to 360°. The red dashed and blue solid lines represent the fitting lines found to match the various CA values for noncomposite and composite states, respectively. L0 ) 10-2 m, a ) b ) 2 µm, h ) 1 µm; intrinsic CA, θY) 120°.

relationships between the various wetting parameters and space orientations such as cutting angle used in the present study. Based on the above numerical results, it is adequate to develop a phenomenological correlation. To this end, it is helpful to employ polar coordinates to better show the symmetry. Here we take the above results, such as the variations of ECA and CAH as shown in Figures 6 and 8b, as a typical example. These figures are plotted in Cartesian coordinates and can be changed into polar coordinates, as illustrated in panels a and b of Figure 14, where polar angle and polar axis represent cutting angle and magnitude of CA and CAH, respectively. From Figure 14a, one can see that the curves of ECA for both composite and noncomposite states exhibit regular symmetrical shapes; the former is spherical whereas the latter is almost elliptic. Such regular shapes mean that there is a definite singular correlation between wettability and orientation. In other words, for the investigated surface geometry, ECA is a simple function of cutting angle. If this function is obtained, ECA at any orientations can be easily derived, which avoids the tedious numerical calculations for each cutting angle. Accordingly, based on the function, the physical meaning of the wetting phenomenon, such as CAH, can be further analyzed; in particular, the concerned anisotropic behavior can be understood completely. Based on the shapes of the ECA curves, we propose an elliptic equation to establish this function. The equation is expressed as follows:

(θH + θY)cos(R) A′

) ( 2

+

(θH + θY)sin(R) B′

)

2

)1

(10)

where θH represents contact angle hysteresis; θY is the intrinsic CA (θY ) 120° in the present study). Parameters A′ and B′ are constants, which can be determined by the use of θa values in the parallel and perpendicular directions, respectively. For the given surface texture, A′ ) 127° and B′ ) 176° for the noncomposite state and A′ ) B′ ) 179.5° for the composite state. Therefore a simple quantitative correlation between CAH and cutting angle can be finally established. Figure 14b also shows the fitting curves using the above derived correlation. As seen, the modeling results are very consistent with the numerical calculations, except for the regions close to the parallel and perpendicular orientations (i.e., cutting angle of 0° or 180° and 90° or 270°), where the CAH curve becomes smooth, indicating a small CAH compared with the numerical results. This also indicates that the fitting function can compensate the neglect of the continuity of the three-phase contact line in the 2D model, where the sharp change in wetting parameters can occur for these orientations. 4. Conclusions In the present study, a thermodynamic methodology is proposed to theoretically investigate anisotropic wetting behavior of superhydrophobic grooved surfaces. The effect of surface texture geometry on FE and FEB as well as various CAs and CAH is discussed in detail. A fitting approximation is further developed to establish the correlations between the wetting parameters and space orientations. Based on numerical calculations, it is demonstrated that for this grooved texture ECA and CAH for the noncomposite state exhibit obvious anisotropic behavior whereas for the composite state exhibit almost isotropic behavior. Based on thermodynamic analysis, it is found that such anisotropic wetting behavior results mainly from the anisotropic distribution of number and magnitude of FEB, which retains a droplet in various metastable states and also makes the droplet hardly reach its final equilibrium state. Furthermore, the anisotropic behavior for the noncomposite state depends strongly on the surface geometry. For the investigated groove structure, all the geometrical parameters such as groove width, depth, and spacing play an important role, which makes ECA and CAH more sensitive in the perpendicular direction much than that in the parallel direction. It is particularly noted that groove width and depth have quite similar effect. However, the surface geometry hardly affects these wetting parameters for the composite state. In addition, using a fitting approximation,

Wetting Behavior of Superhydrophobic Surfaces a simple quantitative correlation between wettability and orientation can be well established, which is consistent with the numerical calculations, and can be readily used to further analyze other anisotropic wetting parameters. Acknowledgment. This work was supported by the National Natural Science Foundation of China (No. 50772086) and the High-Tech R & D Program of China (863). Nomenclature FE FEB CA θa θr ECA (θE) CAH (θH) com non R, R1, R2, R3 a, b, h aR, bR γla,

γsa,

γls θ0, θC, θD L0, LC, LD θY con ∆F CfD , non ∆F CfD

Free energy. Free energy barrier. Contact angle. Advancing contact angle. Receding contact angle. Equilibrium contact angle. Contact angle hysteresis. Composite state, i.e., entrapment of air in the troughs between asperities. Noncomposite state, i.e., complete liquid penetration into the troughs between asperities. Cutting angle and cutting angles in various orientations in X-Y reference frame. Groove spacing, groove width, and groove depth, respectively. Corresponding pillar width and spacing for the cross section of cutting angle R. Free surface energy (surface tension) at liquid-air, solid-air and liquid-solid interfaces, respectively. CA of 2D droplet at the reference state, at instantaneous position C, and at instantaneous position D, respectively. Drop size of 2D droplet at the reference state, at instantaneous position C, and at instantaneous position D, respectively. Intrinsic contact angle. The change in FE per unit length of contact line for a drop moving from C to D for the composite and noncomposite states, respectively.

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JP712019Y

Anisotropic Wetting Behavior Arising from Superhydrophobic Surfaces

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