Article pubs.acs.org/Langmuir
Drops on Hydrophilic Conical Fibers: Gravity Effect and Coexistent States Yu-En Liang,† Heng-Kwong Tsao,*,‡,§ and Yu-Jane Sheng*,† †
Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 106 Department of Chemical and Materials Engineering, National Central University, Jhongli, Taiwan 320 § Department of Physics National, Central University, Jhongli, Taiwan 320 ‡
ABSTRACT: Controlling the droplet equilibrium location and shape on a conical fiber is essential to industrial applications such as dip-pen nanolithography. In this work, the equilibrium conformations of a drop on a vertical, conical fiber has been investigated by the finite element method, Surface Evolver simulations. Similar to the morphology of a drop on a cylinder, two different types (barrel shape and clam-shell shape) can be obtained. In the absence of gravity, the droplet moves upward (lower curvature) and the total surface energy decays as the drop ascends. Whatever the initial conformation of the drop on a conical fiber, the rising drop exhibits the clam-shell shape eventually and there is no equilibrium location. However, in the presence of gravity, the drop can stop at the equilibrium location stably. For a given contact angle, the clam-shell shape is generally favored for smaller drops but the barrel shape is dominant for larger drops. In a certain range of drop volume, the coexistence of both barrel and clam-shell shapes is observed. For large enough drops, the falling-off state is seen.
I. INTRODUCTION Manipulation of droplet motion is crucial for design and fabrication of droplet microfluidics.1−4 The typical approach associated with the spontaneous motion of droplet is to make the solid surfaces with wettability gradient and thus the droplet moves toward the region with the lower contact angle.5−12 The directional movement of droplets can also be induced on some geometrical substrate with a curvature gradient.13−22 It is observed that a liquid slug placed in a hydrophilic conical tube will move toward the thin end (small local radius or high curvature).13,16,18,19,21 The conical tube provides a curvature gradient so that the droplet motion is driven by the Laplace pressure difference between the two ends of the slug. For some applications such as dip-pen nanolithography (DPN), the drop is placed on a conical fiber which has a curvature gradient opposite to a conical tube.15,17,18,20 In DPN, migration of the drop away from the needle tip impedes the capability of the process. The ability to precisely control the droplet configuration, including the equilibrium location and shape, on a conical fiber such as an AFM tip is essential to the success of the application.17,20 The present work focuses mainly on understanding the fate of droplets on conical fibers. The morphologies of a liquid drop on a cylindrical fiber have been investigated thoroughly.23−29 Depending on the drop volume (V), fiber radius (a), and contact angle (θ), two different conformations are identified: axisymmetric barrel and asymmetric clam-shell for drops with negligible gravity. In general, clam-shell shapes exist for small droplets (V/a3) or high contact angles, while barrel shapes take place for large © XXXX American Chemical Society
droplets or low contact angles. The crossover between the two conformations indicates the existence of a regime associated with the coexistence of the two conformations in the phase diagram (V/a3 vs θ).26,29 As the gravitational effect is important, the axisymmetric barrel drop becomes asymmetric on a horizontal fiber.26 In the phase diagram, the regime of the barrel drop shrinks significantly and the regime of falling-off appears for large droplets. On a conical surface, the drop may be axisymmetrically engulfing the needle, just like a barrel shape drop on a cylinder. The drop may also stick to the side of the cone, similar to a clam-shell shape drop on a cylinder. Although the wetting behavior on a cylinder is well-understood, it is anticipated that the drop behavior on conical fibers differs significantly from that on cylindrical fibers. For a small drop of wetting liquid, its motion on a conical fiber has been experimentally studied.15 It is observed that the barrel shape drop moves toward the thick end of the fiber (large local radius or low curvature). Again, the spontaneous movement of the drop is caused by the Laplace pressure difference along the conical fiber.15 By solving the Young−Laplace equation for the droplet shape on a conical fiber with zero contact angle numerically, it is also found that the drop has the least interfacial energy at the broadest end.18 Both aforementioned experimental and theoretical results imply that the droplet cannot stop, and there is no equilibrium Received: November 21, 2014 Revised: December 28, 2014
A
DOI: 10.1021/la504552d Langmuir XXXX, XXX, XXX−XXX
Langmuir
Article
conical fibers for different drop volumes. To avoid such limitations, a reference length a is used. That is, all lengths are scaled by the reference length a, and the energy by γLGa2 in this study. Note that our reference length is related to the capillary length through the gravitational constant G. It will be described later. The dimensionless form of eq 1 is then given by
position for a barrel drop on an infinitely long conical fiber in the absence of external forces such as gravity or contact angle hysteresis. When a spherical liquid-gas interface is assumed,17 the equilibrium shape and location of a barrel drop on a conical fiber was predicted in the absence of gravitational effect. A similar conclusion was obtained by calculating the Gibbs free energies of an axisymmetric drop at different locations on a conical fiber based on extending Carroll’s equations for a drop on a cylindrical fiber.20 The presence of a minimum in the energy-position plot indicates that the drop will stop at this location on a conical fiber. If the drop shape is always restricted to axisymmetric barrel conformation, the equilibrium location may exist without the help of external forces. Nonetheless, a drop with the shape of asymmetric clam-shell on a conical fiber is frequently observed, revealing that the barrel drop may suffer the shape instability. Unfortunately, the study of the transformation between barrel and clam-shell shapes on a conical fiber and the wetting behavior associated with a clam-shell drop on a conical fiber is scarce.22 Moreover, according to previous studies of drop-on-fiber, it is anticipated that gravity plays an important role and the coexistence of the drop conformations appears for a drop on a conical fiber. In this work, the equilibrium shape of a drop on a vertical, conical fiber has been investigated by the finite element method, Surface Evolver (SE) simulations.29 Similar to the morphology of a drop on a cylinder, two different types can be identified. In the absence of gravity, the droplet moves upward (lower curvature) and the variation of the total free energy with the drop position is calculated. On the basis of this result, whether or not the equilibrium shape exists eventually is studied. In the presence of gravity, both the equilibrium shape and position are determined for a given drop morphology. An analysis of force equilibrium of a stationary barrel drop is provided as well. The influences of the capillary length and droplet volume through the gravitational constant (G) on the equilibrium morphology are explored as well. The possibility of the coexistence of the two types of the drop morphology is examined finally.
Ft = ALG − cos θCASL + G
∫ ∫ ∫v
∗
(2)
where the dimensionless gravitational constant is defined as G = (a/lc)2. The zero point of the gravitational energy is set at the tip of the conical fiber (h = 0). Note that the reference length (a) can be determined once lc and G are specified. For instance, one has a = 0.1lc if G = 0.01. For the dimensionless volume V = 40 used in this study, the volume V* = 40a3 = 0.04lc3. If lc is fixed, one can acquire different drop volumes (V*) by adjusting the gravitational constant (G). A larger value of G is corresponding to a larger drop. It is worthy of mentioning that the choice of the reference length scale does not affect our quantitative results as long as the numerical accuracy of SE is kept. In this study, we consider a drop on a vertical cone with halfcone angle α = 2.5° and the contact angle θC = 30°. The evolution of the energy minimization process starts with an initial drop shape. For a cylindrical fiber, the initial drop can be simply constructed by a cuboid engulfing a fiber for barrel and triangular prism contacting the side of a fiber.26 Although the initial barrel drop on a conical fiber can be done just like that on a cylindrical fiber, the construction of the initial clam-shell drop on a cone is much more complicated because the evolving process diverges very fast most of the time. Our approach is to adopt a clam-shell-like conformation (Figure 1d), which is
II. SIMULATION METHODS In this work, the (equilibrium) shape and position of a drop on a conical fiber can be obtained by the public domain Surface Evolver (SE) package.30 It is a finite element method of freeenergy minimization subject to constraints. The total free energy (Ft*) of a drop on a conical fiber can be expressed as26,29 Ft * = γLGALG* − γLG cos θCASL∗ +
∫ ∫ ∫ v (z) dV
Figure 1. SE evolution process for a drop placed at the lower end of the conical fiber with the barrel conformation in the absence of gravity (α = 2.5°, V = 40, and θC = 30°). The conformation change occurs eventually (from barrel to clam-shell shape).
acquired from an upward moving barrel drop under the condition G = 0. Figure 1 shows the spontaneous process for the transition from barrel to clam-shell. Note that the initial clam-shell drop is actually obtained after one cuts the thread connecting the deformed clam-shell in Figure 1d. Compared to a drop on a cylindrical fiber, it takes much more time to obtain an equilibrium state for that on a conical fiber. In general, it takes about 1−2 weeks to reach an equilibrium barrel drop but 4 weeks for the clam-shell drop on an Intel Core i7-3770 processor.
(ρgz∗) dV ∗ (1)
where γLG is the surface tension. ALG* and ASL* represent the liquid-gas and solid-liquid interfacial areas, respectively. The gravitational energy is given in the third term, and the case without gravity corresponds to g = 0. The contact angle on the cone is θC, which is related to the interfacial tensions by the Young’s equation. For a conical fiber, there is no characteristic length. There are two length scales in the system, the capillary and the radius of the droplet. In order to study the effect of the drop volume, the radius of the droplet is not a good choice. If the capillary length lc = (γLG/ρg)1/2 is used, the numeric value of the free energy is very small. Moreover, it is very troublesome to perform SE on
III. RESULTS AND DISCUSSION A. Drop on Hydrophilic Conical Fiber without Gravity. A barrel drop on a conical fiber moves toward the region of lower curvature spontaneously. While some previous studies B
DOI: 10.1021/la504552d Langmuir XXXX, XXX, XXX−XXX
Langmuir
Article
show that the barrel drop possesses the minimum free energy at the broadest end of a conical fiber,15,18 the others suggest that the barrel drop should stop at a certain location on a conical fiber.17,20 The tendency of the upward motion can be observed through SE simulations by placing a drop with the barrel conformation at the lower end of the conical fiber initially, as shown in Figure 1a. As the iteration proceeds, the drop moves up along the conical fiber and the drop shape remains axissymmetrical (Figure 1b). However, as the local radius of the conical fiber becomes comparable to the drop size, the drop exhibits an asymmetrical shape (Figure 1c). The deformation of the barrel shape grows even more significantly as the center of mass of the drop ascends. Eventually, the barrel shape vanishes and a drop with the clam-shell conformation appears on the conical fiber (Figure 1e). Obviously, the barrel shape is not stable in the absence of gravity according to the evolution of free energy. Figure 2 illustrates the evolution process in the plot of the total surface free energy (Ft) versus the vertical position of the
Figure 3. SE evolution process for a clam-shell drop on the conical fiber without gravity (α = 2.5°, V = 40, and θC = 30°).
Figure 2, panels c and d), the total surface free energy continues decaying, while the clam-shell drop keeps ascending (see Figure 3, panels e−h). Similar to the rising process of the barrel drop, FLG of the clam-shell drop grows but FSL declines upon ascending, as shown in the inset of Figure 4. Since the decrease
Figure 4. Variation of the surface energy (Ft) with the position of the clam-shell drop (h) during the SE evolution process.
Figure 2. Variation of the surface energy (Ft) with the drop position (h) during the SE evolution process for a barrel drop placed initially at the lower end of the conical fiber. The conformation change occurs between (c and d).
of FSL dominates over the increase of FLG, Ft decays with increasing h. According to SE simulations for a drop on a flat plane under the same condition of V and θC, one has Ft′ = 13.2, FLG′ = 68.5, and FSL′ = −55.3 for the region of zero curvature. Since Ft of the clam-shell drop on the conical fiber is much greater than Ft′, it is anticipated that the clam-shell drop will ascend endlessly toward the region of zero curvature in the absence of gravity and contact angle hysteresis. B. Drop on a Hydrophilic Conical Fiber with Gravity. In this section, the gravitational force is taken into account to investigate its effect on the equilibrium drop conformation on a hydrophilic conical fiber. Figure 5 illustrates the SE evolution process of a barrel drop, which is initially placed at the lower end of the conical fiber, as shown in Figure 5a. In order to compare the results with and without gravity, all the parameters associated with Figure 5 are the same as those of Figure 1. Moreover, the barrel shape in Figure 5a is the same as that in Figure 1a. Subject to the downward gravitational field with G = 0.01, the barrel drop moves up along the conical fiber as the iteration process proceeds, as illustrated in Figure 5b. Eventually, the upward movement stops and the drop shape remains as an axis-symmetrical barrel conformation, as demonstrated in Figure 5c. Figure 6 illustrates the plot of Ft versus h for the evolution process of the upward movement of a barrel drop under gravity. Just like the case without gravity, Ft continues decaying, while h keeps rising (Figure 6, panels a and b). However, the decrease
center of mass (h). A drop with the volume V = 40 is deposited on the conical fiber with half-cone angle α = 2.5°. The contact angle is set as θC = 30°. When the drop evolves from the barrel (Figure 1, panels a−c) to clam-shell (1e) conformation, Ft continues decaying, while h keeps rising. Upon ascending of the drop, the liquid-gas surface energy FLG rises, but the solid-liquid surface energy FSL declines, as shown in the inset of Figure 2. Since the decrease of FSL wins over the increase of FLG, Ft decays with increasing h. Note that the crossover from deformed barrel (Figure 1c) to deformed clam-shell with a connected thread (Figure 1d)26,29 corresponds to a sudden jump of Ft associated with a drop in FLG and a rise in FSL. Evidently, upon moving toward the region of lower curvature of the conical fiber, the surface free energy declines. Ft decreases even further by the shape transformation from barrel to clamshell when the local radius of the conical fiber is large enough.26 The drop with the barrel shape always moves up but eventually turns into the one with the clam-shell shape. However, just like the barrel drop does, the clam-shell drop on the conical fiber keeps rising. Figure 3 illustrates the evolution process after the shape change from barrel to clam-shell. The clam-shell drop also moves toward the region of lower curvature of the conical fiber. Such ascending tendency can be revealed in the plot of Ft versus h. After the fall in Ft (see C
DOI: 10.1021/la504552d Langmuir XXXX, XXX, XXX−XXX
Langmuir
Article
Figure 5. SE evolution process for a barrel shape drop with gravity (G = 0.01, α = 2.5°, V = 40, and θC = 30°). Finally the drop stops at an equilibrium location. Figure 7. SE evolution process for a clam-shell drop with gravity (G = 0.01, α = 2.5°, V = 40, and θC = 30°). The drop stops at an equilibrium location eventually.
(Figure 7, panels a and b) as the iteration process proceeds. Finally, the upward movement ceases at an equilibrium position, as depicted in Figure 7c. The variation of the free energy, including FLG, FSL, and Fg, with the vertical position h is shown in the inset of Figure 8 for the upward process of a clam-
Figure 6. Variation of the total free energy (Ft) with the drop position (h) during the SE evolution process for a barrel drop placed initially at the lower end of the conical fiber. Ft arrives at the minimum (c) at the equilibrium location.
of Ft with increasing h becomes smaller gradually. Eventually, the upward movement stops and Ft remains essentially unchanged (Figure 6c). That is, the barrel drop reaches its equilibrium position. During the evolution process, the liquidgas surface energy FLG and the gravitational energy Fg rise, but the solid-liquid surface energy FSL declines, as shown in the inset of Figure 6. Note that both the liquid-gas area and the solid-liquid area grow upon upward movement. As long as the gain of FSL dominates over the loss of FLG and Fg, the barrel drop keeps rising. The balance of FSL gain with FLG and Fg loss corresponds to the local minimum of the total free energy. In comparison with the case without gravity, an equilibrium barrel conformation can occur at a particular position due to gravity. The force balance around a stationary barrel drop (Figure 5c) has been performed to examine the equilibrium state. Scaled by γLGlc, the vertical components of the capillary forces [2πr± cos(θ ∓ α/2)] associated with the upper and lower ends are 0.72 and −0.55, respectively. r± (0.1296 and 0.1037) represent the radii associated with the upper and lower ends. The net capillary force always pushes the barrel drop upward and exceeds the weight of drop (−0.04). However, the force of reaction exerted by the conical fiber on the drop, which balances the hydrostatic pressure (Pc) along the fiber surface (−∫ PcndA·g/g), provides an additional downward force (−0.13) to resist the upward motion of the drop. When a clam-shell drop is placed on a conical fiber initially, the SE evolution process in the presence of gravity is shown in Figure 7. Similar to the case of a barrel drop, the clam-shell drop subject to G = 0.01 also ascends along the conical fiber
Figure 8. Variation of the total free energy (Ft) with the drop position (h) during the SE evolution process for a clam-shell drop. Ft at the minimum (c) at the equilibrium location.
shell drop under gravity. Their general trends are similar to those of a barrel drop. Figure 8 also shows that Ft decreases with increasing h and becomes fluctuating around a mean value as the clam-shell drop reaches its equilibrium position. Note that intrinsic perturbations (vertex averaging) are regularly provided to homogenize the mesh size so that the free energy always fluctuates during simulations. The amplitudes of freeenergy fluctuations are similar for both clam-shell and barrel shapes. Consequently, the convergence of iterations corresponds to the condition that both h and mean Ft of the drop remains unchanged. Because of the asymmetric nature of the clam-shell drop, it takes many more iteration steps to reach equilibrium than the barrel drop. Therefore, the initial position (Figure 7a) is chosen not far from the equilibrium position. It is worth mentioning that both barrel and clam-shell conformations can coexist subject to the gravitational field with G = 0.01. D
DOI: 10.1021/la504552d Langmuir XXXX, XXX, XXX−XXX
Langmuir
Article
Compared to the barrel drop, the vertical position h of equilibrium clam-shell drop is higher but the total free energy Ft is slightly lower. In the absence of contact angle hysteresis, the drop on a conical fiber always moves toward the thicker end. For a vertical, conical fiber as shown in Figure 1, the gravitational force is the only one to resist the upward motion of the drop driven by the capillary force. That is, the equilibrium position for a drop on a conical fiber can exist due to gravity. C. Drop Morphology and Coexistence State. The stable state of a drop on a hydrophilic conical fiber can be acquired only when gravity resists the tendency of the upward motion. Evidently, the equilibrium drop position (h) depends on the gravitational constant (G) which corresponds to the drop volume. As G is small, the equilibrium position is anticipated to be away from the tip of the conical fiber. If the drop size is small compared to the local radius of the fiber, only the clam-shell drop can exist.26 On the contrary, as G is large, the equilibrium position is expected to be near the tip of the fiber. If the drop size is much larger than the local radius, the barrel drop is much more favorable. Figure 9 shows the variation of the equilibrium drop position with a different gravitational constant (G). As illustrated in Figure 9a, only the clam-shell drop is observed at G = 0.008, but only the barrel drop is seen at G = 0.025. In fact, as indicated in Figure 9b, the barrel drop is absent and the clam-shell drop dominates for G ≤ 0.008. In contrast, the clamshell drop cannot be seen and the barrel drop is dominant for G ≥ 0.025. However, the barrel drop cannot stay on the conical fiber anymore but falls down when G is further increased to exceed 0.027. That is, both barrel drop and clam-shell drop cannot exist on the conical fiber when the gravitational contribution corresponding to large G overwhelms. In the absence of gravity, the drop on a cylindrical fiber can have two morphologies for a given contact angle and drop volume. That is, both barrel and clam-shell drop can coexist on a specified radius of a cylinder. Similarly, the coexistent state can also be observed on a conical fiber for a given contact angle and drop volume in the presence of gravity. Figure 9a shows both drop morphologies at G = 0.009. As G is increased to 0.01 and 0.02, both shapes still coexist but the equilibrium position (h) of the drop decreases. According to Figure 9b, the equilibrium position of the clam-shell drop is always higher than that of the barrel drop at a given G. The total free energy of the drop (Ft) varies with G and is shown in Figure 9c. Ft of both morphologies grows with increasing G because of the increment of FSL associated with the decrease of h. In the coexistent regime (0.009 ≤ G ≤ 0.024), one can compare the total free energy between barrel drop and clam-shell drop and a crossover point is clearly seen. When 0.009 ≤ G < 0.017, clamshell drop has lower Ft than barrel drop. As G is increased, however, Ft of clam-shell drop grows faster than that of barrel drop and barrel drop has lower Ft than clam-shell drop when 0.017 ≤ G ≤ 0.024. Note that regardless of the difference of Ft, both morphologies are numerically stable in the coexistent regime. The numerical instability appears for barrel drop at G ≤ 0.008 and for the clam-shell drop at G ≥ 0.025. Although the coexistent state is seen for α = 2.5° and θC = 30°, the qualitatively similar results are anticipated for a certain range of α and θC. Nonetheless, the barrel drop is favored for smaller α and θC, but the clam-shell one is dominant for larger α and θC.
Figure 9. (a) Influence of the gravitational constant (G) on the equilibrium shape of the drop. The coexistence state is observed for a range of G. (b) Influence of the gravitational constant (G) on the equilibrium position of the drop. The coexistence state is observed for a range of G. (c) Variation of the total free energy with the gravitational constant. A crossover between barrel and clam-shell conformation is seen in the coexistent regime.
IV. CONCLUSIONS Controlling the equilibrium location and shape of a drop on a conical fiber is crucial for industrial applications such as dip-pen nanolithography. The droplet morphology and whether or not it is equilibrium in a drop-on-conical fiber system depend on the half-cone angle (α), contact angle (θC), and gravitational constant G = (a/lc)2. The effect of varying the gravitational constant G corresponds to the change of the drop volume. The change of G indicates the variation of a or lc. While the variation E
DOI: 10.1021/la504552d Langmuir XXXX, XXX, XXX−XXX
Langmuir
Article
(3) Zhao, B.; Moore, J. S.; Beebe, D.L. Surface-Directed Liquid Flow Inside Microchannels. Science 2001, 291, 1023−1026. (4) Yue, R. F.; Wu, J. G.; Zeng, X. F.; Kang, M.; Liu, L. T. Demonstration of Four Fundamental Operations of Liquid Droplets for Digital Microfluidic Systems Based on an Electrowetting-onDielectric Actuator. Chin. Phys. Lett. 2006, 23, 2303. (5) Ichimura, K.; Oh, S. K.; Nakagawa, M. Light-Driven Motion of Liquids on a Photoresponsive Surface. Science 2000, 288, 1624−1626. (6) Daniel, S.; Chaudhury, M. K.; Chen, J. C. Fast Drop Movements Resulting from the Phase Change on a Gradient Surface. Science 2001, 291, 633−636. (7) Stone, H. A.; Stroock, A. D.; Adjari, A. Engineering Flows in Small Devices: Microfluidics Toward a Lab-on-a-Chip. Annu. Rev. Fluid Mech. 2004, 36, 381−411. (8) De Gennes, P. G.; Brochard-Wyart, F.; Quéré, D. Capillarity and Wetting Phenomena, Drops, Bubbles, Pears, Waves; Springer: New York, 2004. (9) Lazar, P.; Riegler, H. Reversible Self-Propelled Droplet Movement: A New Driving Mechanism. Phys. Rev. Lett. 2005, 95, 136103. (10) Sumino, Y.; Magome, N.; Hamada, T.; Yoshikawa, K. SelfRunning Droplet: Emergence of Regular Motion from Nonequilibrium Noise. Phys. Rev. Lett. 2005, 94, 068301. (11) Zheng, Y.; Bai, H.; Huang, Z.; Tian, X.; Nie, F. Q.; Zhao, Y.; Zhai, J.; Jiang, L. Directional Water Collection on Wetted Spider Silk. Nature 2010, 463, 640−643. (12) Hancock, M. J.; He, J.; Mano, J. F.; Khademhosseini, A. SurfaceTension-Driven Gradient Generation in a Fluid Stripe for Bench-Top and Microwell Applications. Small 2011, 7, 892−901. (13) Bouasse, H. Capillarite, Phenomenes Superficiels; Delagrave: Paris, 1924. (14) Bico, J.; Quéré, D. Self-Propelling Slugs. J. Fluid Mech. 2002, 467, 101−127. (15) Lorenceau, É.; Quéré, D. Drops on a Conical Wire. J. Fluid Mech. 2004, 510, 29−45. (16) Darhuber, A. A.; Troian, S. M. Principles of Microfluidic Actuation by Modulation of Surface Stresses. Annu. Rev. Fluid Mech. 2005, 37, 425−455. (17) Hanumanthu, R.; Stebe, K. J. Equilibrium shapes and locations of axisymmetric, liquid drops on conical, solid surfaces. Colloids Surf., A 2006, 282, 227−239. (18) Liu, J. L.; Xia, R.; Li, B. W.; Feng, X. Q. Directional Motion of Droplets in a Conical Tube or on a Conical Fibre. Chin. Phys. Lett. 2007, 24, 3210−3213. (19) Renvoise, P.; Bush, J. W. M.; Prakash, M.; Quéré, D. Drop propulsion in tapered tubes. Europhys. Lett. 2009, 86, 64003. (20) Michielsen, S.; Zhang, J.; Du, J.; Lee, H. J. Gibbs Free Energy of Liquid Drops on Conical Fibers. Langmuir 2011, 27, 11867−11872. (21) Wang, Z.; Chang, C. C.; Hong, S. J.; Sheng, Y. J.; Tsao, H. K. Trapped Liquid Drop in a Microchannel: Multiple Stable States. Phys. Rev. E 2013, 87, 062401. (22) Lv, C.; Chen, C.; Chuang, Y. C.; Tseng, F. G.; Yin, Y.; Grey, F.; Zheng, Q. Substrate Curvature Gradient Drives Rapid Droplet Motion. Phys. Rev. Lett. 2014, 113, 026101. (23) 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, 248−250. (24) McHale, G.; Newton, M. I.; Carroll, B. J. The Shape and Stability of Small Liquid Drops on Fibers. Oil Gas Sci. Technol. 2001, 56, 47−54. (25) McHale, G.; Newton, M. I. Global Geometry and the Equilibrium Shapes of Liquid Drops on Fibers. Colloids Surf., A 2002, 206, 79−86. (26) Chou, T. H.; Hong, S. J.; Liang, Y. E.; Tsao, H. K.; Sheng, Y. J. Equilibrium Phase Diagram of Drop-on-Fiber: Coexistent States and Gravity Effect. Langmuir 2011, 27, 3685−3692. (27) Eral, H. B.; de Ruiter, J.; de Ruiter, R.; Oh, J. M.; Semprebon, C.; Brinkmann, M.; Mugele, F. Drops on Functional Fibers: From Barrels to Clamshells and Back. Soft Matter 2011, 7, 5138−5143.
of the former (a) describes the change of the drop volume, the variation of the latter [lc = (γLG/ρg)1/2] depicts the change of surface tension or liquid density. In this work, the equilibrium conformations and positions of a drop on a vertical, conical fiber have been investigated by the finite element method, Surface Evolver simulations for different gravitational constant (G) of α = 2.5° and θC = 30°. Similar to the morphology of a drop on a cylindrical fiber, two different types can be obtained: barrel shape and clam-shell shape. In the absence of gravity, a barrel drop initially put at the lower end of a conical fiber (high curvature) moves upwardly toward lower curvature. The total surface energy decreases as the drop rises. When the local radius of the conical fiber becomes comparable to the drop size, the barrel drop is deformed to an asymmetrical shape. Eventually, the barrel shape drop becomes unstable and transforms to the clam-shell drop, which continues to ascend. That is, one is unable to obtain an equilibrium location of the drop. Whatever the initial conformation of the drop on a conical fiber is, the rising drop exhibits the clam-shell shape eventually and there is no equilibrium location without gravity and hysteresis effects. However, in the presence of gravity such as G = 0.01, the drop can stop at an equilibrium location stably. The tendency of the upward motion can be resisted by the gravity force. Both equilibrium barrel and clam-shell drops can be acquired. Dependent on the value of G (drop volume), the equilibrium drop position (h) varies. Note that because of the lack of the characteristic length, varying G corresponds to the change of the drop volume, as described in Simulation Methods. Four regimes are identified: clam-shell only, barrel only, coexistence of clam-shell and barrel, and falling-off. As G is small, the equilibrium drop position is away from the tip of the conical fiber. If the drop size is small compared to the local radius of the fiber, only the clam-shell drop can be observed (clam-shell only regime). In contrast, as G is large, the equilibrium position is near the tip of the fiber. If the drop size is much larger than the local radius, only the barrel drop can be seen (barrel only regime). In a certain range of G, both barrel and clam-shell conformations can exist (coexistent regime). In this regime, regardless of the difference of the total free energy, both morphologies are numerically stable, and an energy crossover between the two conformations appears. At a given G, the equilibrium position of the clam-shell drop is always higher than that of the barrel drop. For large enough drops, the drop falls off from the conical fiber (falling-off regime).
■
AUTHOR INFORMATION
Corresponding Authors
*E-mail: [email protected]. *E-mail: [email protected]. Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS Y.-J.S. and H.-K.T. thank the Ministry of Science and Technology of Taiwan for financial support.
■
REFERENCES
(1) Gilet, T.; Terwagne, D.; Vandewalle, N. Digital Microfluidics on a Wire. Appl. Phys. Lett. 2009, 95, 014106. (2) Gilet, T.; Terwagne, D.; Vandewalle, N. Droplets Sliding on Fibres. Eur. Phys. J. E: Soft Matter Biol. Phys. 2010, 31, 253−262. F
DOI: 10.1021/la504552d Langmuir XXXX, XXX, XXX−XXX
Langmuir
Article
(28) de Ruiter, R.; de Ruiter, J.; Eral, H. B.; Semprebon, C.; Brinkmann, M.; Mugele, F. Buoyant Droplets on Functional Fibers. Langmuir 2012, 28, 13300−13306. (29) Liang, Y. E.; Chang, C. C.; Tsao, H. K.; Sheng, Y. J. An Equilibrium Phase Diagram of Drops at the Bottom of a Fiber Standing on Superhydrophobic Flat Surfaces. Soft Matter 2013, 9, 9867−9875. (30) Brakke, K. The Surface Evolver. Exp. Math. 1992, 1, 141−165.
G
DOI: 10.1021/la504552d Langmuir XXXX, XXX, XXX−XXX
Drops on Hydrophilic Conical Fibers: Gravity Effect and Coexistent States Yu-En Liang,† Heng-Kwong Tsao,*,‡,§ and Yu-Jane Sheng*,† †
Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 106 Department of Chemical and Materials Engineering, National Central University, Jhongli, Taiwan 320 § Department of Physics National, Central University, Jhongli, Taiwan 320 ‡
ABSTRACT: Controlling the droplet equilibrium location and shape on a conical fiber is essential to industrial applications such as dip-pen nanolithography. In this work, the equilibrium conformations of a drop on a vertical, conical fiber has been investigated by the finite element method, Surface Evolver simulations. Similar to the morphology of a drop on a cylinder, two different types (barrel shape and clam-shell shape) can be obtained. In the absence of gravity, the droplet moves upward (lower curvature) and the total surface energy decays as the drop ascends. Whatever the initial conformation of the drop on a conical fiber, the rising drop exhibits the clam-shell shape eventually and there is no equilibrium location. However, in the presence of gravity, the drop can stop at the equilibrium location stably. For a given contact angle, the clam-shell shape is generally favored for smaller drops but the barrel shape is dominant for larger drops. In a certain range of drop volume, the coexistence of both barrel and clam-shell shapes is observed. For large enough drops, the falling-off state is seen.
I. INTRODUCTION Manipulation of droplet motion is crucial for design and fabrication of droplet microfluidics.1−4 The typical approach associated with the spontaneous motion of droplet is to make the solid surfaces with wettability gradient and thus the droplet moves toward the region with the lower contact angle.5−12 The directional movement of droplets can also be induced on some geometrical substrate with a curvature gradient.13−22 It is observed that a liquid slug placed in a hydrophilic conical tube will move toward the thin end (small local radius or high curvature).13,16,18,19,21 The conical tube provides a curvature gradient so that the droplet motion is driven by the Laplace pressure difference between the two ends of the slug. For some applications such as dip-pen nanolithography (DPN), the drop is placed on a conical fiber which has a curvature gradient opposite to a conical tube.15,17,18,20 In DPN, migration of the drop away from the needle tip impedes the capability of the process. The ability to precisely control the droplet configuration, including the equilibrium location and shape, on a conical fiber such as an AFM tip is essential to the success of the application.17,20 The present work focuses mainly on understanding the fate of droplets on conical fibers. The morphologies of a liquid drop on a cylindrical fiber have been investigated thoroughly.23−29 Depending on the drop volume (V), fiber radius (a), and contact angle (θ), two different conformations are identified: axisymmetric barrel and asymmetric clam-shell for drops with negligible gravity. In general, clam-shell shapes exist for small droplets (V/a3) or high contact angles, while barrel shapes take place for large © XXXX American Chemical Society
droplets or low contact angles. The crossover between the two conformations indicates the existence of a regime associated with the coexistence of the two conformations in the phase diagram (V/a3 vs θ).26,29 As the gravitational effect is important, the axisymmetric barrel drop becomes asymmetric on a horizontal fiber.26 In the phase diagram, the regime of the barrel drop shrinks significantly and the regime of falling-off appears for large droplets. On a conical surface, the drop may be axisymmetrically engulfing the needle, just like a barrel shape drop on a cylinder. The drop may also stick to the side of the cone, similar to a clam-shell shape drop on a cylinder. Although the wetting behavior on a cylinder is well-understood, it is anticipated that the drop behavior on conical fibers differs significantly from that on cylindrical fibers. For a small drop of wetting liquid, its motion on a conical fiber has been experimentally studied.15 It is observed that the barrel shape drop moves toward the thick end of the fiber (large local radius or low curvature). Again, the spontaneous movement of the drop is caused by the Laplace pressure difference along the conical fiber.15 By solving the Young−Laplace equation for the droplet shape on a conical fiber with zero contact angle numerically, it is also found that the drop has the least interfacial energy at the broadest end.18 Both aforementioned experimental and theoretical results imply that the droplet cannot stop, and there is no equilibrium Received: November 21, 2014 Revised: December 28, 2014
A
DOI: 10.1021/la504552d Langmuir XXXX, XXX, XXX−XXX
Langmuir
Article
conical fibers for different drop volumes. To avoid such limitations, a reference length a is used. That is, all lengths are scaled by the reference length a, and the energy by γLGa2 in this study. Note that our reference length is related to the capillary length through the gravitational constant G. It will be described later. The dimensionless form of eq 1 is then given by
position for a barrel drop on an infinitely long conical fiber in the absence of external forces such as gravity or contact angle hysteresis. When a spherical liquid-gas interface is assumed,17 the equilibrium shape and location of a barrel drop on a conical fiber was predicted in the absence of gravitational effect. A similar conclusion was obtained by calculating the Gibbs free energies of an axisymmetric drop at different locations on a conical fiber based on extending Carroll’s equations for a drop on a cylindrical fiber.20 The presence of a minimum in the energy-position plot indicates that the drop will stop at this location on a conical fiber. If the drop shape is always restricted to axisymmetric barrel conformation, the equilibrium location may exist without the help of external forces. Nonetheless, a drop with the shape of asymmetric clam-shell on a conical fiber is frequently observed, revealing that the barrel drop may suffer the shape instability. Unfortunately, the study of the transformation between barrel and clam-shell shapes on a conical fiber and the wetting behavior associated with a clam-shell drop on a conical fiber is scarce.22 Moreover, according to previous studies of drop-on-fiber, it is anticipated that gravity plays an important role and the coexistence of the drop conformations appears for a drop on a conical fiber. In this work, the equilibrium shape of a drop on a vertical, conical fiber has been investigated by the finite element method, Surface Evolver (SE) simulations.29 Similar to the morphology of a drop on a cylinder, two different types can be identified. In the absence of gravity, the droplet moves upward (lower curvature) and the variation of the total free energy with the drop position is calculated. On the basis of this result, whether or not the equilibrium shape exists eventually is studied. In the presence of gravity, both the equilibrium shape and position are determined for a given drop morphology. An analysis of force equilibrium of a stationary barrel drop is provided as well. The influences of the capillary length and droplet volume through the gravitational constant (G) on the equilibrium morphology are explored as well. The possibility of the coexistence of the two types of the drop morphology is examined finally.
Ft = ALG − cos θCASL + G
∫ ∫ ∫v
∗
(2)
where the dimensionless gravitational constant is defined as G = (a/lc)2. The zero point of the gravitational energy is set at the tip of the conical fiber (h = 0). Note that the reference length (a) can be determined once lc and G are specified. For instance, one has a = 0.1lc if G = 0.01. For the dimensionless volume V = 40 used in this study, the volume V* = 40a3 = 0.04lc3. If lc is fixed, one can acquire different drop volumes (V*) by adjusting the gravitational constant (G). A larger value of G is corresponding to a larger drop. It is worthy of mentioning that the choice of the reference length scale does not affect our quantitative results as long as the numerical accuracy of SE is kept. In this study, we consider a drop on a vertical cone with halfcone angle α = 2.5° and the contact angle θC = 30°. The evolution of the energy minimization process starts with an initial drop shape. For a cylindrical fiber, the initial drop can be simply constructed by a cuboid engulfing a fiber for barrel and triangular prism contacting the side of a fiber.26 Although the initial barrel drop on a conical fiber can be done just like that on a cylindrical fiber, the construction of the initial clam-shell drop on a cone is much more complicated because the evolving process diverges very fast most of the time. Our approach is to adopt a clam-shell-like conformation (Figure 1d), which is
II. SIMULATION METHODS In this work, the (equilibrium) shape and position of a drop on a conical fiber can be obtained by the public domain Surface Evolver (SE) package.30 It is a finite element method of freeenergy minimization subject to constraints. The total free energy (Ft*) of a drop on a conical fiber can be expressed as26,29 Ft * = γLGALG* − γLG cos θCASL∗ +
∫ ∫ ∫ v (z) dV
Figure 1. SE evolution process for a drop placed at the lower end of the conical fiber with the barrel conformation in the absence of gravity (α = 2.5°, V = 40, and θC = 30°). The conformation change occurs eventually (from barrel to clam-shell shape).
acquired from an upward moving barrel drop under the condition G = 0. Figure 1 shows the spontaneous process for the transition from barrel to clam-shell. Note that the initial clam-shell drop is actually obtained after one cuts the thread connecting the deformed clam-shell in Figure 1d. Compared to a drop on a cylindrical fiber, it takes much more time to obtain an equilibrium state for that on a conical fiber. In general, it takes about 1−2 weeks to reach an equilibrium barrel drop but 4 weeks for the clam-shell drop on an Intel Core i7-3770 processor.
(ρgz∗) dV ∗ (1)
where γLG is the surface tension. ALG* and ASL* represent the liquid-gas and solid-liquid interfacial areas, respectively. The gravitational energy is given in the third term, and the case without gravity corresponds to g = 0. The contact angle on the cone is θC, which is related to the interfacial tensions by the Young’s equation. For a conical fiber, there is no characteristic length. There are two length scales in the system, the capillary and the radius of the droplet. In order to study the effect of the drop volume, the radius of the droplet is not a good choice. If the capillary length lc = (γLG/ρg)1/2 is used, the numeric value of the free energy is very small. Moreover, it is very troublesome to perform SE on
III. RESULTS AND DISCUSSION A. Drop on Hydrophilic Conical Fiber without Gravity. A barrel drop on a conical fiber moves toward the region of lower curvature spontaneously. While some previous studies B
DOI: 10.1021/la504552d Langmuir XXXX, XXX, XXX−XXX
Langmuir
Article
show that the barrel drop possesses the minimum free energy at the broadest end of a conical fiber,15,18 the others suggest that the barrel drop should stop at a certain location on a conical fiber.17,20 The tendency of the upward motion can be observed through SE simulations by placing a drop with the barrel conformation at the lower end of the conical fiber initially, as shown in Figure 1a. As the iteration proceeds, the drop moves up along the conical fiber and the drop shape remains axissymmetrical (Figure 1b). However, as the local radius of the conical fiber becomes comparable to the drop size, the drop exhibits an asymmetrical shape (Figure 1c). The deformation of the barrel shape grows even more significantly as the center of mass of the drop ascends. Eventually, the barrel shape vanishes and a drop with the clam-shell conformation appears on the conical fiber (Figure 1e). Obviously, the barrel shape is not stable in the absence of gravity according to the evolution of free energy. Figure 2 illustrates the evolution process in the plot of the total surface free energy (Ft) versus the vertical position of the
Figure 3. SE evolution process for a clam-shell drop on the conical fiber without gravity (α = 2.5°, V = 40, and θC = 30°).
Figure 2, panels c and d), the total surface free energy continues decaying, while the clam-shell drop keeps ascending (see Figure 3, panels e−h). Similar to the rising process of the barrel drop, FLG of the clam-shell drop grows but FSL declines upon ascending, as shown in the inset of Figure 4. Since the decrease
Figure 4. Variation of the surface energy (Ft) with the position of the clam-shell drop (h) during the SE evolution process.
Figure 2. Variation of the surface energy (Ft) with the drop position (h) during the SE evolution process for a barrel drop placed initially at the lower end of the conical fiber. The conformation change occurs between (c and d).
of FSL dominates over the increase of FLG, Ft decays with increasing h. According to SE simulations for a drop on a flat plane under the same condition of V and θC, one has Ft′ = 13.2, FLG′ = 68.5, and FSL′ = −55.3 for the region of zero curvature. Since Ft of the clam-shell drop on the conical fiber is much greater than Ft′, it is anticipated that the clam-shell drop will ascend endlessly toward the region of zero curvature in the absence of gravity and contact angle hysteresis. B. Drop on a Hydrophilic Conical Fiber with Gravity. In this section, the gravitational force is taken into account to investigate its effect on the equilibrium drop conformation on a hydrophilic conical fiber. Figure 5 illustrates the SE evolution process of a barrel drop, which is initially placed at the lower end of the conical fiber, as shown in Figure 5a. In order to compare the results with and without gravity, all the parameters associated with Figure 5 are the same as those of Figure 1. Moreover, the barrel shape in Figure 5a is the same as that in Figure 1a. Subject to the downward gravitational field with G = 0.01, the barrel drop moves up along the conical fiber as the iteration process proceeds, as illustrated in Figure 5b. Eventually, the upward movement stops and the drop shape remains as an axis-symmetrical barrel conformation, as demonstrated in Figure 5c. Figure 6 illustrates the plot of Ft versus h for the evolution process of the upward movement of a barrel drop under gravity. Just like the case without gravity, Ft continues decaying, while h keeps rising (Figure 6, panels a and b). However, the decrease
center of mass (h). A drop with the volume V = 40 is deposited on the conical fiber with half-cone angle α = 2.5°. The contact angle is set as θC = 30°. When the drop evolves from the barrel (Figure 1, panels a−c) to clam-shell (1e) conformation, Ft continues decaying, while h keeps rising. Upon ascending of the drop, the liquid-gas surface energy FLG rises, but the solid-liquid surface energy FSL declines, as shown in the inset of Figure 2. Since the decrease of FSL wins over the increase of FLG, Ft decays with increasing h. Note that the crossover from deformed barrel (Figure 1c) to deformed clam-shell with a connected thread (Figure 1d)26,29 corresponds to a sudden jump of Ft associated with a drop in FLG and a rise in FSL. Evidently, upon moving toward the region of lower curvature of the conical fiber, the surface free energy declines. Ft decreases even further by the shape transformation from barrel to clamshell when the local radius of the conical fiber is large enough.26 The drop with the barrel shape always moves up but eventually turns into the one with the clam-shell shape. However, just like the barrel drop does, the clam-shell drop on the conical fiber keeps rising. Figure 3 illustrates the evolution process after the shape change from barrel to clam-shell. The clam-shell drop also moves toward the region of lower curvature of the conical fiber. Such ascending tendency can be revealed in the plot of Ft versus h. After the fall in Ft (see C
DOI: 10.1021/la504552d Langmuir XXXX, XXX, XXX−XXX
Langmuir
Article
Figure 5. SE evolution process for a barrel shape drop with gravity (G = 0.01, α = 2.5°, V = 40, and θC = 30°). Finally the drop stops at an equilibrium location. Figure 7. SE evolution process for a clam-shell drop with gravity (G = 0.01, α = 2.5°, V = 40, and θC = 30°). The drop stops at an equilibrium location eventually.
(Figure 7, panels a and b) as the iteration process proceeds. Finally, the upward movement ceases at an equilibrium position, as depicted in Figure 7c. The variation of the free energy, including FLG, FSL, and Fg, with the vertical position h is shown in the inset of Figure 8 for the upward process of a clam-
Figure 6. Variation of the total free energy (Ft) with the drop position (h) during the SE evolution process for a barrel drop placed initially at the lower end of the conical fiber. Ft arrives at the minimum (c) at the equilibrium location.
of Ft with increasing h becomes smaller gradually. Eventually, the upward movement stops and Ft remains essentially unchanged (Figure 6c). That is, the barrel drop reaches its equilibrium position. During the evolution process, the liquidgas surface energy FLG and the gravitational energy Fg rise, but the solid-liquid surface energy FSL declines, as shown in the inset of Figure 6. Note that both the liquid-gas area and the solid-liquid area grow upon upward movement. As long as the gain of FSL dominates over the loss of FLG and Fg, the barrel drop keeps rising. The balance of FSL gain with FLG and Fg loss corresponds to the local minimum of the total free energy. In comparison with the case without gravity, an equilibrium barrel conformation can occur at a particular position due to gravity. The force balance around a stationary barrel drop (Figure 5c) has been performed to examine the equilibrium state. Scaled by γLGlc, the vertical components of the capillary forces [2πr± cos(θ ∓ α/2)] associated with the upper and lower ends are 0.72 and −0.55, respectively. r± (0.1296 and 0.1037) represent the radii associated with the upper and lower ends. The net capillary force always pushes the barrel drop upward and exceeds the weight of drop (−0.04). However, the force of reaction exerted by the conical fiber on the drop, which balances the hydrostatic pressure (Pc) along the fiber surface (−∫ PcndA·g/g), provides an additional downward force (−0.13) to resist the upward motion of the drop. When a clam-shell drop is placed on a conical fiber initially, the SE evolution process in the presence of gravity is shown in Figure 7. Similar to the case of a barrel drop, the clam-shell drop subject to G = 0.01 also ascends along the conical fiber
Figure 8. Variation of the total free energy (Ft) with the drop position (h) during the SE evolution process for a clam-shell drop. Ft at the minimum (c) at the equilibrium location.
shell drop under gravity. Their general trends are similar to those of a barrel drop. Figure 8 also shows that Ft decreases with increasing h and becomes fluctuating around a mean value as the clam-shell drop reaches its equilibrium position. Note that intrinsic perturbations (vertex averaging) are regularly provided to homogenize the mesh size so that the free energy always fluctuates during simulations. The amplitudes of freeenergy fluctuations are similar for both clam-shell and barrel shapes. Consequently, the convergence of iterations corresponds to the condition that both h and mean Ft of the drop remains unchanged. Because of the asymmetric nature of the clam-shell drop, it takes many more iteration steps to reach equilibrium than the barrel drop. Therefore, the initial position (Figure 7a) is chosen not far from the equilibrium position. It is worth mentioning that both barrel and clam-shell conformations can coexist subject to the gravitational field with G = 0.01. D
DOI: 10.1021/la504552d Langmuir XXXX, XXX, XXX−XXX
Langmuir
Article
Compared to the barrel drop, the vertical position h of equilibrium clam-shell drop is higher but the total free energy Ft is slightly lower. In the absence of contact angle hysteresis, the drop on a conical fiber always moves toward the thicker end. For a vertical, conical fiber as shown in Figure 1, the gravitational force is the only one to resist the upward motion of the drop driven by the capillary force. That is, the equilibrium position for a drop on a conical fiber can exist due to gravity. C. Drop Morphology and Coexistence State. The stable state of a drop on a hydrophilic conical fiber can be acquired only when gravity resists the tendency of the upward motion. Evidently, the equilibrium drop position (h) depends on the gravitational constant (G) which corresponds to the drop volume. As G is small, the equilibrium position is anticipated to be away from the tip of the conical fiber. If the drop size is small compared to the local radius of the fiber, only the clam-shell drop can exist.26 On the contrary, as G is large, the equilibrium position is expected to be near the tip of the fiber. If the drop size is much larger than the local radius, the barrel drop is much more favorable. Figure 9 shows the variation of the equilibrium drop position with a different gravitational constant (G). As illustrated in Figure 9a, only the clam-shell drop is observed at G = 0.008, but only the barrel drop is seen at G = 0.025. In fact, as indicated in Figure 9b, the barrel drop is absent and the clam-shell drop dominates for G ≤ 0.008. In contrast, the clamshell drop cannot be seen and the barrel drop is dominant for G ≥ 0.025. However, the barrel drop cannot stay on the conical fiber anymore but falls down when G is further increased to exceed 0.027. That is, both barrel drop and clam-shell drop cannot exist on the conical fiber when the gravitational contribution corresponding to large G overwhelms. In the absence of gravity, the drop on a cylindrical fiber can have two morphologies for a given contact angle and drop volume. That is, both barrel and clam-shell drop can coexist on a specified radius of a cylinder. Similarly, the coexistent state can also be observed on a conical fiber for a given contact angle and drop volume in the presence of gravity. Figure 9a shows both drop morphologies at G = 0.009. As G is increased to 0.01 and 0.02, both shapes still coexist but the equilibrium position (h) of the drop decreases. According to Figure 9b, the equilibrium position of the clam-shell drop is always higher than that of the barrel drop at a given G. The total free energy of the drop (Ft) varies with G and is shown in Figure 9c. Ft of both morphologies grows with increasing G because of the increment of FSL associated with the decrease of h. In the coexistent regime (0.009 ≤ G ≤ 0.024), one can compare the total free energy between barrel drop and clam-shell drop and a crossover point is clearly seen. When 0.009 ≤ G < 0.017, clamshell drop has lower Ft than barrel drop. As G is increased, however, Ft of clam-shell drop grows faster than that of barrel drop and barrel drop has lower Ft than clam-shell drop when 0.017 ≤ G ≤ 0.024. Note that regardless of the difference of Ft, both morphologies are numerically stable in the coexistent regime. The numerical instability appears for barrel drop at G ≤ 0.008 and for the clam-shell drop at G ≥ 0.025. Although the coexistent state is seen for α = 2.5° and θC = 30°, the qualitatively similar results are anticipated for a certain range of α and θC. Nonetheless, the barrel drop is favored for smaller α and θC, but the clam-shell one is dominant for larger α and θC.
Figure 9. (a) Influence of the gravitational constant (G) on the equilibrium shape of the drop. The coexistence state is observed for a range of G. (b) Influence of the gravitational constant (G) on the equilibrium position of the drop. The coexistence state is observed for a range of G. (c) Variation of the total free energy with the gravitational constant. A crossover between barrel and clam-shell conformation is seen in the coexistent regime.
IV. CONCLUSIONS Controlling the equilibrium location and shape of a drop on a conical fiber is crucial for industrial applications such as dip-pen nanolithography. The droplet morphology and whether or not it is equilibrium in a drop-on-conical fiber system depend on the half-cone angle (α), contact angle (θC), and gravitational constant G = (a/lc)2. The effect of varying the gravitational constant G corresponds to the change of the drop volume. The change of G indicates the variation of a or lc. While the variation E
DOI: 10.1021/la504552d Langmuir XXXX, XXX, XXX−XXX
Langmuir
Article
(3) Zhao, B.; Moore, J. S.; Beebe, D.L. Surface-Directed Liquid Flow Inside Microchannels. Science 2001, 291, 1023−1026. (4) Yue, R. F.; Wu, J. G.; Zeng, X. F.; Kang, M.; Liu, L. T. Demonstration of Four Fundamental Operations of Liquid Droplets for Digital Microfluidic Systems Based on an Electrowetting-onDielectric Actuator. Chin. Phys. Lett. 2006, 23, 2303. (5) Ichimura, K.; Oh, S. K.; Nakagawa, M. Light-Driven Motion of Liquids on a Photoresponsive Surface. Science 2000, 288, 1624−1626. (6) Daniel, S.; Chaudhury, M. K.; Chen, J. C. Fast Drop Movements Resulting from the Phase Change on a Gradient Surface. Science 2001, 291, 633−636. (7) Stone, H. A.; Stroock, A. D.; Adjari, A. Engineering Flows in Small Devices: Microfluidics Toward a Lab-on-a-Chip. Annu. Rev. Fluid Mech. 2004, 36, 381−411. (8) De Gennes, P. G.; Brochard-Wyart, F.; Quéré, D. Capillarity and Wetting Phenomena, Drops, Bubbles, Pears, Waves; Springer: New York, 2004. (9) Lazar, P.; Riegler, H. Reversible Self-Propelled Droplet Movement: A New Driving Mechanism. Phys. Rev. Lett. 2005, 95, 136103. (10) Sumino, Y.; Magome, N.; Hamada, T.; Yoshikawa, K. SelfRunning Droplet: Emergence of Regular Motion from Nonequilibrium Noise. Phys. Rev. Lett. 2005, 94, 068301. (11) Zheng, Y.; Bai, H.; Huang, Z.; Tian, X.; Nie, F. Q.; Zhao, Y.; Zhai, J.; Jiang, L. Directional Water Collection on Wetted Spider Silk. Nature 2010, 463, 640−643. (12) Hancock, M. J.; He, J.; Mano, J. F.; Khademhosseini, A. SurfaceTension-Driven Gradient Generation in a Fluid Stripe for Bench-Top and Microwell Applications. Small 2011, 7, 892−901. (13) Bouasse, H. Capillarite, Phenomenes Superficiels; Delagrave: Paris, 1924. (14) Bico, J.; Quéré, D. Self-Propelling Slugs. J. Fluid Mech. 2002, 467, 101−127. (15) Lorenceau, É.; Quéré, D. Drops on a Conical Wire. J. Fluid Mech. 2004, 510, 29−45. (16) Darhuber, A. A.; Troian, S. M. Principles of Microfluidic Actuation by Modulation of Surface Stresses. Annu. Rev. Fluid Mech. 2005, 37, 425−455. (17) Hanumanthu, R.; Stebe, K. J. Equilibrium shapes and locations of axisymmetric, liquid drops on conical, solid surfaces. Colloids Surf., A 2006, 282, 227−239. (18) Liu, J. L.; Xia, R.; Li, B. W.; Feng, X. Q. Directional Motion of Droplets in a Conical Tube or on a Conical Fibre. Chin. Phys. Lett. 2007, 24, 3210−3213. (19) Renvoise, P.; Bush, J. W. M.; Prakash, M.; Quéré, D. Drop propulsion in tapered tubes. Europhys. Lett. 2009, 86, 64003. (20) Michielsen, S.; Zhang, J.; Du, J.; Lee, H. J. Gibbs Free Energy of Liquid Drops on Conical Fibers. Langmuir 2011, 27, 11867−11872. (21) Wang, Z.; Chang, C. C.; Hong, S. J.; Sheng, Y. J.; Tsao, H. K. Trapped Liquid Drop in a Microchannel: Multiple Stable States. Phys. Rev. E 2013, 87, 062401. (22) Lv, C.; Chen, C.; Chuang, Y. C.; Tseng, F. G.; Yin, Y.; Grey, F.; Zheng, Q. Substrate Curvature Gradient Drives Rapid Droplet Motion. Phys. Rev. Lett. 2014, 113, 026101. (23) 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, 248−250. (24) McHale, G.; Newton, M. I.; Carroll, B. J. The Shape and Stability of Small Liquid Drops on Fibers. Oil Gas Sci. Technol. 2001, 56, 47−54. (25) McHale, G.; Newton, M. I. Global Geometry and the Equilibrium Shapes of Liquid Drops on Fibers. Colloids Surf., A 2002, 206, 79−86. (26) Chou, T. H.; Hong, S. J.; Liang, Y. E.; Tsao, H. K.; Sheng, Y. J. Equilibrium Phase Diagram of Drop-on-Fiber: Coexistent States and Gravity Effect. Langmuir 2011, 27, 3685−3692. (27) Eral, H. B.; de Ruiter, J.; de Ruiter, R.; Oh, J. M.; Semprebon, C.; Brinkmann, M.; Mugele, F. Drops on Functional Fibers: From Barrels to Clamshells and Back. Soft Matter 2011, 7, 5138−5143.
of the former (a) describes the change of the drop volume, the variation of the latter [lc = (γLG/ρg)1/2] depicts the change of surface tension or liquid density. In this work, the equilibrium conformations and positions of a drop on a vertical, conical fiber have been investigated by the finite element method, Surface Evolver simulations for different gravitational constant (G) of α = 2.5° and θC = 30°. Similar to the morphology of a drop on a cylindrical fiber, two different types can be obtained: barrel shape and clam-shell shape. In the absence of gravity, a barrel drop initially put at the lower end of a conical fiber (high curvature) moves upwardly toward lower curvature. The total surface energy decreases as the drop rises. When the local radius of the conical fiber becomes comparable to the drop size, the barrel drop is deformed to an asymmetrical shape. Eventually, the barrel shape drop becomes unstable and transforms to the clam-shell drop, which continues to ascend. That is, one is unable to obtain an equilibrium location of the drop. Whatever the initial conformation of the drop on a conical fiber is, the rising drop exhibits the clam-shell shape eventually and there is no equilibrium location without gravity and hysteresis effects. However, in the presence of gravity such as G = 0.01, the drop can stop at an equilibrium location stably. The tendency of the upward motion can be resisted by the gravity force. Both equilibrium barrel and clam-shell drops can be acquired. Dependent on the value of G (drop volume), the equilibrium drop position (h) varies. Note that because of the lack of the characteristic length, varying G corresponds to the change of the drop volume, as described in Simulation Methods. Four regimes are identified: clam-shell only, barrel only, coexistence of clam-shell and barrel, and falling-off. As G is small, the equilibrium drop position is away from the tip of the conical fiber. If the drop size is small compared to the local radius of the fiber, only the clam-shell drop can be observed (clam-shell only regime). In contrast, as G is large, the equilibrium position is near the tip of the fiber. If the drop size is much larger than the local radius, only the barrel drop can be seen (barrel only regime). In a certain range of G, both barrel and clam-shell conformations can exist (coexistent regime). In this regime, regardless of the difference of the total free energy, both morphologies are numerically stable, and an energy crossover between the two conformations appears. At a given G, the equilibrium position of the clam-shell drop is always higher than that of the barrel drop. For large enough drops, the drop falls off from the conical fiber (falling-off regime).
■
AUTHOR INFORMATION
Corresponding Authors
*E-mail: [email protected]. *E-mail: [email protected]. Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS Y.-J.S. and H.-K.T. thank the Ministry of Science and Technology of Taiwan for financial support.
■
REFERENCES
(1) Gilet, T.; Terwagne, D.; Vandewalle, N. Digital Microfluidics on a Wire. Appl. Phys. Lett. 2009, 95, 014106. (2) Gilet, T.; Terwagne, D.; Vandewalle, N. Droplets Sliding on Fibres. Eur. Phys. J. E: Soft Matter Biol. Phys. 2010, 31, 253−262. F
DOI: 10.1021/la504552d Langmuir XXXX, XXX, XXX−XXX
Langmuir
Article
(28) de Ruiter, R.; de Ruiter, J.; Eral, H. B.; Semprebon, C.; Brinkmann, M.; Mugele, F. Buoyant Droplets on Functional Fibers. Langmuir 2012, 28, 13300−13306. (29) Liang, Y. E.; Chang, C. C.; Tsao, H. K.; Sheng, Y. J. An Equilibrium Phase Diagram of Drops at the Bottom of a Fiber Standing on Superhydrophobic Flat Surfaces. Soft Matter 2013, 9, 9867−9875. (30) Brakke, K. The Surface Evolver. Exp. Math. 1992, 1, 141−165.
G
DOI: 10.1021/la504552d Langmuir XXXX, XXX, XXX−XXX
