ARTICLE pubs.acs.org/Langmuir
Repellency of the Lotus Leaf: Resistance to Water Intrusion under Hydrostatic Pressure C. W. Extrand Entegris, Inc., 101 Peavey Road, Chaska, Minnesota 55318, United States ABSTRACT: In an attempt to better understand the repellency of the lotus leaf, a model was constructed from hydrophobic hemispheres arranged on a hexagonal array. Two scenarios were considered. In the first, the hemispheres were smooth. In the second, the hemispheres had a secondary roughness. The model shows that, without the secondary structure, the repellency of this surface geometry is relatively poor. The secondary structure directs the surface tension upward, allowing much greater resistance to penetration of water and prevents the loss of repellency. From the proposed model, the maximum intrusion pressure (or so-called CassieWenzel transition) of the lotus leaf is estimated to be 1215 kPa. The predicted maximum pressure agrees well with reported values from experimental measurements.
’ INTRODUCTION Efforts to understand water repellency began in the 1940s, when investigators examined waterfowl1 and insects.2 More recent work on plants by Barthlott and co-workers3 has captured the imagination of casual observers and career scientists alike. Today, the biggest star in repellency is arguably the lotus plant (Nelumbo nucifera). Revered in Asian culture for its purity, its leaves are superhydrophobic and self-cleaning3,4 via a combination of inherent hydrophobicity and surface geometry. The surface of the lotus leaf, like many naturally repellent surfaces, is covered with protuberances that have structural hierarchy (Figure 1). The primary structure of these protuberances or papillae takes the form of prolate hemispheroids. The protuberances are covered with a secondary structure of wax crystalloids that project orthogonally outward, creating a myriad of sharp edges. A number of attempts have been made to capture the key elements of its design. Marmur5 constructed a model of the lotus leaf by approximating its surface as a regular array of smooth paraboloids and then analyzing its wettability using free energy methods. Also using a free energy approach, Patankar6 explored the wettability of regular arrays of square pillars covered with a secondary roughness. Several groups have attempted to model the repellent capillary forces of the lotus protuberances by approximating their shape as textured frusta.7,8 Most recently, Sheng and Zhang have employed a hexagonal array of smooth, round pillars to estimate intrusion pressures.9 These various theoretical constructs examine two different aspects of liquid repellency: (a) the ability of a superlyophobic solid to minimize contact by suspending a liquid on small surface features and (b) the contact angles on these structured surfaces that determine the extent to which a liquid adheres. In the scientific literature on wetting, the term “repellency” has become synonymous with the retention forces that scale with the r 2011 American Chemical Society
hysteresis in contact angles.10 However, the root of “repellency” is “repel”, which according to www.merriam-webster.com means to “repulse”, “resist”, “reject” or “to be incapable of adhering to, mixing with, taking up, or holding.” Both of these factors, the ability to resist intrusion of liquids and the ability to shed them readily, are important aspects of the repellency of the lotus leaf and more generally, super lyophobic surfaces. These two factors also are intricately coupled. If a liquid fully penetrates and wets a structured surface, it adheres tenaciously and consequently is difficult to remove. In this study, the principal goal is to analyze the ability of the lotus leaf to repel the intrusion of water. The various models described earlier are unrealistic, because they lack either the curvature of the primary protuberances or the secondary roughness. Here, a model that incorporates both has been constructed and analyzed to address the following questions. How large are the capillary forces acting around the protuberances? How does the secondary roughness affect those forces? And, at what hydrostatic pressure is repellency lost due to intrusion of water into the surface structure of the lotus leaf?
’ THEORY Description of the Model Surface. Figure 2 shows the model surface employed in this study. The surface is covered with hemispheres centered within a hexagonal array. The base diameter of the protuberances is 2R. The unit cell dimension is 2y. From Figure 1, 2R and 2y for the lotus leaf are estimated to be 11.0 ( 1.4 μm and 18.6 ( 3.3 μm, respectively, in agreement with previously reported values.9,11,12 (The measured value of 2R includes the secondary features which are about 1 μm long.13) Received: March 18, 2011 Revised: April 25, 2011 Published: May 05, 2011 6920
dx.doi.org/10.1021/la201032p | Langmuir 2011, 27, 6920–6925
Langmuir
ARTICLE
Figure 3. Intrusion of water into the structured surface.
Figure 1. Scanning electron micrograph of the top surface of the lotus (Nelumbo nucifera) leaf. The white scale bar is 20 μm long. Image kindly provided by Prof. Barthlott.
Figure 2. Plan view of the lotus leaf model.
Figure 3 depicts the extent of intrusion of water into the structured surface, as described by the angle φ, which ranges from 0 radians (0°) for no intrusion to π/2 radians (90°) for complete intrusion. While the orientation of the airliquid interface at the contact line varies locally with φ, it is assumed that, globally, the airliquid interface can be approximated as flat and horizontal. Depictions of the two types of hemispherical protuberances examined here are shown in Figure 4. In one, the hemispheres are assumed to be smooth (Figure 4a). In the other, the hemispheres have a secondary roughness that radiates orthogonially (Figure 4b). For the rough hemispheres, 2R extends to the tips of secondary features on the lotus leaf protuberances. The model assumes the secondary features are densely packed such that most of the length of the contact line touches them.14 It also assumes that only the outmost portions of the secondary features interact with water. The sharp edges of these features pin water7,15,16 and prevent intrusion into the hierarchical structure. Increasing pressure is expected to drive water downward between the rough hemispheres, where the advancing contact line jumps from one secondary feature to the next. At each step, the liquid is pinned at the outermost edge. With a further increase in pressure, the process repeats. (In practice, the incremental jumps from one secondary feature to the next is quite small, on the order of tens of nanometers. The density of the secondary features depicted in Figure 4b is intentionally sparse to allow for clear depiction of the liquid orientation and the various parameters.)
Figure 4. Close-up side views of the hemispherical protuberances. (a) Smooth hemisphere. (b) Rough hemisphere. Curvature of the air liquid interface is exaggerated to emphasize the liquid orientation and local inherent contact angle. The density of the secondary features depicted in (b) is intentionally sparse to allow for clear depiction of the liquid orientation and the various parameters.
The liquid has a surface tension of γ and advances across the solid structures with an inherent contact angle of θ0. The value of θ0 for water on the surface wax of the lotus leaf is reported to be 105°.17 Forces and Pressure. In the absence of trapped or compressed gas, the ability of the lotus leaf to resist intrusion of liquid can be cast as a competition between the capillary forces acting at the contact line18,19 and the gravitational force acting on the mass of the liquid. The vertical component of the capillary force (fc) that can oppose (or aid) the downward pull of gravity on a liquid can be calculated from the product of the length of the contact line on the hemispheres, 2πR sin φ, and the vertical component of the liquid surface tension, γ sin(θ0 þ φ). For the smooth hemispheres depicted in Figure 4a,2033 fc ¼ πð2RÞγ sin φ sinðθ0 þ φÞ ð1Þ For the rough hemispheres in Figure 4b, the secondary features redirect the surface tension vector,7,34 increasing the effective intrusion angle by π/2, fc ¼ πð2RÞγ sin φ sinðθ0 þ φ þ π=2Þ ¼ πð2RÞγ sin φ cosðθ0 þ φÞ
ð2Þ
The gravitational force can be described as the product of the hydrostatic pressure (Δp) and the unwetted cross-sectional area between the hemispheres (Au As), fg ¼ ΔpðAu As Þ where Au is the area of the unit cell, pffiffiffi Au ¼ 2 3y2
ð3Þ ð4Þ
and As is the wetted cross-sectional area of each hemisphere, As ¼ πR 2 sin2 φ
ð5Þ
Combining eqs 25 yields an expression that relates the hydrostatic pressure (Δph) and extent of intrusion, Δph ¼ 6921
2Rγ sin φ cosðθ0 þ φÞ pffiffiffi ð2 3=πÞy2 R 2 sin2 φ
ð6Þ
dx.doi.org/10.1021/la201032p |Langmuir 2011, 27, 6920–6925
Langmuir
ARTICLE
Limiting Values for Maintaining Repellency. Maximum values of the immersion angle (φmax) can be determined by differentiating eq 2 with respect to φ, Dfc =Dφ ¼ πð2RÞγ cos φ cosðθ0 þ φÞ sin φ sinðθ0 þ φÞ
ð7Þ simplifying with the appropriate angle-sum relation,35 Dfc =Dφ ¼ πð2RÞγ cosðθ0 þ 2φÞ
ð8Þ
and then setting eq 8 equal to zero, cosðθ0 þ 2φÞ ¼ 0
ð9Þ
Equation 9 has solutions or critical points where (θ0 þ 2φ) is equal to multiples of π/2 (π/2, 3π/2,...). The immersion angle where the capillary force exhibits a maximum vertical value occurs at φmax
3π θ0 ¼ 4 2
ð10Þ
Incorporating eq 10 into eqs 2 and 6 and simplifying gives expressions for maximum capillary force (fc,max) and the critical hydrostatic pressure for complete intrusion (Δph,c), 2 fc, max ¼ πRγ cosðθ0 =2Þ þ sinðθ0 =2Þ ð11Þ and Δph, c ¼
2Rγ ð12Þ 2 pffiffiffi ð4 3=πÞy2 = cosðθ0 =2Þ þ sinðθ0 =2Þ R 2
’ RESULTS AND DISCUSSION Capillary Forces. The ability of structured surfaces to repel liquids often hinges on the capillary force. Therefore, let us examine the magnitude and direction of the capillary force as water interacts with the hemispheres. Water initially contacts the model surface at the apex of the hemispheres where φ = 0°. As liquid intrudes, φ increases. Consequently, the length of the contact line around each hemisphere grows. The invading water produces an advancing contact angle at each step.36 The local orientation of water on the surface of the hemispheres determines the magnitude and direction of the capillary force. Figure 5a shows the variation in the vertical component of the dimensionless capillary force (fc/2Rγ) versus intrusion angle (φ) for each smooth hemisphere (eq 1). As water intrudes and φ increases, the magnitude of fc/2Rγ initially decreases from zero. Here, the capillary force is directed downward, acting in concert with gravity. The fc/2Rγ values pass through a minimum (i.e., the largest downward capillary force) and then climb. For θ0 e 90°, the capillary force never attains a positive value, and thus, one could conclude that it is not possible to construct a super repellent surface using smooth hydrophilic hemispheres, regardless of their dimensions or spacing. More hydrophobic smooth hemispheres are expected to perform marginally better. For θ0 > 90°, the capillary force becomes positive for large φ values. Here, the capillary force is directed upward and has the potential to retard the intrusion pressure of water. However, the repellency of smooth hemispheres is not especially robust. The capillary force cannot oppose the intrusion until water has mostly engulfed the smooth hemispheres; for
Figure 5. Variation of the capillary force (fc) with the immersion angle (φ). (a) Smooth hemispheres, according to eq 1. (b) Hemispheres with orthogonal surface features, eq 2.
θ0 = 105°, φ must be >75° for the surface tension to be directed upward. The presence of the secondary orthogonal features has a huge impact on the direction and magnitude of the capillary force. Figure 5b shows the variation in the vertical component of the dimensionless capillary force (fc/2Rγ) versus intrusion angle (φ) for the model with rough hemispheres (eq 2). If θ0 > 90°, the surface tension vector is directed upward for all φ values. Therefore, the capillary force is always positive and always exhibits some resistance to intrusion. For θ0 = 105°, the maximum of fc/2Rγ = 3.09 occurs at φ = 82.5°. At the maximum, the surface tension vector is directed straight up. In comparison, the largest fc/2Rγ value for the smooth hemispheres with θ0 = 105° occurs at φ = 90° and it is only 0.813, nearly 4 times smaller than the corresponding value from the rough ones. Thus, the secondary features allow an otherwise mediocre primary structure to exhibit extraordinary breadth and magnitude of upwardly directed capillary forces. Intrusion Pressures. Now let us turn to the pressure required to push water into the structured surface of the lotus leaf against the resistance of the capillary force. Hydrostatic pressures (Δph) were estimated as a function of intrusion angle (φ) for water (γ = 72 mN/m) on the model surface with rough hemispheres. The results are plotted in Figure 6. Even though the open space between the rough hemispheres closes as water intrudes, the general trend for Δph resembles that of the capillary force; values are positive and the curves exhibit a maximum. Next, let us examine the maximum hydrostatic pressure that a lotus leaf can withstand before the capillary force is overtaken and repellency is lost. For θ0 = 105°, the critical pressure from eq 12 is estimated to be Δph,c = 12 kPa, which is equivalent to a water column of 1.2 m. In contrast, if the hemispheres were smooth and water penetrated to nearly their base where φ ≈ 90°, the intrusion pressure would be only 3 kPa. Examination of the model surface with secondary roughness reveals that the critical intrusion pressure is relatively indifferent with respect to liquid surface tension and inherent contact angle, but is quite sensitive to small differences between the diameter of the hemispheres and unit cell spacing. A variation of (5° in θ0 or (2 mN/m in γ resulted in less than 3% variation in Δph,c. On the other hand, a 10% variation in the size and spacing of the 6922
dx.doi.org/10.1021/la201032p |Langmuir 2011, 27, 6920–6925
Langmuir
Figure 6. Hydrostatic pressure (Δph) versus immersion angle (φ), according to eq 6.
hemispheres produced a wide distribution in the Δph,c values that ranged from 9 to 16 kPa. Uncertainty in eq 12 was also estimated by standard error propagation techniques involving the root mean square of partial derivatives.37 Using the same inputs for θ0, γ, R, and y listed above, the relative uncertainty in Δph,c is approximately (4 kPa or (35%. How do the results from the proposed model compare with experimental observations? Sheng and Zhang studied the critical intrusion pressure (CassieWenzel transition) of the lotus leaf.9 They placed a portion of a leaf in a special apparatus and poured a 5 mm layer of water on top of the leaf. To eliminate complications associated with displacement and compression of air, their apparatus was evacuated and then pressure was applied. Initially, water was suspended on top of the protuberances. They observed that water began to collapse into the surface structure at a hydrostatic pressure of 13.5 kPa. Intrusion was complete at 15.0 kPa. They confirmed the loss of repellency with receding contact angle measurements. Before collapse, the surface of the leaf had a receding contact angle of 140150°. After collapse, the receding angle fell to 0°, verifying complete infiltration of water. The measured value is slightly greater than the Δph,c value estimated from the rough hemisphere model presented here (eq 12). Even though Sheng and Zhang removed air from the experimental apparatus, the ambient pressure almost certainly would not have been zero. At room temperature (2025 °C), the vapor pressure of water is 23 kPa.38 If this vapor pressure is added to the predicted resistance from the capillary forces, then the predicted and measured values are nearly identical. When submerged underwater, there may be additional factors that contribute to the repellency of the lotus leaf. Compression of air also could aid surface tension in impeding intrusion. As described in greater detail in the Appendix, if air is uniformly trapped and compressed, capillary forces and vapor pressure of water would play a minor role. At the intrusion angle where the capillary force is a maximum (φmax = 82.5°), the pressure due to air compression would be 790 kPa, which dwarfs the corresponding capillary and vapor pressures, 12 and 23 kPa. It seems that capillary forces are more important for impinging drops than for submersion. The size and velocity of raindrops has been thoroughly studied. They range in diameter from
