pubs.acs.org/Langmuir © 2010 American Chemical Society
Thermodynamic Analysis of the Effect of the Hierarchical Architecture of a Superhydrophobic Surface on a Condensed Drop State Tianqing Liu,* Wei Sun, Xiangyu Sun, and Hongru Ai School of Chemical Engineering, Dalian University of Technology, Dalian 116012, China Received May 10, 2010. Revised Manuscript Received July 21, 2010 Condensed drops usually display a Wenzel state on a superhydrophobic surface (SHS) only with microrough architecture, while Cassie drops easily appear on a surface with micro-nano hierarchical roughness. The mechanism of this is not very clear. It is important to understand how the hierarchical structure affects the states of condensation drops so that a good SHS can be designed to achieve the highly efficient dropwise condensation. In this study, the interface free energy (IFE) of a local condensate, which comes from the growth and combination of numerous initial condensation nuclei, was calculated during its shape changes from the early flat shape to a Wenzel or Cassie state. The final state of a condensed drop was determined by whether the IFE continuously decreased or a minimum value existed. The calculation results indicate that the condensation drops on the surface only with microroughness display a Wenzel state because the IFE curve of a condensed drop first decreases and then increases, existing at a minimum value corresponding to a Wenzel drop. On a surface with proper hierarchical roughness, however, the interface energy curve of a condensed drop will continuously decline until reaching a Cassie state. Therefore, a condensed drop on a hierarchical roughness surface can spontaneously change into a Cassie state. Besides, the states and apparent contact angles of condensed drops on a SHS with different structural parameters published in the literature were calculated and compared with experimental observations. The results show that the calculated condensed drop states are well-coordinated with experimental clarifications. We can conclude that micro-nano hierarchical roughness is the key structural factor for sustaining condensed drops in a Cassie state on a SHS.
1. Introduction Condensation is an important and widely accepted heat transfer type in many industries and engineering processes, such as the petrochemical industry, the power industry, and the air conditioning and refrigerating process. The heat transfer coefficient of dropwise condensation is much higher than that of filmwise condensation. It is therefore undisputed that the size of the heat exchangers and/or the energy consumption rate can be greatly reduced if dropwise condensation can be realized in those industries and processes. A drop deposited on a superhydrophobic surface (SHS) can appear as a Cassie state with an apparent contact angle (CA) larger than 150° and a very small rolling angle, which easily makes people think that good dropwise condensation will take place on a SHS. However, the condensation experiments published in the literature on a SHS show that condensed drops on the SHS only with microroughness usually lost the superhydrophobicity.1 Condensed drops exhibited a Wenzel state2-5 or combined states6,7 of Wenzel and Cassie, and the heat transfer effect with this dropwise condensation was not satisfied.8,9 On the other hand, condensed drops could display a Cassie state if micro-nano hierarchical *To whom correspondence should be addressed. Telephone: 86 411 84706360. E-mail:
[email protected].
(1) Wier, K. A.; McCarthy, T. J. Langmuir 2006, 22, 2433. (2) Narhe, R. D.; Beysens, D. A. Langmuir 2007, 23, 6486. (3) Jung, Y. C.; Bhushan, B. J. Microsc. 2008, 229, 12740. (4) Narhe, R. D.; Beysens, D. A. Phys. Rev. Lett. 2004, 93, 076103. (5) Narhe, R. D.; Beysens, D. A. Europhys. Lett. 2006, 75, 98. (6) Dorrer, C.; Ruhe, J. Langmuir 2007, 23, 3820. (7) Chen, X. L.; Lu, T. Sci. China, Ser. G 2009, 52, 233. (8) Song, Y. J.; Ren, X. G.; Ren, S. M.; Wang, H. Gongcheng Rewuli Xuebao 2007, 28, 95. (9) Chen, L.; Liang, S. Q.; Yan, R. S.; Cheng, Y. J.; Huai, X. L.; Chen, S. L. J. Therm. Sci. 2009, 18, 160.
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structure or only nanoroughness existed on a SHS and the drops easily rolled off the surface.10-12 Why can condensed drops sustain their superhydrophobicity only on the SHS with hierarchical roughness or nanostructure? How does a condensate on a SHS, which originated from the growth and combination of numerous initial condensate nuclei, evolve and change its shape to a final stable Wenzel or Cassie drop? What is the relation between the final state of the condensed drop and the hierarchical structural parameters? All these questions remain unanswered. We thus decided to calculate the interface free energy (IFE) of condensed drops during their shape changes. The final stable state of a condensed drop on a SHS can be determined with the criterion of the drop IFE decreasing until reaching a minimum. The IFE of relatively large drops deposited on SHS has already been analyzed in several reports. The apparent CA and CA hysteresis,13-17 the states of drops and their transform as well as the energy barrier,13,18-20 and the influence of rough structure parameters13-21 were all widely investigated. However, all these studies were not aimed at condensed drops, and it is well-known that the behavior of condensed drops on a SHS is quite different from that of deposited drops. To the best of our knowledge, this is (10) Chen, C. H.; Cai, Q. J.; Tsai, C. L.; et al. Appl. Phys. Lett. 2007, 90, 173108. (11) Dorrer, C.; Ruhe, J. Adv. Mater. 2008, 20, 159. (12) Lau, K. K. S.; Bico, J.; Teo, K. B. K.; Chhowalla, M.; Amaratunga, G. A. J.; Milne, W. I.; McKinley, G. H.; Gleason, K. K. Nano Lett. 2003, 3, 1701. (13) Barbieri, L.; Wagner, E.; Hoffmann, P. Langmuir 2007, 23, 1723. (14) Yamamoto, K.; Ogata, S. J. Colloid Interface Sci. 2008, 326, 471. (15) Li, W.; Amirfazli, A. J. Colloid Interface Sci. 2005, 292, 195. (16) Li, W.; Amirfazli, A. Adv. Colloid Interface Sci. 2007, 132, 51. (17) Li, W.; Cui, X. S.; Fang, G. P. Langmuir 2010, 26, 3194. (18) Carbone, G.; Mangialardi, L. Eur. Phys. J. E 2005, 16, 67. (19) Werner, O.; Wagberg, L. Langmuir 2005, 21, 12235. (20) Patankar, N. A. Langmuir 2004, 20, 7097. (21) Patankar, N. A. Langmuir 2004, 20, 8209.
Published on Web 08/23/2010
DOI: 10.1021/la101845t
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While the free energies of the drop base include 4h Ebase ¼ Abase ð1 - f Þσlg þ Abase f 1 þ ½fn σsl þ ð1 - fn Þσlg L-l 4ðH - hÞf þ ð1 - fn þ rn - 1Þσsg þ Abase 1 - f þ rn σsg ð0eheHÞ L-l
ð3Þ
Figure 1. Schematic of a SHS with micro-nano hierarchical textured pillars.
The solid-gas IFE beyond the drop is EðAtotal - Abase Þ ¼ ðAtotal - Abase Þrrn σsg
ð4Þ
The general IFE becomes 4h cos θn σ lg Esurf ¼ Aext σ lg þ Abase ð1 - f Þσlg - Abase f 1 þ L-l þ Atotal rrn σsg Figure 2. Schematic of a composite or wetted drop on a SHS with
where
micro-nano hierarchical textured pillars: (a) composite and (b) wetted.
cos θn ¼ fn cos θ0 þ fn - 1
the first study to calculate the final states of condensed drops on a SHS by means of the IFE.
cos θ0 ¼
2. Apparent CA of Drops on a SHS with Micro-Nano Hierarchical Architecture The following parameters are defined, taking the hierarchical pillar texture shown in Figure 1 as an example: Cassie rough factor:
Equation 5 can be expressed as
f ¼
1-
l L
2
where
Aext
Wenzel rough factor: f in microtexture L-l
fn in nanotexture Ln - ln
ð1Þ
where the gas-liquid IFE of the external drop surface is Eext ¼ Aext σlg 14836 DOI: 10.1021/la101845t
ð7Þ
2=3 3V 1 ¼ π sin2 θ π 2 - 3 cos θ þ cos3 θ
ð8Þ
ð9Þ
Then eq 6 becomes
1. For Drops in the Composite State. As shown in Figure 2a, the liquid drop contacts the solid only at the tops of nanopillars that sit on all micropillar surfaces, and all other parts of the drop surface make up the gas-liquid interface. The drop penetrates the micropillars to a depth of h in this case (0 e h < H). For a given unchanged project area Atotal, within which a drop is included, the total IFE is Esurf ¼ Eext þ Ebase þ EðAtotal - Abase Þ
" # 4h cos θn þ f - 1 ¼ - σlg f 1 þ L-l
" # 3V 2=3 1 - cos θ ¼ 2π π ð2 - 3 cos θþcos3 θÞ2=3
Abase rn ¼ 1 þ 4Hn
ð6Þ
The external and base areas of a drop can be expressed with its volume and CA
ln 2 fn ¼ 1 in nanotexture Ln
r ¼ 1 þ 4H
σsg - σ sl σ lg
Esurf ¼ Aext σ lg þ Ccomp Abase þ Atotal rrn σ sg
Ccomp
in microtexture
ð5Þ
ð2Þ
Esurf ¼ π
3V π
2=3
1 ð2 - 3 cos θþcos3 θÞ2=3
½2σlg ð1 - cos θÞ
þ Ccomp sin2 θ þ σsg Atotal rrn
ð10Þ
The apparent CA corresponding to the minimum Esurf can be found after the derivative of Esurf to cos θ is found: 4h cos θn þ f - 1 cos θ ¼ f 1 þ ð11Þ L-l It is thus clear that the apparent CA equation for a composite drop on a SHS with hierarchical architecture is similar to the Langmuir 2010, 26(18), 14835–14841
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3. IFE Analysis of a Condensed Drop on a SHS during Its Shape Transformation
Figure 3. Schematic of formation of a condensed drop on SHS: (a) initial nuclei, (b) grown up and combined small drops, (c) liquid spot within pillar gaps formed by the condensate, (d) liquid spot with a flat external surface, (e) condensate with a shrinking base radius, (f) Wenzel state drop, and (g) Cassie state drop.
traditional Cassie-Box formula if only the intrinsic CA is replaced with the apparent CA on a nanotexture. 2. For Drops in the Wenzel State. As shown in Figure 2b, the drop base now penetrates the entire micropillars but not nanopillars; i.e., the drop always shows a Cassie state on nanostructure. The free energies of the drop base in this case include Ebase ¼ Abase r½ fn σsl þ ð1 - fn Þσ lg þ ð1 - fn þ rn - 1Þσ sg þ ðAtotal - Abase Þrrn σ sg
Ebase
ð13Þ
The external drop surface energy and the solid-gas IFE beyond the drop are the same as before. Therefore, the total IFE is Esurf ¼ Aext σ lg þ CWenzel Abase þ Atotal rrn σ sg
ð14Þ
CWenzel ¼ - σ lg r cos θn
ð15Þ
where
and the apparent CA can also be found after the similar derivation of composite case cos θ ¼ r cos θn
ð16Þ
It is thus also clear that the apparent CA equation for a Wenzel drop on a SHS with hierarchical architecture is similar to the traditional Wenzel formula if only the intrinsic CA is replaced with the apparent CA on nanotexture. Langmuir 2010, 26(18), 14835–14841
Esurf ¼ Aext σlg þ Ccomp Abase þ Atotal rrn σ sg
ð6Þ
Wenzel state: Esurf ¼ Aext σlg þ CWenzel Abase þ Atotal rrn σsg
ð14Þ
Condensate spot partially filled pillar gaps: ð12Þ
or ! σ sl - σ sg ¼ σlg A base r fn þ 1 - fn þ Atotal rrn σsg σ lg
The process of formation of a condensed drop on a SHS is shown in Figure 3. The initial liquid nuclei are formed on the surface with a nanometer size (a). The nuclei grow and combine into micrometer-sized drops (b). These small drops continue to grow and combine to fill the microtextured gaps, and a liquid spot with a relatively flat external surface will form (c and d).1,2 The IFE of this shaped condensate, however, is high; the condensate thus will spontaneously change its shape to reduce the IFE and finally become a Wenzel or Cassie state (e and f). One way for a condensed spot shifting its shape is to reduce its base radius. If the IFE of the drop becomes minimal when its base radius decreases to a certain value, the drop state at this moment should be its final stable Wenzel state. On the other hand, if its IFE continuously declines while its base radius decreases until reaching a Cassie state, the final state of the drop is Cassie. The other way for the condensate to alter its state is to move its base upward while the base radius remains the same. However, our calculation and that of Patankar20 all illustrate that the IFE would increase suddenly when a drop departs from its bottom, and this energy barrier will hinder the drop from spontaneously transferring to the Cassie state with this path. Therefore, we calculated the IFE transformation only for a condensed drop while its base radius was shrinking. The formulas for the IFE of condensed drops at different states are Cassie state:
4ðH - hÞf ½ fn σsl þ ð1 - fn Þσ lg Esurf ¼ Abase 1 - f þ L-l 4h rn σ sg - rrn σsg þ ð1 - fn þ rn - 1Þσ sg þ ð1 - f Þσ lg þ f 1 þ L-l 4ðH - hÞf cos θn σ lg þ Atotal rrn σ sg ¼ Abase 1 - f - 1 - f þ L-l
þ Atotal rrn σsg
ð17Þ
The 4/(L - l) term in these formulas is the ratio between the perimeter and sectional area of a micropillar. For other patterned geometric pillars, these formulas are still valid if the corresponding ratio and f, r, etc., are used. Typical IFE changing curves are shown in Figure 4 for a condensed drop on a SHS during its shape transformation (IFE differences are calculated here on the basis of the initial IFE of a condensate spot). Because the initial condensate spot calculated here is in such a state that only 80% of the pillar height is filled with condensate, the IFE decreases first until the condensate spot immerses the top surface of pillars, at which point the IFE will increase suddenly. However, a condensate spot does not need to overcome this energy barrier because condensation continuously takes place and more condensate will form and completely fill the gaps between pillars during condensation. Therefore, we need to DOI: 10.1021/la101845t
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Figure 4. IFE changes of a condensed drop during its shape transformation. L = 16 μm. Ln = 16 nm. V = 1 nL. θ0 = 110°.
consider only the IFE trend after the condensate fills the whole space between the pillars. We can see that the IFE of a condensed drop begins to decrease after the condensate spot completely fills the local gaps between pillars. In many cases, with certain textured parameters, the IFE of a drop will stop declining when its base radius is reduced to a certain value, while in some other cases, the IFE may continuously decline until it reaches the Cassie state with the lowest energy. Generally, the IFE of a condensed drop is influenced greatly by the micro-nano textures of a SHS. The smaller the pitch between micropillars, the more obvious the energy variation of the drop, so that the drop will reach the Cassie state more easily. Furthermore, the IFE of a condensed drop decreases with the increase in the pitch between nanopillars, and the condensate drop will also transform into the Cassie state more easily. The IFE curves for other textured parameters and drop volumes are similar to those in Figure 4.
4. Analysis of the Final State of a Condensed Drop on a SHS The final stable state of a condensed drop on a SHS can be determined according to its IFE curve. The drop will appear in the Cassie state if its IFE continuously decreases while its base radius is reduced. Otherwise, if the slope of the IFE curve becomes zero when the base radius shrink to a certain value, the condensed drop will be in the Wenzel state with the corresponding base radius, and the apparent CA is also determined when the drop state is fixed. The method we applied to analyze and determine the final state of condensed drops on a SHS was briefly described here. For each IFE curve changing similarly as shown in Figure 4, we divided the drop base radius into 500 equal portions and calculated the slopes of the curve at these positions along the direction of drop radius reduction after the condensate spot fills the local gaps between pillars. As the drop base radius decreases from the initial value to zero, the slopes of the curve may continuously decrease without an extremum existing. In this case, we considered the final state of the drop to be Cassie. Alternatively, the slopes of the curve may first decrease and then increase with an extremum presenting. Then the final state of the drop was taken to be Wenzel with the base radius corresponding to the minimum energy. The calculated results of final state and corresponding CA of condensed drops on a SHS with different micro-nano textured parameters are shown in Figure 5 (A1-A3 and B1-B3). One can see that condensed drops can reach the Cassie state on the SHS 14838 DOI: 10.1021/la101845t
with a relatively smaller l/L or a higher H when ln is zero (no nanostructure), but the apparent CAs in these cases are all smaller than 150°. Therefore, the drops in these cases are not superhydrophobic, although they appear as the Cassie state. However, the condensed drops will more easily reach the Cassie state with a larger CA as ln increases or as the hierarchical structure becomes more obvious. To illustrate the condensed drops with the superhydrophobic Cassie state, we further calculated the textured parameters that are needed for a condensed drop to appear with both the Cassie state and a CA larger than 150°. The results are shown in Figure 5 (C1-C3). It is thus clear that the superhydrophobic Cassie drops can form only on the SHS with clearly hierarchical textures. Without nanostructure, a condensed drop will seldom display the superhydrophobic Cassie state. Another interesting result that can be found in Figure 5 is that a condensed drop will always appear in the Wenzel state with a small apparent CA when l/L is large, no matter the surface with or without nano hierarchical structures. Especially when ln/Ln is small, the CA can even be smaller than the intrinsic CA, which should not be true according to the Wenzel equation for the case of an intrinsic CA larger than 90°. Our calculation thus indicates that the final apparent CA of a condensed drop may be quite different from and much smaller than that from the Wenzel equation. It is clear that the water volume within the spaces between pillars cannot be ignored for the very small condensed drop in nanoliters. Moreover, the condensed drop states and apparent CAs in the literature were calculated and compared with experimental observations, as shown in Table 1. One can see that the calculated condensed drop states are all in agreement with those of the experiments. First, the condensation experiments on a SHS with hierarchical texture or with only nanostructure showed that the condensed drops on these surfaces displayed the Cassie state and easily rolled,10-12 while our calculation also indicated that the condensed drops on these surfaces simply form Cassie drops with CA values of >150°. Besides, both Wenzel and Cassie drops were observed on the same surface in the literature,6,7 and our calculation showed that smaller drops can be in the Wenzel state while larger drops are in the Cassie state within the drop size range observed in the literature; our calculation also shows the possibility of the coexistence of the two types of drops. Furthermore, all condensed drops appeared in the Wenzel state in the literature1-3 for all sized drops, and our calculation also gave the same results. Langmuir 2010, 26(18), 14835–14841
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Figure 5. Effect of micro-nano textures of a SHS on final states of condensed drops. V = 1 nL. θ0 = 110°. (A1-A3) Drop states when H is 10, 20, and 40 μm, respectively, above for Cassie, below for Wenzel. (B1-B3) Apparent CA when H is 10, 20, and 40 μm, respectively. (C1-C3) Drop states when H is 10, 20, and 40 μm, respectively, above for Cassie with CA g 150°, below for Wenzel or Cassie with CA < 150°.
As for CA, our calculations do not agree with the experimental measurements for two cases in the literature. For one case, the CAs of relatively larger drops in microliters measured by Chen7 are much smaller than those of our calculations or those from the Wenzel formula. We all know that liquid volume within pillar space can be ignored when a drop size is relatively larger, and the Wenzel formula is valid in this case. Moreover, the apparent CA in this case should not be lower than the intrinsic CA if the intrinsic CA is greater than 90°. However, the measured apparent CAs are all smaller than their given intrinsic CA7 (117.3°), which is difficult to explain unless the hydrophobic coating peeled off. The second case is that in which the CAs reported by Jung et al.3 are all obviously higher than those of our calculation or those from the Wenzel formula. After detailed observation, we found that their condensed drops were all so small that only a few pillars were covered by a drop. Some of these small drops were located on pillars with larger CA values, while others sit between pillars with comparatively smaller CA values. In fact, the CA of very small drops will be very sensitive to their locations on the SHS if a drop is so small that only a few pillars are covered by the drop. In this case, solid fraction f and other parameters under a drop will not be constant but will depend on the positions in which the drop is located. Our model is valid only when the pillar number beneath a drop is greater Langmuir 2010, 26(18), 14835–14841
than 50 to make the fluctuation of f under the drop less than 1%. It is apparent that the drop volumes can affect the CAs of Wenzel drops according to our calculations. The drop volumes affect the CA only when a Wenzel drop is so small that the liquid volume within the rough texture of the SHS cannot be omitted. Condensed drops indeed are very small, so their CAs can be obviously influenced by their volumes. In this case, the Wenzel formula is no longer valid, and the larger the volume ratio of a drop in rough structure, the lower the CA. For a common sessile drop, the volume ratio in the rough texture can be omitted; therefore, its CA is not affected by its volume and can be predicted by the Wenzel formula. This model does not consider the tiny fluctuation in IFE when the contact line passes micropillars during the transformation of the shape of a drop.14-17 The calculation of Yamamoto et al.14 shows that this energy fluctuation is on the order of e10-10 J, while the IFE of a drop in our calculation is on the order of 10-8 J. Therefore, the energy fluctuation when the contact line passes micropillars can be ignored because this tiny energy barrier can be overcome by an extremely small vibration energy or the inertia of drop shape transformation. Furthermore, this tiny energy barrier will become even smaller when hierarchical texture exists. In addition, our calculation also does not consider the drop DOI: 10.1021/la101845t
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3
3
2
1
6
6
Drop volumes were determined according to the sizes of condensed drops observed in the literature.
Wenzel 126 Wenzel H = 30 μm, d = 14 μm, p = 26 μm, Hn = 0, dn = 0, pn = 0, θ0 = 98°
a
Wenzel 129 H = 10 μm, d = 5 μm, p = 12.5 μm, Hn = 0, dn = 0, pn = 0, θ0 = 98°
Wenzel
Wenzel umeasured H = 62 μm, l = 32 μm, L = 64 μm, Hn = 0, ln = 0, Ln = 0, θ0 = 90°
Wenzel
Wenzel 24-120 H = 40 μm, l = 24 μm, L = 32 μm, Hn = 0, ln = 0, Ln = 0, θ0 = 110°
Wenzel
Wenzel (0.1 nL) and Cassie (1 nL) unmeasured Wenzel and Cassie H = 40 μm, l = 8 μm, L = 12 μm, Hn = 0, ln = 0, Ln = 0, θ0 = 118°
unmeasured
14840 DOI: 10.1021/la101845t
H = 40 μm, l = 8 μm, L = 16 μm, Hn = 0, ln = 0, Ln = 0, θ0 = 118°
Wenzel and Cassie
Wenzel (0.1 nL) and Cassie (1 nL)
164.0 (0.1-1 nL) 94.0-100.9 (0.1-1 nL) 168.8 (0.1-1 nL) 84.3-98.6 (0.1-1 nL) 179.5 157.4 143.2 (1 nL) 141.4 (10 nL) 108.9 (0.1 nL) 150.2 (1 nL) 77.7 (0.1 nL) 160.2 (1 nL) 35.2 (0.1 nL) 102.6 (10 nL) 37.6 (1 nL) 61.1 (10 nL) 90.4 (1 nL) 101.3 (10 nL) 77.1 (1 nL) 94.0 (10 nL) unmeasured unmeasured unmeasured unmeasured 179.4 160-170 88.1-115.9 Cassie Wenzel Cassie Wenzel Cassie Cassie Wenzel and Cassie H = 5.2 μm, l = 6.3 μm, L = 11.2 μm, Hn = 0.4 μm, ln = 0.06 μm, Ln = 0.12 μm, θ0 = 101° H = 5.2 μm, l = 6.3 μm, L = 11.2 μm, Hn = 0, ln = 0, Ln = 0, θ0 = 101° H = 8.0 μm, l = 8.3 μm, L = 12 μm, Hn = 0.4 μm, ln = 0.06 μm, Ln = 0.12 μm, θ0 = 101° H = 8.0 μm, l = 8.3 μm, L = 12 μm, Hn = 0, ln = 0, Ln = 0, θ0 = 101° only with nanotexture, fn = 0.01%, θ0 = 117° only with nanotexture, fn = 11%, θ0 = 108° H = 45.2 μm, l = 13.6 μm, L = 37.3 μm, Hn = 0, ln = 0, Ln = 0, θ0 = 117.3°
Cassie Wenzel Cassie Wenzel Cassie Cassie Wenzel (1 nL) and Cassie (10 nL)
CA (deg) Cassie or Wenzel CA (deg) Cassie or Wenzel textured parameters
drop state calculated in this paper drop state observed in experiment
Table 1. Comparison of Calculated Condensed Drop States on a SHS with Experimental Resultsa
10 10 10 10 11 12 7
Liu et al. ref
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gravitational potential energy because of the especially small condensed drops. Moreover, our calculation indicates that condensed drops can spontaneously change to the Cassie state from the Wenzel state on the SHS with suitable micro-nano structures. Zheng et al.22 discovered that condensed Wenzel drops on a lotus leaf could migrate from the valley of the papilla upward to reach the Cassie state, and they proposed a “wettability gradient” along the exterior surface of the papilla driving this migration. However, it has not been proven that there is such a dampening gradient on the surface of the papilla. On the other hand, our calculation shows that the hierarchical textures on a lotus leaf can make a Wenzel drop spontaneously transform into the Cassie state without needing a dampening gradient. Finally, we describe an ideal dropwise condensation process on a SHS as follows. Many initial nuclei form on the microroughness structural surfaces everywhere, and the growing nuclei will combine until the condensate fills partial areas of SHS to form many condensate spots, each with a volume between 0.1 and 1 nL. The condensate spots then begin to reduce their base radii and change their shapes first into the Wenzel and then the Cassie state. Finally, the Cassie drops will depart from the SHS with a very small rolling angle. This process takes place and is repeated very rapidly; the condensation heat transfer is thus enhanced greatly. To realize condensed drops in the Cassie state on the SHS, Varanasi et al.23 applied a novel method with the top surface of pillars being relatively wettable and the side and base surfaces of pillars being hydrophobic, so that condensed drops formed only on the top surfaces of pillars. This method, however, merely used the partial area of the SHS as a condensation surface; i.e., the full surfaces of the SHS were not fully used, which can impact the condensation heat transfer rate.
5. Conclusions (1)
(2)
(3)
(4)
On a SHS with micro-nano hierarchical textures, the apparent CA of relatively larger drops still obeys the Cassie or Wenzel formula, if the intrinsic CA in these formulas is replaced with the apparent CA on a nanotextured surface. The initial small condensed drops on the SHS first combine until filling the space between pillars, and the condensate then will change its shape along the direction of the shrinking base radius. When the IFE of a condensed drop no longer decreases, the corresponding shape is the final stable state for the condensed drop, which may be Wenzel or Cassie, depending on the micro-nano architecture of the SHS. Condensed drops on a SHS only with microroughness texture rarely appear in the Cassie state with an apparent CA of >150°, while on a SHS with suitable micro-nano hierarchical textures, condensed drops can easily reach a superhydrophobic state. Because of the very small size of condensed drops, the liquid volume of a Wenzel drop within the roughness structure can no longer be ignored. Therefore, the apparent CA of a small condensed drop in the Wenzel state no longer obeys the Wenzel equation.
(22) Zheng, Y. M.; Han, D.; Zhai, J.; Jiang, L. Appl. Phys. Lett. 2007, 92, 084106. (23) Varanasi, K. K.; Hsu, M.; Bhate, N.; Yang, W. S.; Deng, T. Appl. Phys. Lett. 2009, 95, 094101.
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(5) The final state, apparent CA, rolling angle, and departure of a condensed drop on SHS are considerably influenced by the micro-nano textures, which should be properly designed so that an ideal dropwise condensation process can be realized.
Nomenclature Abase projected drop base surface area, mm2 Aext external drop surface area, mm2 Atotal selected total solid projected surface area, mm2 Ccomp composite state coefficient CWenzel Wenzel state coefficient d sectional diameter of a micro-cylindrical pillar, μm dn sectional diameter of a nano-cylindrical pillar, nm Ebase interface free energy beneath a drop, J/mm2 Eext gas-liquid interface free energy of an external drop surface, J Esurf general interface free energy of a selected system, J E(Atotal-Abase) solid-gas interface free energy beyond a drop, J f Cassie roughness factor for microtexture fn Cassie roughness factor for nanotexture
Langmuir 2010, 26(18), 14835–14841
h penetration depth of a composite drop along a pillar height, μm H micropillar height, μm Hn nanopillar height, nm l distance between two micropillars, μm ln distance between two nanopillars, nm L pitch of micro-cuboid pillars, μm Ln pitch of nano-cuboid pillars, nm p pitch of micro-cylindrical pillars, μm pn pitch of nano-cylindrical pillars, nm r Wenzel roughness factor for microtexture rn Wenzel roughness factor for nanotexture V drop volume, nL θ apparent CA of a drop on SHS, deg θ0 intrinsic CA of a drop on flat surface, deg θn apparent CA of a drop on nanotextured surface, deg σlg gas-liquid interface tension, N/m σsg solid-gas interface tension, N/m σsl solid-liquid interface tension, N/m Acknowledgment. We acknowledge financial support from the National Natural Science Foundation of China (Grant 50876015).
DOI: 10.1021/la101845t
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