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Contact-Angle Hysteresis on Super-Hydrophobic Surfaces G. McHale,* N. J. Shirtcliffe, and M. I. Newton School of Biomedical and Natural Sciences, The Nottingham Trent University, Clifton Lane, Nottingham NG11 8NS, United Kingdom Received June 1, 2004. In Final Form: August 10, 2004 The relationship between perturbations to contact angles on a rough or textured surface and the superhydrophobic enhancement of the equilibrium contact angle is discussed theoretically. Two models are considered. In the first (Wenzel) case, the super-hydrophobic surface has a very high contact angle and the droplet completely contacts the surface upon which it rests. In the second (Cassie-Baxter) case, the super-hydrophobic surface has a very high contact angle, but the droplet bridges across surface protrusions. The theoretical treatment emphasizes the concept of contact-angle amplification or attenuation and distinguishes between the increases in contact angles due to roughening or texturing surfaces and perturbations to the resulting contact angles. The theory is applied to predicting contact-angle hysteresis on rough surfaces from the hysteresis observable on smooth surfaces and is therefore relevant to predicting roll-off angles for droplets on tilted surfaces. The theory quantitatively predicts a “sticky” surface for Wenzel-type surfaces and a “slippy” surface for Cassie-Baxter-type surfaces.
Introduction How a droplet of a liquid sits and rolls on a surface is determined by both the surface chemistry and the surface roughness or topography.1-3 A flat and smooth hydrophobic surface exhibiting an equilibrium contact angle to water of, say, 115° can be converted into a superhydrophobic surface exhibiting a contact angle of greater than 150° simply by roughening it, even without altering any surface chemistry. Two models of contact-angle enhancement exist. In the Cassie-Baxter4,5 model, a drop rests on the peaks of surface protrusions and bridges the air gaps in between. In the Wenzel6,7 model, the liquid droplet retains contact at all points with the solid surface below it. Experimentally, when the response of a drop to a tilt of the surface is investigated, it is found that all super-hydrophobic surfaces are not equivalent. In some cases, the drop rolls off the surface easily, while for others the drop clings to the surface, even for high tilt angles.8-10 Que´re´ refers to these two contrasting cases as “slippy” and “sticky” super-hydrophobic surfaces.11 The ability of a drop to move on a surface is determined by the contactangle hysteresis.12,13 When a drop resting on a horizontal surface has liquid steadily injected by a syringe until the contact line advances, the contact angle of the drop observed when it * Corresponding author: E-mail:
[email protected]. Tel: +44 (0)115 8483383. (1) Onda, T.; Shibuichi, S.; Satoh, N.; Tsujii, K. Langmuir 1996, 12, 2125-2127. (2) Neinhuis, C.; Barthlott, W. Ann. Bot. 1997, 79, 667-677. (3) Blossey, R. Nat. Mater. 2003, 2, 301-306. (4) Cassie, A. B. D.; Baxter, S. Trans. Faraday Soc. 1944, 40, 546551. (5) Johnson, R. E.; Dettre, R. H. Contact angle, Wettability and Adhesion; Advances in Chemistry Series 43; American Chemical Society: Washington, DC, 1964; p 112. (6) Wenzel, R. N. Ind. Eng. Chem. 1936, 28, 988-994. (7) Wenzel, R. N. J. Phys. Colloid Chem. 1949, 53, 1466-1467. (8) Miwa, M.; Nakajima, A.; Fujishima, A.; Hashimoto, K.; Watanabe, T. Langmuir 2000, 16, 5754-5760. (9) O ¨ ner, D.; McCarthy, T. J. Langmuir 2000, 16, 7777-7782. (10) Feng, L.; Li, S.; Li, Y.; Li, H.; Zhang, L.; Zhai, J.; Song, Y.; Liu, B.; Jiang, L.; Zhu, D. Adv. Mater. 2002, 14, 1857-1860. (11) Que´re´, D.; Lafuma, A.; Bico, J. Nanotechnology 2003, 14, 11091112. (12) Que´re´, D. Physica A 2002, 313, 32-46. (13) Marmur, E. Langmuir 2004, 20, 3517-3519.
Figure 1. (a) Advancing contact angle, (b) receding contact angle, and (c) advancing and receding contact angles determined by tilting experiment.
is just set in motion is defined as the advancing angle, θA (Figure 1a). Similarly, when liquid is steadily retracted, the contact angle observed when the contact line is just set in motion is the receding angle, θR (Figure 1b). If a surface on which a drop rests is tilted slightly, the drop remains at rest but with differing contact angles at each side of the drop (Figure 1c). The difference in forces per unit length at the two sides of the drop is proportional to γLV(cos θL - cos θU) where γLV is the liquid-vapor surface tension and θL and θU are the contact angles at the lower and upper sides of the drop. When the upper angle reaches the receding angle and the lower angle reaches the advancing angle, the drop just begins to move. The range of angles, ∆θH ) (θA - θR), is the contact-angle hysteresis and is an important parameter in understanding drop motion on a surface. Que´re´ has discussed how a CassieBaxter surface with a high contact angle and low hysteresis can result in a reduction in the force required to set a drop into motion.12 Few quantitative attempts have been reported on modeling how contact-angle hysteresis might be influenced in the transition of a surface from hydrophobic to superhydrophobic. Two lines of thought exist in the literature on super-hydrophobicity. In the first case, contact-angle hysteresis is viewed as a consequence of solid surface area, while in the second, it is the contact perimeter of the solid surface area which is viewed as important. That these views are not necessarily equivalent is evident by considering a surface of square pillars across which a drop sits in contact with the tops of the pillars but bridging the gaps between (i.e., the Cassie-Baxter case). For pillars having sides of length D and arranged in a square array of sides L (Figure 2a), the solid contact area is D2 per L2 of area and the perimeter is 4D per L2 of area; the solid
10.1021/la0486584 CCC: $27.50 © 2004 American Chemical Society Published on Web 10/02/2004
Contact-Angle Hysteresis on Super-Hydrophobic Surfaces
Figure 2. (a) Surface texture based on squares of side lengths D embedded in a square array of repeat size L, (b) equivalent texture to (a) but with side length and array length halved, and (c) a further halving of the dimensions in (b). In each case, the solid surface fraction is the same. While the perimeter doubles each time, the dimensions are halved.
fraction is φS ) (D/L) . If we now imagine a similar arrangement, but with the lengths of the pillar sides halved and the array spacing also halved, as shown in Figure 2b, the solid contact area remains D2 per L2 of area but the perimeter doubles to 8D per L2 of area. Significantly, the solid fraction remains unchanged at φS ) (D/L)2 so that, according to a Cassie-Baxter model, both surfaces would exhibit the same super-hydrophobic contact angle despite the pattern size variation. If the dimensions are halved again, the solid fraction remains constant but the perimeter doubles once more to 16D per L2 of area (Figure 2c). We have previously reported a lithographic surface where a pattern size variation was performed for circular pillars arranged in a square array.14 We observed no significant change in hysteresis with pattern scaling despite changing the perimeter by a factor of 8 while maintaining a constant solid contact surface area and surface area fraction; the super-hydrophobic contact angle remained constant to within 2°. In contrast, a perimeter dependence has been suggested both by O ¨ ner and McCarthy,9 based on experiments using different shapes of pillar, and theoretically by Extrand.15 One difference between the results of O ¨ ner and McCarthy and ourselves is that the shapes used in their work involved corners and indented edges rather than simple smooth shapes, such as with circular cross-sectional pillars. 2
Contact-Angle Operating Points On a flat, smooth, and chemically homogeneous surface, the contact angle, θse, is given by Young’s equation
cos θse )
γSV - γSL γLV
(1)
where γLV is the liquid-vapor surface tension and γSL and γSV are the solid-liquid and solid-vapor interfacial tensions, respectively. This formula can be derived by considering a small spherical cap droplet and minimizing the total surface free energy of the drop. When the surface is roughened, the same minimization approach results in CB two different possible contact angles, θW e or θe , through either Wenzel’s equation s cos θW e ) r cos θe
(2)
or the Cassie-Baxter equation s cos θCB e ) φs cos θe - (1 - φs)
(3)
depending on whether an assumption of intimate contact (14) Shirtcliffe, N. J.; Aqil, S.; Evans, C.; McHale, G.; Newton, M. I.; Perry, C. C.; Roach, P. J. Micromech. Microeng. 2004, 14, 1384-1389. (15) Extrand, C. Langmuir 2002, 18, 7991-7999.
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Figure 3. Contact angles predicted by Wenzel’s equation (eq 2) for roughness factors of r ) 1, r ) 1.2, and r ) 2. In the r > 1 cases, a linear region with a slope greater than 1 occurs close to 90°, surrounded by super-linear regions, which then convert to saturation values of 0° or 180°. An operating point, A, on the r ) 1 smooth surface becomes an operating point, B, on the r ) 2 rough surface.
by the liquid of the solid at all points below the drop is maintained (eq 2) or an assumption of air-bridging of some points below the drop is assumed (eq 3). In these formulas, r is the surface roughness, defined as the ratio of the true surface area to the horizontal projection of the surface area, and φS is defined as the solid fraction upon which the drop rests. Both eq 2 and eq 3 predict that under some conditions contact-angle enhancement from that predicted by Young’s law toward values approaching 180° can be achieved, but these equations also suggest significant differences once the contact-angle enhancement has been achieved, as we will illustrate using the concepts of operating points and contact-angle change amplification. To illustrate the effect of Wenzel’s equation (eq 2) consider Figure 3 which plots the contact angle resulting from roughness factors of r ) 1, r ) 1.2, and r ) 2. The curves for r > 1 have three distinct features: (i) a linear region close to θse ) 90°, (ii) saturation to 0° or 180° at θse ) cos-1((1/r), and (iii) super-linear regions intermediate between (i) and (ii). As the roughness increases, the curves always cross at 90°, but the steepness of the curves at this point increases. Imagine that a liquid has a contact angle of 105° on the smooth surface (point A on the r ) 1 curve) which becomes 121.2° on the r ) 2 rough surface (point B on the r ) 2 curve). If the contact angle now increases or decreases by a small amount, ∆θ, the change in position of point B on the r ) 2 curve will be approximately r∆θ. Points, such as A and B, within the linear region can be regarded as operating points for a given liquid (i.e., Young’s-law-determined contact angle) on a given surface roughness, and because r is necessarily greater than unity, always predict that contact angle changes occurring on the smooth surface will be amplified by the surface roughness; the extent of amplification can be greater than r depending on the operating point. Thus, for θse > 90°, Wenzel’s equation (eq 2) predicts both an increase in the contact angle and an amplification of any changes from that value of contact angle. Figure 4 shows how the contact angle on a smooth surface is modified by surface roughness according to the Cassie-Baxter equation (eq 3) using solid surface fractions of φs ) 1, 0.65, and 0.1. The curves for φs < 1 have three distinct features: (i) a constant region close to θse ) 0°, (ii) a linear region approaching 180°, and (iii) a crossover region intermediate between (i) and (ii). In particular, as the solid fraction is reduced, the initial contact angle on the rough surface increases, but the slope of the curve approaching 180° decreases. If we now imagine a liquid
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Figure 4. Contact angles predicted by the Cassie-Baxter equation (eq 3) using solid surface fractions of φs ) 1, 0.65, and 0.1. The initially constant curves tend to a maximum slope of xφs < 1 at 180°. An operating point, A, on the φs ) 1 smooth surface becomes an operating point, C, on the φs ) 0.65 rough surface.
McHale et al.
Figure 6. Gain factors, corresponding to curves in Figure 4, determining response of the contact angle on a Cassie-Baxter surface to a perturbation of the smooth-surface contact angle.
Wenzel equation (eq 2) gives a change in the Wenzel contact angle ∆θW caused by a change in the contact angle on the smooth surface, ∆θe, as
∆θW e ) r
( ) sin θse
sin θW
∆θe
(4)
Using eq 2 and defining a Wenzel gain factor GW(r, θse) as W
G (r,
θse)
)
rsin θse
x1 - r2 cos2 θse
(5)
Equation 4 can be rewritten as Figure 5. Gain factors, corresponding to curves in Figure 3, determining response of the contact angle on a Wenzel surface to a perturbation of the smooth-surface contact angle.
which has a contact angle of 110° on the smooth surface (point A on the φs ) 1 curve), then on the φs ) 0.65 surface this again becomes 121.2° (point C on the φs ) 0.65), which is the same as that of the r ) 2 rough surface. If the contact angle now increases or decreases by a small amount, ∆θ, the change in position of point C on the φs ) 0.65 curve will be less than ∆θ. Points, such as A and C, can again be regarded as operating points for a given liquid (i.e., Young’s-law-determined contact angle) on a given surface, but because φS is necessarily smaller than unity, we can predict that contact-angle changes occurring on the smooth surface will be diminished rather than increased by the surface roughness. This is evident from Figure 4 because the slopes for all curves with φs < 1 are necessarily less than unity. Thus, the Cassie-Baxter equation predicts an increase in the contact angle but a reduction (i.e., attenuation) of any changes from that value of contact angle. Gain and Attenuation. The concept of gain and attenuation factors for changes in contact angles are visualized in Figure 5 and in Figure 6 by numerical computations of the slopes of the curves in Figures 3 and 4. In the Wenzel case (Figure 5), the region close to θse ) 90° shows an approximately constant gain equal to the roughness factor, r. On either side of this region of approximately constant gain, the gain increases rapidly. The range of angles, θse, (i.e., the contact-angle bandwidth) for which constant gain holds reduces as the roughness increases. The equivalent Cassie-Baxter case (Figure 6) has a gain which increases from zero up to a maximum value less than 1. The numerically determined gains can be considered analytically for a change in contact angle about any operating point. Using derivatives of the
s W ∆θW e ) G (r, θe)∆θe
(6)
For contact angles, θse ≈ 90°, the Wenzel gain factor is approximately unity and the effect of roughness is to linearly scale-up (i.e., amplify) any contact-angle changes. As the roughness increases, such that θse f cos-1(-1/r) and θW e f 180°, the Wenzel gain factor becomes infinitely large. In a similar manner, a gain factor can be derived from eq 3 for changes, ∆θCB, in a Cassie-Baxter contact angle caused by changes in the contact angle on the smooth surface ∆θe
∆θCB e ) φs
(
sin θse
)
sin θCB
∆θe
(7)
which, using eq 3, gives a Cassie-Baxter gain factor, GCB(φs, θse) of
G
CB
(φs, θse)
)
φs sin θse
x1 - [-1 + φs(cos θse + 1)]2
(8)
Thus, a change in contact angle on a smooth solid surface becomes a change on the Cassie-Baxter surface of CB s ∆θCB e ) G (φs, θe)∆θe
(9)
Because solid surface fractions are by definition less than one, the Cassie-Baxter gain factor is always less than 1 so that the gain factor is in fact an attenuation of any contact-angle changes occurring on the smooth surface. In contrast to the Wenzel case, we cannot approximate the Cassie-Baxter gain factor to φs because of the effect on sin θCB e in eq 7 caused by the (1 - φs) offset term on the right-hand side in eq 3. The largest value of the Cassie-
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Baxter gain is in the limit of θse f 180° when we find GCB(φs, θse) f xφs; this is an analytical result derived from eq 8. As the solid surface fraction φs f 0, the CassieBaxter gain factor also diminishes toward zero. Contact-Angle Hysteresis. The concept of a contactangle gain factor is useful because it separates the idea of an increase in equilibrium contact angle due to surface topography from the response of the observed contact angle due to perturbations to the contact angle on the smooth surface. One example of a perturbation could be a contamination of the liquid (or the surface) which alters the interfacial tensions and, hence, the Young’s-lawdefined contact angle. A contact angle resulting from a Wenzel-type super-hydrophobicity would show increased sensitivity to the contamination, whereas a contact angle resulting from a Cassie-Baxter type super-hydrophobicity would show decreased sensitivity. Physically, we would expect this type of behavior because, in the Wenzel case, the liquid increases its contact with the solid, whereas in the Cassie-Baxter case, the contact is decreased. A second example of a perturbation could be to increase the contact angle on the surface until the contact line is set in motion so that advancing or receding contact angles are obtained. In this case, simple general arguments can be used to show that the gain factors might be applicable to ∆θA ) (θA - θse) and ∆θR ) (θR - θse) and, hence, to the contact-angle hysteresis, ∆θH. One method of deriving the equilibrium contact angle on a smooth and flat surface (i.e., Young’s law) is to minimize the surface free energy, F. If the contact angle on the surface changes by ∆θe from the equilibrium but no change in contact area occurs, the change in surface free energy, ∆F, will be
assumed that over the range of angles involved in the hysteresis the gain is approximately a constant. The assumption that the extra perimeter does not contribute any additional energy barrier may hold best on surface topographies that are smooth and differentiable, e.g., sinetype curves or granular materials with spherical or elliptical grains. This may be relevant to differences in experimental results published in the literature using lithographic pillars and size scaling.9,14 Since the gain factor is larger than unity for the Wenzel case and less than unity for the Cassie-Baxter case, hysteresis will be increased on a Wenzel-type surface and decreased on a Cassie-Baxter-type surface. As a numerical example, if the average contact angle on the smooth surface is 110° and the contact-angle hysteresis is 16°, then a roughness factor of 1.4 gives a Wenzel contact angle of 119° and a gain factor of 1.5 so that the hysteresis on the rough surface will be increased to 24°. For the same smooth surface parameters, a Cassie-Baxter surface with a solid surface fraction of 0.1 will give a Cassie-Baxter contact angle of 159° and a gain factor of 0.26, reducing the contact-angle hysteresis on the rough surface to 4°. This example illustrates in a quantitative manner the notions of “sticky” and “slippy”. The model we have presented does not seek to explain at a microscopic level where contact-angle hysteresis arises but simply uses the Cassie-Baxter and Wenzel concepts to predict how a preexisting hysteresis on a flat, smooth material is transformed when a super-hydrophobic surface is created from the material. An advantage of this approach is that it may be possible, using a series of liquids with steadily varying contact angles on the flat, smooth surface, to determine experimentally analogues of Figures 3 and 4 and numerically determine the gain factors required for predicting changes in contact-angle hysteresis. An advantage of experimentally determined gain curves is that they should be applicable to estimating hysteresis via eq 12 even if the surface is behaving as a mixed Wenzel and Cassie-Baxter surface such that eq 5 and eq 8 are not directly applicable. Knowing the hysteresis should enable the forces required to move droplets to be estimated when designing a super-hydrophobic surface.
∆F )
(∂F∂θ)
θ)θes
∆θe
(10)
Assuming this change must exceed a minimum energy barrier, ∆Fmin, before contact line motion is initiated gives
) ∆θmin e
(∂F∂θ)
-1 θ)θes
∆Fmin
(11)
is the minimum change in contact angle on where ∆θmin e the smooth and flat surface for the initiation of contact line motion. Using eq 6 or eq 9 gives the equivalent change in contact angle on the rough Wenzel-type and CassieBaxter-type surfaces
(∂F∂θ)
> G(κ, θse) ∆θW,CB e
-1
θ)θes
∆Fmin ) G(κ, θse)∆θmin (12) e
where the gain factor is either the Wenzel or the CassieBaxter formula (eq 5 or eq 8) and κ is either r or φs, respectively, if the superscript on the RHS of eq 12 is W or CB. In both cases, the appropriate gain factor multiplies the change in contact angle on the smooth surface to either increase (Wenzel case) or reduce (Cassie-Baxter case) the change in contact angle on the rough surface. A consequence of eq 12 is that the difference between the advancing and receding angles, i.e., the contact-angle hysteresis, will be multiplied by the appropriate gain factor. There is no requirement for the energy barriers to be the same for the receding and advancing contact angles nor is there any requirement for the free energy curve to be symmetric about the energy minimum. However, it has been assumed that there is no significant additional energy barrier arising from the extra perimeter of contact on a rough or structured surface, and it has also been
Conclusion In this work, we have considered theoretically how super-hydrophobic surfaces can either enhance or diminish contact-angle changes occurring on smooth, flat surfaces. In the Wenzel case, an increase in the equilibrium contact angle on a rough surface is accompanied by an amplification effect for perturbations to the contact angle. When the contact angle on the smooth surface is around 90°, the gain factor for the amplification is approximately equal to the roughness factor, r. In contrast, in the CassieBaxter case, an increase in the equilibrium contact angle on a rough surface is accompanied by an attenuation effect for perturbations to the contact angle. The idea of amplification and attenuation has been applied to contactangle hysteresis, thus giving analytical formulas predicting that a Wenzel surface will become “sticky”, while a Cassie-Baxter surface will become “slippy”. Acknowledgment. The authors’ acknowledge the financial support of the UK EPSRC and MOD/Dstl (Grants GR/R02184/01 and GR/S34168/01). LA0486584
