Langmuir 2008, 24, 11651-11660
11651
Droplet Motion on Designed Microtextured Superhydrophobic Surfaces with Tunable Wettability Guoping Fang,† Wen Li,*,‡ Xiufeng Wang,‡ and Guanjun Qiao*,† State Key Laboratory for Mechanical BehaVior of Materials, Xi’an Jiaotong UniVersity, Xi’an 710049, PR China, The Key Laboratory of Terrain-Machine Bionics Engineering, Ministry of Education, Jilin UniVersity (Nanling Campus), 5988 Renmin Street, Changchun 130025, PR China, and Department of Mechanical Engineering, UniVersity of Alberta, Edmonton AB T6G 2G8, Canada ReceiVed July 3, 2008. ReVised Manuscript ReceiVed July 25, 2008 Superhydrophobic surfaces have shown promising applications in microfluidic systems as a result of their waterrepellent and low-friction properties over the past decade. Recently, designed microstructures have been experimentally applied to construct wettability gradients and direct the droplet motion. However, thermodynamic mechanisms responsible for the droplet motion on such regular rough surfaces have not been well understood such that at present specific guidelines for the design of tunable superhydrophobic surfaces are not available. In this study, we propose a simple but robust thermodynamic methodology to gain thorough insight into the physical nature for the controllable motion of droplets. On the basis of the thermodynamic calculations of free energy (FE) and the free-energy barrier (FEB), the effects of surface geometry of a pillar microtexture are systematically investigated. It is found that decreasing the pillar width and spacing simultaneously is required to lower the advancing and receding FEBs to effectively direct droplets on the roughness gradient surface. Furthermore, the external energy plays a role in the actuation of spontaneous droplet motion with the cooperation of the roughness gradient. In addition, it is suggested that the so-called “virtual wall” used to confine the liquid flow along the undesired directions could be achieved by constructing highly advancing FEB areas around the microchannels, which is promising for the design of microfluidic systems.
1. Introduction Rapid improvements in micro/nanofabrication techniques have allowed for the great development for microfluidic or lab-ona-chip systems over the past decade. Because of system integration and miniaturization, a series of chemical analyses and bioassays can now be carried out just on a square-centimeter-large chip with high efficiency and low cost. In particular, there has been strong interest in droplet-based microfluidics in several academic and industrial fields in recent years.1-3 Compared to the traditional continuous fluid flow systems, discrete microdroplets with isolated and confined reagents have small convecting volumes, fast mixing velocity, and (axial) dispersionless transportation.4,5 Microdroplets as ideal chemical reactors, therefore, show promising applications in the field of microfluidics.6 However, in such microscale systems, control of tiny droplet transport in a precise way becomes extremely difficult. Hence, the surface-tensionbased manipulation of droplet motion has been developed and plays a crucial role in directing the microdroplets on the microscale.7,8 In particular, considerable approaches of surface tension or wettability gradient actuation have been proposed to control the motion of droplets, which has been recently reviewed * Corresponding authors. E-mail:
[email protected], mail.xjtu.edu.cn. † Xi’an Jiaotong University. ‡ Jilin University and University of Alberta.
gjqiao@
(1) Pollack, M. G.; Fair, R. B.; Shenderov, A. D. Appl. Phys. Lett. 2000, 77, 1725. (2) Pollack, M. G.; Shenderov, A. D.; Fair, R. B. Lab Chip 2002, 2, 96. (3) Teh, S. Y.; Lin, R.; Hung, L. H.; Lee, A. P. Lab Chip 2008, 8, 198. (4) Burns, M. A.; Mastrangelo, C. H.; Sammarco, T. S.; Man, F. P.; Webster, J. R.; Johnson, B. N.; Foerster, B.; Jones, D.; Fields, Y.; Kaiser, A. R.; Burke, D. T. Proc. Natl. Acad. Sci. U.S.A. 1996, 93, 5556. (5) Song, H.; Tice, J. D.; Ismagilov, R. F. Angew. Chem., Int. Ed. 2003, 42, 768. (6) Stone, H. A.; Stroock, A. D.; Ajdari, A. Annu. ReV. Fluid Mech. 2004, 36, 381. (7) Kotz, K. T.; Noble, K. A.; Faris, G. W. Appl. Phys. Lett. 2004, 85, 2658. (8) Baviere, R.; Boutet, J.; Fouillet, Y. Microfluid. Nanofluid. 2008, 4, 287.
in detail by Genzer et al.9 and Morgenthaler et al.10 However, conventional methods involved in thermal, chemical, electrostatic, and optical principles may cause a new problem: the compatibility of reagents with the change of biophysical or biochemical properties. Furthermore, delivered materials might be adsorbed on the solid walls as a result of the nature of the contact, leading to mass loss in liquid transport and contamination or clogging of channels.11,12 It is interesting that during almost the same period, superhydrophobic surfaces, inspired by the so-called lotus effect,13,14 have been extensively investigated since the end of the last century. Now it has been well recognized that surface roughness is necessary to create superhydrophobic surfaces and form Fakir droplets or composite states (the droplet is suspended on top of the surface roughness).15-20 Because of the very large contact angle (CA) and small contact angle hysteresis (CAH) (i.e., the difference between the advancing or maximum and receding or minimum contact angles), superhydrophobic surfaces would have ideal liquid-shedding or droplet-sliding properties.21,22 Such unique wettability has shown great potential applications in microfluidics and no lost transport. Exploratory research using tunable or controllable superhydrophobic surfaces by micro(9) Genzer, J.; Bhat, R. R. Langmuir 2008, 24, 2294. (10) Morgenthaler, S.; Zink, C.; Spencer, N. D. Soft Matter 2008, 4, 419. (11) Hong, X.; Gao, X.; Jiang, L. J. Am. Chem. Soc. 2007, 129, 1478. (12) Blossey, R. Nat. Mater. 2003, 2, 301. (13) Barthlott, W.; Neinhuis, C. Planta 1997, 202, 1. (14) Neinhuis, C.; Barthlott, W. Ann. Bot. 1997, 79, 667. (15) Patankar, N. A. Langmuir 2003, 19, 1249. (16) He, B.; Patankar, N. A.; Lee, J. Langmuir 2003, 19, 4999. (17) Miwa, M.; Nakajima, A.; Fujishima, A.; Hashimoto, K.; Watanabe, T. Langmuir 2000, 16, 5754. (18) Nosonovsky, M.; Bhushan, B. Microsyst. Technol. 2005, 11, 535. (19) Quere, D. Nat. Mater. 2002, 1, 14. (20) Yoshimitsu, Z.; Nakajima, A.; Watanabe, T.; Hashimoto, K. Langmuir 2002, 18, 5818. (21) Della Volpe, C.; Siboni, S.; Morra, M. Langmuir 2002, 18, 1441. (22) Feng, L.; Li, S.; Li, Y.; Li, H.; Zhang, L.; Zhai, J.; Song, Y.; Liu, B.; Jiang, L.; Zhu, D. AdV. Mater. (Weinheim, Ger.) 2002, 14, 1857.
10.1021/la802033q CCC: $40.75 2008 American Chemical Society Published on Web 09/13/2008
11652 Langmuir, Vol. 24, No. 20, 2008
structures designed to direct the motion of droplets has been carried out, which may greatly facilitate the development of microfluidic systems.11,12,23,24 For example, Sandre et al. originally introduced the direction-dependent motion of liquid droplets by structuring the surface roughness as ratchetlike topographical structures.25 Such a ratcheting motion26 of droplets stems from local spatial asymmetry; however, it has not been involved in roughness-induced wettability and superhydrophobicity. Subsequently, He et al.27 reported the droplet motion across the boundary between flat and rough surfaces. Moreover, Petrie et al.28 first introduced Fakir droplets into the field of fast directed motion of microdroplets along porous substrates coated with chemical wettability gradients to reduce friction at the liquid/ substrate interface. Following Petrie et al.’s work, Shastry et al.29 and Zhu et al.30 successively showed the manipulation of droplet motion with the genuine conception of the surface geometry design of micropillar structures to form wettability gradients. In addition, Yang et al.31 and Sun et al.32 recently investigated droplet manipulation on wettability gradients actuated by the alignment of parallel patterned microtextures. However, fundamental principles for the design of wettability gradients to direct droplet motion are lacking in the literature. In particular, although the correlations between roughness and CA were well established almost 60 years ago33-35 (e.g., the classical Wenzel and Cassie equations), it is still inadequate to gain complete understanding of the role of surface geometry in wetting properties because roughness represents only a composite measure of all surface texture parameters. For instance, for a pillar microtexture, the same roughness can result from different combinations of pillar width, spacing, and height and may correspond to different contact angles. Moreover, systematic investigations of the mechanism for roughness-induced droplet motion on finely designed surfaces are very rare. Consequently, some experimental phenomena described in the literature can be hardly understood or are even controversial. For instance, Shastry et al.29 emphasized the important role of mechanical vibrations in the spontaneous motion of a droplet, whereas Zhu et al.30 achieved such motion without mentioning any assistance from vibrations. In our previous studies,36-39 we proposed a simple, robust thermodynamic approach to analyze the surface free energy (FE) and free-energy barrier (FEB) of various metastable wetting states. The model has been used to investigate symmetrically and periodically aligned surface profiles, such as homogeneous surface pillar and parallel grooved microstructures. In the present study, an improved FE analysis and formulation is proposed to (23) Ma, M.; Hill, R. M. Curr. Opin. Colloid Interface Sci. 2006, 11, 193. (24) Feng, X.; Jiang, L. AdV. Mater. (Weinheim, Ger.) 2006, 18, 3063. (25) Sandre, O.; Gorre-Talini, L.; Ajdari, A.; Prost, J.; Silberzan, P. Phys. ReV. E 1999, 60, 2964. (26) Daniel, S.; Sircar, S.; Gliem, J.; Chaudhury, M. K. Langmuir 2004, 20, 4085. (27) He, B.; Lee, J. Proceedings of the 16th IEEE Annual International Conference on MEMS;Kyoto, Japan; Jan 2003; 120. (28) Petrie, R. J.; Bailey, T.; Gorman, C. B.; Genzer, J. Langmuir 2004, 20, 9893. (29) Shastry, A.; Case, M. J.; Boehringer, K. F. Langmuir 2006, 22, 6161. (30) Zhu, L.; Feng, Y.; Ye, X.; Zhou, Z. Sens. Actuators, A 2006, A130-A131, 595. (31) Yang, J. T.; Chen, J. C.; Huang, K. J.; Yeh, J. A. J. Microelectromech. Syst. 2006, 15, 697. (32) Sun, C.; Zhao, X.-W.; Han, Y.-H.; Gu, Z.-Z. Thin Solid Films 2008, 516, 4059. (33) Wenzel, R. N. J. Ind. Eng. Chem. 1936, 28, 988. (34) Cassie, A. B. D.; Baxter, S. Trans. Faraday Soc. 1944, 40, 546. (35) Johnson, R. E., Jr.; Dettre, R. H. AdV. Chem. Ser. 1964, 43, 112. (36) Li, W.; Amirfazli, A. J. Colloid Interface Sci. 2005, 292, 195. (37) Li, W.; Amirfazli, A. AdV. Colloid Interface Sci. 2007, 132, 51. (38) Li, W.; Amirfazli, A. Soft Matter 2008, 4, 462. (39) Li, W.; Fang, G.; Li, Y.; Qiao, G. J. Phys. Chem. B 2008, 112, 7234.
Fang et al.
Figure 1. Typical 2D pillar surface microtextures. (a) Noncomposite wetting state on a uniformly rough surface. (b) Composite wetting state on a uniformly rough surface. (c) Composite wetting state on a nonuniformly rough surface.
investigate nonuniformly rough surfaces. Consequently, a detailed knowledge of the surface FE landscapes and metastable states is gained, and mechanisms responsible for the motion of Fakir droplets on such topographically heterogeneous surfaces can be understood. Such an understanding is important to the design of surface microtextures in order to direct the motion of droplets.
2. Model 2.1. Wetting Theories on Rough Surfaces. It is well known that the CA of a liquid droplet on an ideal smooth solid surface can be predicted by Young’s equation
γla cos θY ) γsa - γls
(1)
where γla, γsa, and γls are the surface tension at liquid-air, solid-air, and liquid-solid interfaces, respectively. For a rough surface, two wetting states may occur if a droplet is deposited on the surface: the noncomposite (i.e., complete liquid penetration into the troughs of a rough surface, Figure 1a) and the composite (Fakir droplets) (i.e., the entrapment of air in the troughs of a rough surface, Figure 1b). Note that the surface microtexture is uniformly constructed from the constant geometrical parameters of pillar width (a), spacing (b), and height (h) and the length scale of the posts is much smaller than the droplet size. The apparent CA of the noncomposite is given by Wenzel’s equation33
cos θw ) r cos θY
(2)
where r is the roughness factor as the ratio between the actual area and geometric projected area for a wetting surface; for the 2D model, r ) 1 + 2h/(a + b). However, the apparent CA of
Droplet Motion on Superhydrophobic Surfaces
Langmuir, Vol. 24, No. 20, 2008 11653
the composite, θCB, can be calculated using the Cassie-Baxter equation34
cos θCB ) f1 cos θ1 + f2 cos θ2 f1 + f2 ) 1
(3)
where f1 and f2 are the fractions of the liquid-solid and liquid-air interfaces with intrinsic equilibrium contact angles θ1 and θ2, respectively. For one aqueous droplet on the rough surface, θ1 equals the intrinsic CA (θY) and θ2 equals 180°, leading to a changed form of the Cassie-Baxter equation
cos θCB ) fcos θY - (1 - f)
(4)
where f is the solid-liquid contact area fraction of the substrate; for the 2D model, f ) a/(a + b). It is of interest to note the recent intensive debate on the validity of the Wenzel and Cassie equations.40-43 The debate mainly focuses on the range of applicability of the two classical equations. For example, a droplet is deposited on a rough surface consisting of nonuniform arrangements of geometrical parameters, as illustrated in Figure 1c, and the pillar width and spacing are described in arrays below,
a ) (a1, a2 · · · ak, ak+1 · · · an), b ) (b1, b2 · · · bk, bk+1 · · · bn) (5) where ak and bk represent the pillar width and pillar spacing at the three-phase contact line (the triple line). Although the 2D model of the nonuniformly rough surface concerned here is not completely the same as that in previous studies,40-42 a general issue is posted: Can the apparent CA on such a nonuniformly rough surface can be predicted by the Cassie equation for the composite state (or by the Wenzel’s equation)? Gao and McCarthy40 experimentally demonstrated that it would be completely misleading to use eq 4 if the surface fraction is taken to be the average of the contact area beneath the droplet, as expressed by k
f )
∑ ai i)1
k
k
i)1
i)1
∑ ai + ∑ bi
(6)
Therefore, the authors concluded that the Cassie equation will be invalid on such a nonuniformly rough surface. However, other researchers41-43 argued that the choice of the surface fraction plays a key role in the application of the Cassie equation. For instance, Nosonovsky42 indicated that the Cassie equation can be generalized as
cos θCB ) f1(x, y)cos θ1 + f2(x, y)cos θ2
(7)
To validly use the Cassie equation, f1(x, y) and f2(x, y) should be chosen as the local densities of the two components that compose the heterogeneous surface at the triple line. For example, in the present case (Figure 1c) the local solid surface fraction at the triple line can be defined as
f)
ak ak + bk
(8)
Note that the droplet size is much larger than the dimension of surface asperities. (40) Gao, L.; McCarthy, T. J. Langmuir 2007, 23, 3762. (41) McHale, G. Langmuir 2007, 23, 8200. (42) Nosonovsky, M. Langmuir 2007, 23, 9919. (43) Panchagnula, M. V.; Vedantam, S. Langmuir 2007, 23, 13242.
Figure 2. Illustration of FE analysis for a drop on a surface with nonuniform microtextures. The dashed line represents the initial state of the drop at the very start of wetting whereas the solid line signifies an arbitrary state of the drop.
In conclusion, there is a general viewpoint that it is the roughness in the vicinity of the triple line rather than the overall surface area within the contact perimeter that determines the apparent CA. However, it is worth pointing out that the position where the three-phase contact line of a sessile droplet will rest is unknown and the local solid surface fraction can hardly be determined. Accordingly, various CAs (e.g., the equilibrium CA, the advancing and receding CAs) cannot be calculated, and the analysis of droplet motion on a nonuniformly rough surface remains unsolved. 2.2. Thermodynamic Model for Microstructure-Patterned Surfaces. To obtain the various CAs as well as the surface FE and FEB of nonuniformly aligned pillar textures, a 2D model is first proposed in Figure 2 (the geometrical parameters are set to be consistent with those in Figure 1c). Superhydrophobic behavior is based on the composite state, or in other words, in the present work the noncomposite state is less significant and the composite state is our main concern. To this end, the pillar height is set to be large enough to form the composite state.20,36,37 For the analysis of the 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 on the millimeter scale and is much larger than the dimensions 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 line44,45) becomes extremely small and makes little contribution to the wettability for such a macroscopic droplet. In fact, Pompe et al.46 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. On the basis of the above assumptions, the 2D droplet profile can be considered to be a spherical cap and have a constant area (analogue of constant volume in the 3D model).47,48 (44) Boruvka, L.; Neumann, A. W. J. Chem. Phys. 1977, 66, 5464. (45) Li, D. Colloids Surf., A 1996, 116, 1. (46) Pompe, T.; Herminghaus, S. Phys. ReV. Lett. 2000, 85, 1930. (47) Marmur, A. J. Colloid Interface Sci. 1994, 168, 40.
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For a wetting composite state, when the three-phase contact line moves from a reference position O (with a CA of 180° when a droplet touches a solid surface at one point at the very start of wetting and a droplet radius of R0) to an arbitrary position such as C (with θar and Lar) (see Figure 2), the magnitude of FE per unit length of the droplet contact line for each of the two states can be written as
(
FO ) γlaπR0 + γsa(Lar + 2kh) + C k
) (∑
Lar Far ) γ θar + b + γsa sin θar i)1 i
∑
la
(9)
)
k
∑
ai + C (10)
i)1
(
)
k k Lar + bi - πR0 + ai(γls - γsa) sin θar i)1 i)1 (11)
∑
∑
Young’s equation is locally valid. Substituting eq 1 into eq 11 gives
∆FO far γla
(
) θar
)
k k Lar + bi - πR0 ai cos θY sin θar i)1 i)1 (12)
∑
n
γsa
n-k′
bi +
i)1
∑
)
n
ai + 2nh + γls
i)1
∑
ai + C (15)
i)n-k′
Then the FE change for the two wetting systems can be expressed as eq 16.
(
n
∑
i)n - k′
k
bi -
) ( (∑
∑ bi i)1
+ γsa
n - k′
n
i)1 n
i) k
)
∑ ai - ∑ ai i)n - k′
k
ai -
+
)
∑ ai i)1
(16)
3. Results and Discussion
where C is the FE of the unchanged portion for the system. Thus, the system FE changes from O to an arbitrary position, which can be expressed as
∆FO far ) γla θar
(∑
∑
γls
k
γls
)
n Lar + b + sin θar i)n-k′ i
ar ∆F OfO ) γla
bi + 2kh +
i)1
(
ar F O ) γla θar
∑
3.1. Typical FE and FEB Curves. To make our results and discussion easily understandable, first we give a typical example to show how to obtain various CAs and CAHs from the analysis and calculations of FE and FEB. Figure 4 shows the normalized FE as a function of the instantaneous contact angle for composite states. One can see that the FE curve for the droplet sitting on the weak hydrophobic area is much lower than that for the strong hydrophobic area, implying a preferred energy state. This thermodynamically indicates that the droplet has a tendency to move from a strong hydrophobic area to a weak hydrophobic area in order to minimize the system free energy. Moreover, it is worth pointing out that in practice the motion of the droplet is also involved with releasing the geopotential due to the change in the center of gravity (Figure 3a). However, whether the droplet
Note that the unit of energy (J/m) has been normalized with respect to γ (J/m2). Therefore, the unit of free energy is the meter. According to the assumption, eq 13 is derived from geometrical analysis.
s ) θar
Lar2 sin2 θar
- Lar2ctgθar ) πR02
(13)
Similarly, the geometrical constraint and FE equations can be derived between other arbitrary instantaneous positions. Here it should be noted that the reference FE state O is assigned a value of zero. Such a simplification cannot affect the results because the FE barrier is a relative value with respect to its neighbors. Because it is not possible to obtain θar as an explicit expression, ∆FOfar is determined by solving eqs 12 and 13 via successive approximations, and numerical computations can be implemented. To compare FE between the states of droplets sitting on the strong and weak hydrophobic regions of a nonuniformly rough surface (Figure 3a), the correlations of all of the geometrical parameters between the two regions are schematically shown (Figure 3b). From Figure 3, the magnitude of FE per unit length of the droplet contact line for each of the two states can be written as
(
la F ar o ) γ θar
) (∑
k Lar + b + γsa sin θar i)1 i
∑
n
i)1
)
n
bi +
∑ ai + 2nh i)k
+
k
γls
∑ ai + C (14) i)1
(48) Long, J.; Hyder, M. N.; Huang, R. Y. M.; Chen, P. AdV. Colloid Interface Sci. 2005, 118, 173.
Figure 3. (a) Schematic showing droplets sitting on the strong and weak hydrophobic regions of a nonuniformly rough surface. (b) Geometrical correlations between strong and weak hydrophobic surfaces for composite wetting states.
Droplet Motion on Superhydrophobic Surfaces
Figure 4. Normalized free energy as a function of the instantaneous contact angle of composite wetting states for the droplets sitting on strong (solid blue curve) and weak (dashed black curve) hydrophobic regions. R0 ) 0.0145 m, a1 ) 54 × 10-6m, b1 ) 4 × 10-6m, ai+1 ) ai - (0.1 × 10-6) (i ) 1, 2 · · · 500), bi+1 ) bi + 0.1 × 10-6 (i ) 1, 2 · · · 500), h ) 40 µm; intrinsic CA, θY ) 120°. The insets shows enlarged views of three segments of the FE curve illustrating the determination of equilibrium (θE), advancing (θa,) and receding (θr) CAs; ∆Fadv, ∆Frec represent the advancing and receding FEBs, respectively; positions A-C correspond to those in Figure 2, showing the two different energy barriers; red arrows illustrate that when external energy is employed θa and θr will gradually change to values close to θE.
motion could be actuated also depends on the dynamic analysis (below). One can also see that the two FE curves contain multivalued local minimum and maximum FEs. 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/θE) and exactly corresponds to the generalized Cassie CA resulting from eqs 7 and 8. For example, when the droplet is deposited on the strong hydrophobic position, the equilibrium CA could be easily determined to be148.32° for the FE that reaches its global minimum (Figure 4). At the moment, the average surface fraction within the contact perimeter gained from eq 6 is 0.184, and the resultant Cassie CA is 155.2°, which shows no agreement with our thermodynamic analysis. Correspondingly, if the surface fraction in the vicinity of the triple line, about 0.300, is adopted, then the resultant Cassie CA of 148.21° is very close to the value obtained from the FE curve. A similar conclusion can also be drawn for other nonuniformly rough surfaces in the model. It is also noted that the two curves intersect at CA ) 180°. This indicates that there is no difference in FE between the two 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°). The insets of Figure 4 also illustrate that there are two different energy barriers related to one local energy minimum position: the advancing ((∆Fadv)) and receding ((∆Frec)) FEBs. Figure 5 shows the detailed landscapes of advancing and receding FEBs of the composite state for the same surface geometry used in Figure 2. Here it should be pointed out that the external energy (∆E), which was first reported by Johnson and Dettre,35 is assumed to have a value of zero. As a result, the advancing (θa ) 180°) and receding (θr ) θY ) 120°) CAs as well as the maximum theoretical CAH defined as (θa - θr ) 60°) can be determined by the intersecting values of advancing and receding curves with the x axis, respectively (Figure 5). Meanwhile, θa and θr are also illustrated in the inset of Figure 4, where advancing and receding FEBs are zero. Here it should also be noted that in practice, because various external sources such as gravity, surface
Langmuir, Vol. 24, No. 20, 2008 11655
Figure 5. Determining receding and advancing CAs as well as CAH from the typical curves of advancing and receding FE barriers for a composite wetting state. R0 ) 0.0145 m, a1 ) 54 × 10-6 m, b1 ) 4 × 10-6 m, ai+1 ) ai - (0.1 × 10-6) (i ) 1, 2 · · · 500), bi+1 ) bi+ (0.1 × 10-6) (i ) 1,2 · · · 500), h ) 40 µm; intrinsic CA, θY ) 120°. The maximum CAH shown is the value associated with zero FEB on the advancing and receding branches of the FE curve (Figure 4). When external energy is employed, modulated CAH will be gained as a result of the counterbalance with energy barriers. Note that FEB is per unit length of the contact line and is normalized with respect to the surface tension of the liquid.
imperfections, electrowetting,49 mechanical vibration,50 and thermal fluctuation51 inevitably exist around a wetting system, the external energy may not be zero, which will help the system to overcome FEB, leading to a small or even a zero value of CAH. For instance, if the external energy is set to be ∆E/γla ) 4 × 10-6 m (for consistency, the external energy is also normalized), then the FEBs below this value can be overcome such that the advancing and receding CAs are modulated to be θa′ () 142.3°) and θr′ () 128.4°), as illustrated in Figure 4. Furthermore, if an external energy is large enough, then both θa and θr will approach θE (Figures 4 and 5), leading to a zero value of CAH. Hence, the resistant force against the droplet motion arising from CAH29,31,52 may disappear, and the droplet motion may become much easier. 3.2. Requirements for the Initiation of Droplet Motion. The above analysis shows that the droplet has a thermodynamic tendency to move from an area of strong hydrophobicity to another area of weak hydrophobicity. It is therefore possible to introduce the ladderlike wettability gradient, which has been used in others’ experimental studies (Figure 6a)29,30 to cause a driving force for the motion of a droplet. As long as the length of the droplet baseline (LR) is larger than the width of each roughness portion (e.g., UO or OU′), the front and back edges of the droplet will be settled on different roughness portions wherever it is deposited. This indicates that a large enough droplet is necessary to create the wettability gradient on the designed ladderlike surface textures. Note that as mentioned earlier, the apparent CA is determined by the roughness in the vicinity of the triple line rather than the overall surface area within the contact perimeter. Therefore, the CAs for the left and right edges of the droplet should be calculated respectively by the local advancing and receding FEBs for the two different roughness portions. For example, as illustrated in Figure 6a, the CAs for the left and right edges of the droplet are indicated as θL,a, θR,a, θL,r, and θR,r, respectively, leading to the different CAHs of droplets moving rightward (θR,a - θL,r) and leftward ((θL,a - θR,r), as illustrated in Figure 6b. (49) Krupenkin, T. N.; Taylor, J. A.; Schneider, T. M.; Yang, S. Langmuir 2004, 20, 3824. (50) Decker, E. L.; Frank, B.; Suo, Y.; Garoff, S. Colloids Surf., A 1999, 156, 177. (51) Kong, B.; Yang, X. Langmuir 2006, 22, 2065. (52) Daniel, S.; Chaudhury, M. K. Langmuir 2002, 18, 3404.
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Figure 6. (a) Schematic illustration of a ladderlike microtextured surface with wettability gradient. (b) Droplets moving to the right (R) and left (L) show different CAHs on a nonuniformly rough surface. (c) Normalized advancing and receding FEB with apparent CA for the left (black curves) and right (blue curves) parts of the roughness, respectively (left part: a ) 15 µm, b ) 40 µm; right part: a ) 35 µm, b ) 20 µm; h ) 40 µm; intrinsic CA, θY ) 120°). Subscripts r, a, and E denote receding, advancing, and equilibrium, respectively; θL and θR represent the contact angles at the left and right edges of the 2D droplet, respectively. Note that the length of the droplet baseline (LR) is always larger than the width of each roughness portion (e.g., the length of UO).
To start droplet motion on a surface, it is necessary to overcome the resistant force, which is related to the difference between the advancing and receding contact angles.52-54 In the 2D model, this force will be
F ) γLV(cos θr - cos θa)
(17)
Figure 6c shows the local advancing and receding FEB curves for the two edges of the droplet deposited on the boundary between (53) Brochard, F. Langmuir 1989, 5, 432. (54) Ionov, L.; Houbenov, N.; Sidorenko, A.; Stamm, M.; Minko, S. AdV. Funct. Mater. 2006, 16, 1153.
the left (a ) 15 µm, b ) 40 µm) and right roughness portions (a ) 35 µm, b ) 20 µm), respectively. On the basis of the FEB curves, the advancing and receding CAs for the left and right edges of the droplet will remain constant (i.e., 180° and θY, respectively) if the external energy is neglected. As a result, no matter which direction the droplet is supposed to move, the initial force required to actuate the motion should be a constant positive value such as γLV(cos θY + 1). Thus, the droplet will remain stationary on the microtextured surface with a wettability gradient if there is no additional driving force. Meanwhile, because of the effect of the Laplace pressure,55 the droplet will show itself to be a spherical cap, resulting in a constant mean curvature surface on the liquid-air interface and the same CAs (θY < θL ) θR < 180°) for the left and right edges, respectively (Figure 6a). To start the tunable motion of a droplet, the external energy can be introduced. As illustrated in Figure 6c, if the external energy is set to be energy level one, then an analogous ratcheting motion26 will appear in the dynamic wetting behavior because the contact angle hysteresis CAHR () θR,a1 - θL,r1) rightward is smaller than CAHL () θL,a1-θR,r1) leftward. Therefore, from eq 17, the supposed initial force required to counteract the resistance against the rightward motion (γLV(cos θL,r1 - cos θR,a1)) is smaller than that of the leftward motion ((γLV(cos θR,r1 - cos θL,a1)). As a result, as long as the droplet moves to the right under a random force, moving backward to the left will be difficult. If the external energy increases to energy level two (∆E/γla ) 4.3 × 10-6 m exactly in this case), then the energy level line shown in Figure 6c will cross the intersecting point of the receding curve for the left edge and the advancing curve for the right edge (i.e., the receding CA of the left droplet edge will be equivalent to the advancing CA of the right droplet edge (θR,a2 ) θL,r2)). Therefore, CAH as well as the resistant force of moving rightward will thus decrease to zero, whereas CAH for moving leftward remains a large value (θL,a2 > θR,r2). We can mark a red arrow on energy level two for the subsequent comparison because this level is necessarily required for the initial spontaneous motion (below). Interestingly, if the external energy increases to energy level three (∆E/γla > 4.3 × 10-6 m), then the receding CA of the left droplet edge θL,r3 will be larger than the advancing CA of the right droplet edge θR,a3. As a result, the spontaneous droplet motion without an additional driving force will occur because the receding CA of the back (receding) edge of the droplet is larger than the advancing CA of the front (advancing) edge of the droplet.55,56 In this case, the resistant force against the initial motion to the right (i.e., γLV(cos θL,r3 – cos θR,a3)) will become negative. This can cause a nonequilibrium Laplace pressure on the opposite sides of the droplet, leading to a driving force (-γLV(cos θL,r3 - cos θR,a3)) and hence spontaneous motion of the droplet. In addition, if the external energy increases to energy level four, then the advancing and receding CAs of both the right and left edges of the droplet will reach their extreme values (i.e., the equilibrium CAs for the two edges of the droplet (θR,E and θL,E, respectively). Further increases in the external energy will then have no influence on the CAs. Therefore, the maximum driving force acting on the droplet for the rightward motion can be derived as (-γLV(cos θL,E - cos θR,E)), whereas the maximum additional force required to overcome the resistant force for the leftward motion can be derived as (γLV(cos θL,E - cos θR,E)) In summary, a large enough external energy is required to overcome the hysteresis effect, and then spontaneous droplet (55) Daniel, S.; Chaudhury, M. K.; Chen, J. C. Science 2001, 291, 633. (56) Chaudhury, M. K.; Whitesides, G. M. Science 1992, 256, 1539.
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Figure 7. (a) Schematic illustration of droplets on a ladderlike microtextured surface with gradually changed pillar width. (b) Normalized advancing and receding FEBs with apparent CA for the left (L), center (C), and right (R) parts of the roughness. Left part: a ) 10 µm; center part: a ) 15 µm; right part: a ) 20 µm; b () 20 µm) and h () 40 µm) remain constant; intrinsic CA, θY ) 120°. Subscripts r, a, and E denote receding, advancing, and equilibrium, respectively; θL, θC, and θR represent the contact angles when the triple lines of the droplet are settled at the left, center, and right portions of the roughness, respectively.
Figure 8. (a) Schematic illustration of droplets on a ladderlike microtextured surface with gradually changing pillar spacing. (b) Normalized advancing and receding FEBs with apparent CA for the left (L), center (C), and right (R) parts of the roughness. Left part: b ) 10 µm; center part: b ) 15 µm; right part: b ) 20 µm; a () 15 µm) and h () 40 µm) remain constant; intrinsic CA, θY ) 120°. Subscripts r, a, and E denote receding, advancing, and equilibrium, respectively; θL, θC, and θR represent the contact angles when the triple line of the droplet is settled at the left, center, and right portions of the roughness, respectively.
motion can be actuated. To this end, the use of mechanical vibration together with the roughness gradient to direct droplet motion is necessary.29 Although other studies30-32 achieved such motion without mentioning any assistant of vibrations, in fact mechanical vibrations from environments or experimental processes (e.g., the vibration arising from moving the droplet off of the syringe) acted as the external stimulus. Because of the small hysteresis effect along the parallel patterned microtextures,20,39 a very small external energy can be large enough to overcome the hysteresis and droplet motion on such parallel patterned microtextures,31,32 can be much easier than it is on the micropillar structures. 3.3. Effects of Surface Microtextures on the FEB. 3.3.1. Effect of Pillar Width. To investigate the role of pillar width, parameter a is gradually changed on the roughness gradient constructed by three different portions whereas the other geometrical parameters remain unchanged (Figure 7a). For instance, b is set equal to 20 µm whereas a varies in a sequence of 10, 15, and 20 µm. Correspondingly, the advancing and receding FEB curves for the left, central, and right parts of the roughness are given in Figure 7b. One can see that the advancing FEB curves overlap for the three portions whereas the receding FEB curves shift upward with increasing pillar width. As a result, if the droplet is deposited on the boundary between the left and central roughness portions, then energy level one (∆E/γla ) 3.3 × 10-6 m in the case), which crosses the intersection point of the receding curve of the left edge and the advancing curve of the central edge (θC,a)θL,r), is needed for the initial spontaneous motion. Accordingly, the CAH as well as the resistant force
against moving rightward will decrease to zero. Therefore, as the external energy increases to a value larger than energy level one, the droplet will accelerate to move to the right. Furthermore, if the droplet is deposited on the boundary between the central and right roughness portions, then energy level two (∆E/γla ) 4.3 × 10-6 m, as illustrated in Figure 7b) is needed for the initial spontaneous motion. Note that energy level two is larger than energy level one, indicating that increasing the pillar width can enhance the critical energy level for the initial droplet motion. In addition, it is worth pointing out that, for the whole system, external energy larger than energy level three (∆E/γla ) 5.0 × 10-6 m) is required to actuate the spontaneous droplet motion. In this case, the advancing and receding CAs of both the right and left edges of the droplet will reach their extreme values wherever the droplet is deposited. Thus, if the droplet rests on the two boundaries, then the maximum driving forces along the roughness gradient surface will reach -γLV(cos θL,E - cos θC,E) and -γLV(cos θC,E - cos θR,E), respectively (section 3.2). 3.3.2. Effect of Pillar Spacing. As illustrated in Figure 8a, parameter b is gradually changed on the roughness gradient constructed of three different portions whereas the other geometrical parameters remain constant in order for us to investigate the role of pillar spacing (e.g., a ) 15 µm and b varies in a sequence of 20, 15, and 10 µm). The advancing and receding FEB curves for the left, central, and right parts of the roughness are correspondingly given in Figure 8b. One can see that the receding FEB curves overlap for the three portions whereas the advancing FEB curves shift downward with decreasing pillar spacing. Similar to the results shown in section 3.3.1, energy level one (∆E/γla ) 3.8 × 10-6 m) and energy level two (∆E/γla
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) 3.0 × 10-6 m, as illustrated in Figure 8b) are required for the initial spontaneous motions of the droplet deposited on the two boundaries, respectively. Note that energy level two is smaller than energy level one, indicating that decreasing the pillar spacing can lower the critical energy level for the initial droplet motion. In addition, for an effective motion of the droplet throughout the whole roughness gradient, external energy larger than energy level three (∆E/γla ) 4.3 × 10-6 m) is preferred. In this case, the driving forces for the droplet will reach their extreme values along the roughness gradient surface (i.e., -γLV(cos θL,E - cos θC,E) and -γLV(cos θC,E - cos θR,E), respectively). 3.4. Guidelines for the Surface Geometry Design of Droplet Motion. On the basis of the above analysis of FE and FEB, it is possible to design the surface geometry to control the droplet motion precisely. In particular, because the pillar width and spacing have different effects on FEB, a suitable combination is helpful for the tunable motion of droplets. 3.4.1. Analysis of the MoVing BehaVior of Droplets Based on the FEB CurVes. In the present work, the typical values of pillar width (a) and pillar spacing (b) in a pillar microtexture with a roughness gradient presented in the experimental work29 have been taken as an example by which to analyze the droplet motion. For this microtexture, six combinations of surface geometry {(a1,b1), (a2,b2)...(a6b6)} with gradually increased solid-liquid contact area fractions {f1, f2...f6} are used, as shown in Figure 9a; note that although the calculated fraction values are different compared to those used in a 3D surface, this will not affect the analysis of the droplet motion. The FEB curves for the six areas are given correspondingly in Figure 9b; note that the advancing and receding FEB curves, whose intersections are marked with blue arrows with six numerals, are related to the six areas, respectively. First, we will investigate the droplet deposited on the boundary of area one and area two. According to section 3.2, the energy level at the intersection (marked by the red triangle besides the numeral 1) of the receding FEB curve for area 1 and the advancing FEB curve for area 2 is just the critical external energy required for the initial spontaneous motion of the droplet (denoted as critical energies 1 and 2 for the purpose of this discussion). Similarly, other critical intersections can also be found and marked by the red triangles, as shown in Figure 9b. One can see that except for critical energies 4, 5 and 5, 6 all others are located below the given energy level if the external energy is given as energy level one. Therefore, a positive driving force acting on the droplet on the boundaries between areas 1 and 2, 2 and 3, and 3 and 4 will be produced, implying an accelerating motion along the surface. However, if the droplet moves to the boundaries between areas 4, 5 and 5, 6, then a hysteresis force will occur, leading to decelerating motion. As a result, the external energy above the critical energy levels (e.g., level two in Figure 9b) is required to guarantee the spontaneous motion of droplets throughout the roughness gradient surface. It is of interest to point out that if the droplet volume is increased to extend across three areas of roughness (e.g., areas 3-5, as shown in Figure 9a) then the critical energy point will be the intersection of the receding FEB curve for area 3 and the advancing FEB for area 5. This critical energy level is lower than that for the smaller droplet extending across areas 3 and 4. Similarly, other critical energy points for the larger droplet that extends across three roughness areas along the gradient surface can also be found; these are indicated by the green circular marks, as shown in Figure 9b. As seen, compared to the red triangular marks, all of the green circular marks are located below energy level one. Such a low critical energy can lead to the motion of the larger droplet without any resistance force all along the
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Figure 9. (a) Schematic illustration of droplets on a ladderlike microtextured surface with gradually changing wettability. Note that (an, bn) represents the original surface geometrical combinations in ref 29 whereas (an′, bn′) represents the improved surface geometrical combinations; fn represents the solid-liquid contact area fractions for the six roughness areas, respectively; the six Arabic numerals represent the six ladderlike areas of roughness. (b) Normalized advancing and receding FEBs with apparent CA for the six ladderlike roughness areas of the original surface geometrical combinations (an, bn). h ) 40 µm; intrinsic CA, θY ) 120°. Note that the advancing and receding FEB curves, whose intersections are marked with the blue arrows with six numerals, are related to the six areas; the red triangular marks illustrate the critical energy points for droplets sitting between areas 1 and 2, 2 and 3, 3 and 4, 4 and 5, and 5 and 6; the green circular marks illustrate the critical energy points for droplets sitting between areas 1 and 3, 2 and 4, 3 and 5, and 4 and 6.
gradient surface under the actuation of energy level one, thus leading to spontaneous motion throughout the wetting system. Therefore, the related phenomena observed in the work of Shastry et al.29 can be interpreted well using the above thermodynamic analysis. 3.4.2. Surface Geometry Design for Directing Droplets. On the basis of the above discussion, for the initiation of droplet motion, lowering of the critical energy is expected. Whereas a large droplet volume is useful for this purpose, a suitable surface geometry is more practical. As discussed in section 3.3, because decreasing the pillar width and spacing simultaneously can lead to both low receding and advancing FEB curves, a combination of decreased pillar width and spacing {(a′1,b′1),(a′2,b′2)...(a′6,b′6)} can be used to reconstruct the six roughness portions while the wettability gradient {f1, f2...f6} can be kept unchanged, as illustrated in Figure 9a. For this surface geometry, the corresponding FEB can be calculated, as illustrated in Figure 10. By comparing Figure 10 to Figure 9b, one can see that the critical energy points for droplets along the roughness gradient surface all decline to values below energy level one. Meanwhile, the equilibrium CAs remain unchanged as a result of the constant solid-liquid contact area fractions for each rough region. As a result, although the wettability gradients for the two surfaces are
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Figure 10. Normalized advancing and receding FEBs with the apparent CA for the six ladderlike roughness areas of the improved surface geometrical combinations. The improved combinations of (an′, bn′) are shown in Figure 9a. h ) 40 µm; intrinsic CA, θY ) 120°. Note that the advancing and receding FEB curves, whose intersections are marked with the blue arrows with six numerals, are related to the six areas, respectively; the six Arabic numerals represent the six ladderlike areas of roughness, respectively; the red triangular marks illustrate the critical energy points for droplets sitting between areas 1 and 2, 2 and 3, 3 and 4, 4 and 5, and 5 and 6.
the same, the droplets on the designed gradient surface should have lower FEBs, and spontaneous droplet motion can happen more easily under a given external energy. On the basis of the above discussion, for a given solid-liquid contact area fraction extremely small values of pillar width and spacing (e.g., nanoscale surface geometry) are preferred for constructing the roughness gradient in order to obtain low FEBs, leading to a very small or even zero values of hysteresis. As a result, spontaneous droplet motions on such a rough microtexture can be autonomously generated under a small external energy. Interestingly, such motions (or ideal droplet-sliding behavior) resulting from nanoscale microstructure can be seen in nature (e.g., lotus leaves).57-59 In particular, recent observations of the dynamic suspending of microdroplets on lotus leaves can be mainly attributed to the wettability gradient formed on the nanoscale hairs.60 However, this phenomenon can hardly be observed because of experimental limitations because it is still difficult to prepare a nanoscale roughness gradient using the current nanofabrication techniques. 3.4.3. Guidelines for the Design of a Three-Dimensional Surface. Figure 11a schematically shows the top view of the proposed surface geometrical design for microfluidic channels used to transport droplets. Side views of droplets on the surface from the x and y axis directions are illustrated in Figure 11b. One can see that combinations of surface geometry {(a1, b1), (a2, b2)...(a5b5)...} with gradually increased solid-liquid contact area fractions {f1, f2...f5...} are distributed along the x-axis direction. The {f1, f2...f5...} set related to equilibrium CAs of each roughness portion determines the maximum driving force for the motion of a droplet. For a given wettability gradient, pillar width and spacing for each rough area along the gradient surface should be chosen to be as small as possible while their ratio remains unchanged in order to gain the lowest FEBs. Such low FEBs are (57) Lee, W.; Jin, M. K.; Yoo, W. C.; Lee, J. K. Langmuir 2004, 20, 7665. (58) Zheng, Y.; Gao, X.; Jiang, L. Soft Matter 2007, 3, 178. (59) Gao, X.; Yan, X.; Yao, X.; Xu, L.; Zhang, K.; Zhang, J.; Yang, B.; Jiang, L. AdV. Mater. (Weinheim, Germany) 2007, 19, 2213. (60) Zheng, Y.; Han, D.; Zhai, J.; Jiang, L. Appl. Phys. Lett. 2008, 92, 084106/ 1.
Figure 11. (a) Schematic illustration of the top view of the proposed surface geometry designed for microfluidic channels used to guide the droplets. (b) Side views of droplets on the surface from the x and y axis directions, respectively. (c) Side views of droplets between the top and bottom plates decorated with symmetric surface microtextures from the x and y axis directions, respectively.
necessary, especially by means of external energy, to achieve tunable wettability along the gradient surface or the precise control of droplet motion (e.g., motion in a desired area or direction of the microchannel). In addition, droplet motion is expected to be confined in the direction perpendicular to the gradient (i.e., along the y axis) to avoid liquid loss or mixing. The so-called virtual wall, which means no physical walls on the sides of the liquid but is used to confine the liquid flow,61,62 can be realized by a suitable choice of surface geometrical parameters. For example, as illustrated in Figure 11b, a combination of (a′1, b′1) with low advancing FEB and (a′2, b′2) with high advancing FEB distributed along the y axis can limit the droplet motion along the y axis because of large advancing CAs on the two sides of the droplet. (61) Zhao, B.; Moore, J. S.; Beebe, D. J. Science 2001, 291, 1023. (62) Zhao, B.; Moore, J. S.; Beebe, D. J. Anal. Chem. 2002, 74, 4259.
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In fact, Zhao et al.61,62 first realized the virtual wall by the patterned chemical heterogeneity and created pressure-sensitive switches inside channel networks. Compared to the chemically heterogeneous surfaces, the present microtextured surfaces can have more effective control of the droplets because a large range of CAs (θY ≈ 180°) can be tuned by the adjustment of surface geometry. In particular, the undesirable side effect arising from the changes in biochemical properties can be avoided, and the friction and adhesion caused by the solid-liquid area contact can also be further reduced. The above analysis therefore indicates that the proposed approach should be useful for the design of microfluidic channels. Here, a typical example can be suggested. As shown in Figure 11c, in the microfluidic channels consisting of the bottom and top plates decorated with symmetrically designed microtextures, the motions of droplets along the roughness-induced wettability gradient surface (i.e., x axis) can be actuated by external energy whereas the motions of droplets along the undesired directions (i.e., y axis) can be avoided by the so-called virtual wall.
4. Conclusions In the present study, a thermodynamic methodology is proposed to investigate the motion of droplets on nonuniformly rough surfaces. On the basis of thermodynamic calculations of advancing and receding FEBs, it is demonstrated that the roughness in the
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vicinity of the triple line rather than the overall surface area within the contact perimeter determines the apparent CA. Furthermore, it is found from local FEB landscapes on the roughness gradient that the receding CA can be increased to a value larger than the advancing CA under the actuation of the external energy, leading to spontaneous droplet motion. Moreover, decreasing the pillar width and spacing simultaneously is required to lower the advancing and receding FEBs to effectively direct droplets. In addition, with high or low FEBs by different combinations of surface geometry, tunable wettability of a gradient surface or the precise control of droplet motion (e.g., motion in a desired area or direction of the microchannel) can be achieved along with the external energy. In particular, the so-called virtual wall used to confine the liquid flow along the undesired directions could also be realized by constructing highly advancing FEB areas around the microchannels, which is promising for the design of microfluidic systems. Acknowledgment. This work was supported by the National Natural Science Foundation of China (nos. 50772086 and 50635030), the High-Tech R & D Program of China (863, no. 2007AA03Z558), and the National Basic Research Program (973 program) (no. 2007CB616913). LA802033Q
