Article pubs.acs.org/Langmuir
Theoretical Exploration of Barrel-Shaped Drops on Cactus Spines Cheng Luo* Department of Mechanical and Aerospace Engineering, University of Texas at Arlington 500 West First Street, Woolf Hall 226, Arlington, Texas 76019, United States ABSTRACT: To survive an arid environment, desert cacti are capable of harvesting water from fog by transporting condensed water drops using their spines. Cactus spines have a conical shape. In this work, on the basis of the difference of liquid pressure, a new theoretical model has been developed for a barrel-shaped liquid drop on a conical wire. This model is further simplified to interpret the effects of contact angles, conical angle, surface microgrooves, and gravity on the drop movement along a cactus spine.
1. INTRODUCTION Cacti are capable of harvesting water from fog by transporting condensed water drops using their spines.1,2 This may be part of the reason why they can survive an arid environment, such as deserts. A cactus spine can be considered as a thin conical wire, which has a small conical angle. Its side surface is covered by microgrooves.1,3 These specific structures of the cactus spine should affect the drop motions. In addition, a water drop may overcome gravity and move upward on a cactus spine.1−3 To understand the effects of conical angle, surface microgrooves, contact angles, and gravity on the transport of a drop along a cactus spine, it is necessary to know the corresponding driving mechanism. On a conical wire, such as a cactus spine, liquid drops may exhibit two basic shapes: barrel and clamshell. The barrelshaped drop is axisymmetric with respect to the central axis of the wire, while the clamshell type has a more complicated profile, which is asymmetric. Our focus here is on the behavior of barrel-shaped drops. The driving mechanism of a barrel-shaped drop on a conical wire has been previously explored in ref 4 when the apparent contact angle is 0°. The corresponding model is not applicable to the case of cactus spines, which have non-zero contact angles.1−3 Thus, in this work, to explore the drop movement on a cactus spine, we develop a new theoretical model, in which the apparent contact angle is not limited to be 0°. The outline of this paper is as follows. The difference of liquid pressure is first formulated for a general conical wire in section 2 and then simplified in section 3 for the case of cactus spines. In sections 4 and 5, the simplified relations are further applied to cactus spines. Finally, in section 6, this work is summarized and concluded.
Figure 1. Cross-sectional schematics of liquid drops that are (a) at rest and (b) at the moment to move.
solid/liquid) contact points at the side surface of the wire. Set c to be a representative point on ab. Let o denote the edge of the wire tip. Set θa and θb to denote equilibrium contact angles at a and b, respectively. When the wire surface is smooth, θa and θb are intrinsic contact angles. Otherwise, they are apparent contact angles. Take a water drop as an example. When θa and θb are greater than 90°, it means that the wire surface is hydrophobic. If they are smaller than 90°, then the surface is hydrophilic. Let θadv and θrec stand for advancing and receding contact angles, respectively. (θadv − θrec) is the so-called contact angle hysteresis. Then, both θa and θb vary between θrec and θadv. A triple contact line is pinned on a surface until the corresponding contact angle increases to θadv or decreases to θrec. Use α to stand for conical angle of the wire.
2. LIQUID PRESSURE DIFFERENCE INSIDE A DROP ON A CONICAL WIRE Consider a barrel-shaped drop on a conical wire. The wire surface is not limited to be smooth and wetting. It may be, for example, rough and non-wetting. Let ab represent one of the meridian curves of the drop (Figure 1a). a and b are triple (air/ © XXXX American Chemical Society
Received: July 20, 2015 Revised: October 16, 2015
A
DOI: 10.1021/acs.langmuir.5b03600 Langmuir XXXX, XXX, XXX−XXX
Article
Langmuir Assume that the maximum thickness of the drop is less than the capillary length of the liquid (2.7 mm for water). Hence, the gravity effect on liquid pressure along the direction perpendicular to the central axis of the wire is neglected. Subsequently, by the Young−Laplace equation,5 we have
pw − pa = 2γB 1 1 + = 2B R1 R2
rb = (lp + ll)sin
α θ2 = − θb 2
(1a)
α 2
)
α
(
(4c)
)
cos θb − 2 1 = α R 2b (lp + ll)sin 2
(1b)
(4d)
Set pwa and pwb to be liquid pressures at a and b, respectively. With the aid of eqs 1b−4d, by eq 1a we obtain α ⎡ cos θa + 2 ⎢ 1 pwa = γ ⎢ + α R lp sin 2 ⎣ 1a
(
) ⎤⎥ + p ⎥ ⎦
α ⎡ cos θb − 2 ⎢ 1 pwb = γ ⎢ + α R (lp + ll)sin 2 ⎣ 1b
a
(5a)
) ⎤⎥ + p
(
⎥ ⎦
a
(5b)
Let Δp = pwa − pwb. Then, it follows from eqs 5a and 5b that ⎡ ⎛ 1 γ ⎢ cos θa + 1 ⎞ Δp = γ ⎜ − ⎟+ α R1b ⎠ sin 2 ⎢ lp ⎝ R1a ⎣
(
(2a)
−
(2b)
(
cos θb − l p + ll
α 2
α 2
)
) ⎤⎥ ⎥ ⎦
(6)
3. SIMPLIFIED RELATIONS FOR CACTUS SPINES Two points have been observed from the existing tests on cactus spines:1−3 (i) small α and (ii) small spine radius in comparison to ll. The cactus spines examined in ref 1 have lengths from 0.8 to 2.5 mm, and their lengths vary from 0.7 to 2.4 mm in ref 2. Also, the values of α in the tested cactus spines varied from 8° to 14°.1−3 Hence, in these reported cases, the largest possible radius of a spine, which is the radius of the root, is 0.15 mm. On the other hand, the barrel-shaped drops that were observed to transport on the spines have the lengths of around 0.5 mm (see panels b and e of Figure 2 in ref 1) and 1 mm (Figure 2 of ref 3). This difference in the dimensions implies that ll is generally much larger than the radius of the cactus spine. These two points enable us to simplify theoretical results for cactus spines as well as for other conical wires, which have these two properties, such as the wires examined in ref 4. On the basis of the first point, by eqs 4a−4d, we can approximate 1/R2a and 1/R2b as
(3)
Let ra and rb denote the radii of the wire at a and b, respectively. They actually equal the y coordinates at these two points, respectively. Use ll to stand for the distance between a and b. Let lp represent that between o and a. Then, we have
ra = lpsin
α
(
A theoretical solution to eq 1b has been found by Carrol for a barrel-shaped drop on a thin cylinder, which has identical cross-sections with diameters in the order of 100 μm.6 The corresponding drop has a size smaller than the liquid capillary length. Accordingly, in his derivation, the gravity effect was neglected. The solution by Carrol has been modified to consider Gibbs free energy of liquid drops on conical wires.4,7 The solution has also been extended to the case of a conical wire to derive the gradient of Laplace pressure, when the drop thickness is much larger than or close to the wire radius.4 However, when the drop thickness is comparable to the wire radius, the extension may not be straightforward, because this thickness has to be numerically determined in the solution of Carrol. Meanwhile, equilibrium shapes and positions of barrelshaped drops on conical wires were directly determined by Hanumanthu and Stebe through numerical calculation.8 Here, we use a different approach to consider the difference of the liquid pressure, which yields simple relations for judging, for example, the moving direction of a barrel-shaped drop on a cactus spine. For an axisymmetric shape, at c, it is known that6,9 cos θt 1 = R2 y
(4b)
cos θa + 2 1 = α R 2a lp sin 2
where pw and pa denote liquid pressure and air pressure at a point of ab, respectively, B represents mean curvature at this point, γ denotes the surface tension of the liquid, R1 is the radius of the curvature of ab at this point, and R2 is the radius of the curvature of the curve that is perpendicular to ab at the same point on the drop surface. (pw − pa) is the Laplace pressure. Set R1a and R1b to be the values of R1 at a and b, respectively, and let R2a and R2b denote the values of R2 at a and b, respectively. Set up an x−y rectangular coordinate system. x and y axes are along the central axis of the wire and the direction perpendicular to the central axis, respectively, and the origin is located at o (Figure 1). Let θt denote the angle formed by the tangent to ab and the x axis at c. Set θ1 and θ2 to be the values of θt at a and b, respectively. By geometric analysis, we obtain α θ1 = θa + 2
α 2
cos θa 1 = R 2a ra
(7a)
cos θb 1 = R 2b rb
(7b)
which indicate that 1/R2a and 1/R2b have the orders of 1/ra and 1/rb, respectively. With the aid of these two relations, by eq 6, Δp can also be written as
(4a) B
DOI: 10.1021/acs.langmuir.5b03600 Langmuir XXXX, XXX, XXX−XXX
Article
Langmuir ⎛ 1 ⎛ cos θa cos θb ⎞ 1 ⎞ Δp = γ ⎜ − − ⎟ ⎟ + γ⎜ R1b ⎠ rb ⎠ ⎝ R1a ⎝ ra
conditions for a drop to move on a cactus spine from its tip to the root. By eqs 4a, 4b, and 10, Δp can be rewritten in terms of ll and lp as
(8)
According to the second point, 1/ra and 1/rb are much larger than 1/ll. Next, we desire to estimate the orders of 1/R1a and 1/ R1b to simplify eq 8. As commented in ref 10, although the middle portion of ab should bend toward air, it is possible that ab bends toward liquid in the neighborhoods of a and b. In other words, 1/R1a and 1/R1b may be negative. However, according to reported experimental results,1 the entire ab appears to bend toward air in the case of cactus spines, which implies that 1/R1 is always positive. When ab bends toward air, two situations were considered in ref 8. In the first situation, θrec and θadv were equal and ab was assumed to be a circular arc. On the basis of this assumption, we have ll R1a = R1b = 2 sin θ
Δp =
cos θrec cos θadv > lp l p + ll
(13)
(9)
Exploration of the inequality in eq 13 leads to the following claim: given that
ll > > l p
(14)
only if θrec
0. Therefore, by the inequality in eq 15, the inequality in eq 13 is satisfied, which means that (Δp)cri > 0. In case the inequality in eq 15 is violated, i.e., θrec > π/2, we obtain (Δp)cri < 0, which implies that the drop does not move toward the root in this case. The above claim shows that only lyophilic drops may move on a cactus spine from its tip toward the root. Given that a water drop may move from the tip to the root of a cactus spine,1−3 this result indicates that the spine surface should be hydrophilic, instead of being hydrophobic. According to Figure 2 of ref 1, which is the in situ observation of water transport on a cactus spine from the tip to the root, the advancing contact angles of water drops are less than 90°. Also, as judged from Figure 2 of ref 3, which gives the movement of a drop of an xanthan solution on a cactus spine from the tip to the root, the advancing contact angle is less than 45°. The xanthan solution was made by mixing deionized water and xanthan gum particles. These two results validate our theoretical predication that a cactus spine surface should be hydrophilic.
(10)
which implies that, on a thin conical wire, the difference in the liquid pressure is mainly related to the change in the wire radius. We may approximate the pressure gradient as (Δp/ll). When θrec = θadv = 0°, which means that θa = θb = 0°, by eq 10, we obtain
Δp γα = ll 2rarb
(12)
Let (Δp)cri denote the critical value of Δp for given α, lp, and ll at the moment that a drop begins to move toward the root. At this moment, θa and θb are changed to θrec and θadv, respectively (Figure 1b). Also, we should have (Δp)cri > 0. To make this inequality hold true, by eq 12, the following inequality should be satisfied:
where θ denotes the values of θrec and θadv. These relations indicate that 1/R1a and 1/R1b have the same order as 1/ll. In the second situation, θrec ≠ θadv, which is actually true in the case of cactus spines. In this situation, the aforementioned assumption may not hold. However, although ab may not be approximated as a circular arc, it is still reasonable to consider that 1/R1a and 1/R1b are in the order of 1/ll, which is much smaller than 1/ra and 1/rb. Consequently, by eq 8, we approximate Δp by ⎛ cos θa cos θb ⎞ Δp = γ ⎜ − ⎟ rb ⎠ ⎝ ra
cos θb ⎞ 2γ ⎛ cos θa ⎜⎜ ⎟ − α ⎝ lp lp + ll ⎟⎠
(11)
Assume that a drop has a small volume such that ll is small. Because rb = ra + llα/2 for small α, this assumption means that rb approximately equals ra. Hence, by eq 11, we have Δp/ll = γα/2ra2). It agrees with eq 6 of ref 4, which was derived essentially on the basis of the same assumption.
4. CONDITIONS FOR DROPS TO MOVE ON A CACTUS SPINE A barrel-shaped drop was seen to move on a conical wire from its tip (i.e., the narrow end) to the root (i.e., the wide end) when the apparent contact angle was 0°.4 Such motions were also observed on a cactus spine, which had a conical shape. The corresponding contact angles were found to be 113.3° and 126.5° on the base and top of the cactus spine, respectively (see Supplementary Figure S6 of ref 1). These two results indicate that lyophilic and lyophobic drops may move along the same direction on a conical wire. On the other hand, it is known that these two types of drops may move along opposite directions, for example, inside a conical capillary11 or between two nonparallel plates.12,13 Hence, we desire to find out why the situation is different in the case of a conical wire. For this purpose, in this section, we determine the corresponding
5. EFFECTS OF SURFACE MICROGROOVES, CONICAL ANGLES, AND GRAVITY ON THE DROP MOVEMENT ON A CACTUS SPINE 5.1. Effects of Surface Microgrooves and Conical Angles. As demonstrated in section 4, the cactus spine surface should be hydrophilic. Thus, the wetting is considered to be in a Wenzel state on the spine surface that is covered by microgrooves. In the Wenzel state, a liquid drop completely fills the roughness grooves underneath this drop. Let θbef and θaft denote the contact angles on a wire surface before and after the C
DOI: 10.1021/acs.langmuir.5b03600 Langmuir XXXX, XXX, XXX−XXX
Article
Langmuir
cos θB ⎞ 2l 2 cos θA 2l 2 cos θA 2l 2 ⎛⎜ cos θA ⎟ − > = lp + ll ⎟⎠ αll ⎜⎝ lp αlp(lp + ll) αl p l p + 3 6v
incorporation of any roughness structures. Then, they are related by the Wenzel equation14 cos θaft = r cos θ bef
(
(16)
cos θb ⎞ 2rγ ⎛ cos θa ⎟ ⎜⎜ − α ⎝ lp lp + ll ⎟⎠
(
αltot ltot +
3
6v π
)
(19)
The relations in eq 19 imply that if 2l 2 cos θA
(
αltot ltot +
3
6v π
)
>1 (20)
then the inequality in eq 18 is satisfied. When we set l to be the water capillary length of 2.7 mm, α to be the maximum reported value of 14°, ltot to be the maximum reported cactus spine length of 2.5 mm, and θA to be 60°, which is estimated from Figure 2e of ref 1, by the inequality in eq 20, we should have v < 440 μL. It is much larger than the volumes of the drops that were reported to move along cactus spines,1−3 indicating that those drops can transport on cactus spines even if the cactus spines are vertically oriented.
(17)
Because of the enhanced wettability on the spine surface, the drop is elongated along the axial direction of the spine after the incorporation of the microgrooves; that is, ll is increased, which, by eq 17, results in the increase of Δp. Furthermore, because r > 1, this equation indicates that the incorporation of microgrooves at least increases the pressure difference by r times. Accordingly, a large value of r is preferred to make a water drop transport faster and also to overcome the gravity effect when the wire is inclined. In the case of cactus spines, the bottoms of microgrooves are structured with sub-microgrooves,1,3 which further increase the value of r. In addition, eq 17 indicates that Δp is inversely proportional to α and that a small value of α results in a large value of Δp in the case of a cactus spine. 5.2. Effect of Gravity. It was noted that a water drop could still be collected even if a cactus spine was oriented along the vertical direction with its tip located at the lower position,1 implying that all of the cactus spines have the capability of collecting water regardless of their orientations. This phenomenon is interpreted below. Let β denote the maximum tilt angle of the wire that the water drop can move upward from the tip to the root of a cactus spine. Then, − 90° ≤ β ≤ 90°. When a cactus spine is inclined, the gravity effect may have to be considered. Along the axial direction of the cactus spine, the gradient of gravitational pressure is ρg sin β. If ρg sin β is less than Δp/ll, then the corresponding drop moves toward the root. When β = 90°, which is the maximum tilt angle that a cactus spine could have, to make water drops move upward, by eqs 16 and 17, we should have cos θB ⎞ 2l 2 ⎛⎜ cos θA ⎟>1 − ⎜ αl l ⎝ l p lp + ll ⎟⎠
)
2l cos θA
>
where r denotes the ratio of the actual surface area of the rough surface to the projected surface area. This equation indicates that, for given θbef, which is less than 90°, θaft decreases with both the increase in r and the decrease in θbef. Consider θa and θb in eq 10 to be the angles on a cactus spine without the incorporation of any rough structures. In addition, because α is small, sin α/2 may be approximated as α/2. Then, after the incorporation of the microgrooves, by eq 12, corresponding Δp can be rewritten in terms of ll and lp as Δp =
π
2
6. SUMMARY AND CONCLUSION In this work, we first derived an expression of pressure difference and then determined conditions for a barrel-shaped liquid drop to move on a conical wire. According to the derived relations, different from the previous understanding, only a lyophilic drop may move on a cactus spine from its tip to the root. We further demonstrated three points: (i) the pressure difference linearly increases with the inverse of the conical angle, (ii) the incorporation of microgrooves at least enhances the pressure difference by r times, and (iii) water drops with volumes less than 440 μL should be able to move along a cactus spine even if the cactus spine is vertically oriented. Our results provide some new understandings about the drop movement on a cactus spine during the process of collecting water from fog, which may be useful in developing artificial fog collectors.2,15
■
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
■
REFERENCES
(1) Ju, J.; Bai, H.; Zheng, Y.; Zhao, T.; Fang, R.; Jiang, L. A multistructural and multi-functional integrated fog collection system in cactus. Nat. Commun. 2012, 3, 1247. (2) Heng, X.; Xiang, M. M.; Lu, Z. H.; Luo, C. Branched ZnO wire structures for water collection inspired by Cacti. ACS Appl. Mater. Interfaces 2014, 6, 8032. (3) Guo, L.; Tang, G. H. Experimental study on directional motion of a single droplet on cactus spines. Int. J. Heat Mass Transfer 2015, 84, 198. (4) Lorenceau, É.; Quéré, D. Drops on a conical wire. J. Fluid Mech. 1999, 510, 29. (5) Adamson, A. V. Physical Chemistry of Surfaces; Wiley: New York, 1990. (6) Carroll, B. J. The accurate measurement of contact angle, phase contact areas, drop volume, and Laplace excess pressure in drop on fiber system. J. Colloid Interface Sci. 1976, 57, 488.
(18)
where l represents liquid capillary length and θA and θB denote apparent contact angles at a and b on a microgroove-covered cactus spine. Let ltot denote the length of a cactus spine. Set v to be an upper limit of the drop volume. A possible value of v is obtained when the drop is considered to be a sphere with ll as the diameter. In addition, note that, when the drop is moving, θA ≤ θB. Accordingly, we have D
DOI: 10.1021/acs.langmuir.5b03600 Langmuir XXXX, XXX, XXX−XXX
Article
Langmuir (7) Michielsen, S.; Zhang, J. L.; Du, J. M.; Lee, H. J. Gibbs free energy of liquid drops on conical fibers. Langmuir 2011, 27, 11867. (8) Hanumanthu, R.; Stebe, K. J. Equilibrium shapes and locations of axisymmetric, liquid drops on conical, solid surfaces. Colloids Surf., A 2006, 282−283, 227. (9) Extrand, C. W.; Moon, S. I. Contact angles of liquid drops on super hydrophobic surfaces: Understanding the role of flattening of drops by gravity. Langmuir 2010, 26, 17090. (10) Carroll, B. J. Equilibrium Conformations of Liquid Drops on Thin Cylinders under Forces of Capillarity. A Theory for the Roll-up Process. Langmuir 1986, 2, 248. (11) Luo, C.; Heng, X.; Xiang, M. Behavior of a Liquid Drop Between Two Nonparallel Plates. Langmuir 2014, 30, 8373. (12) Luo, C.; Heng, X. Separation of Oil from a Water/Oil Mixed Drop Using Two Nonparallel Plates. Langmuir 2014, 30, 10002. (13) Tsori, Y. Discontinuous Liquid Rise in Capillaries with Varying Cross-Sections. Langmuir 2006, 22, 8860. (14) Wenzel, R. N. Resistance of solid surfaces to wetting by water. Ind. Eng. Chem. 1936, 28, 988. (15) Ju, J.; Xiao, K.; Yao, X.; Bai, H.; Jiang, L. Bioinspired Conical Copper Wire with Gradient Wettability for Continuous and Efficient Fog Collection. Adv. Mater. 2013, 25, 5937.
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DOI: 10.1021/acs.langmuir.5b03600 Langmuir XXXX, XXX, XXX−XXX
