Article pubs.acs.org/Langmuir
Trapped Liquid Drop at the End of Capillary Zhengjia Wang,† Hung-Yu Yen,‡ Cheng-Chung Chang,‡ Yu-Jane Sheng,*,‡ and Heng-Kwong Tsao*,† †
Department of Chemical and Materials Engineering, National Central University, Jhongli, Taiwan 320, R.O.C. Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 106, R.O.C.
‡
ABSTRACT: The liquid drop captured at the capillary end, which is observed in capillary valve and pendant drop technique, is investigated theoretically and experimentally. Because of contact line pinning of the lower meniscus, the lower contact angle is able to rise from the intrinsic contact angle (θ∗) so that the external force acting on the drop can be balanced by the capillary force. In the absence of contact angle hysteresis (CAH), the upper contact angle remains at θ∗. However, in the presence of CAH, the upper contact angle can descend to provide more capillary force. The coupling between the lower and upper contact angles determines the equilibrium shape of the captured drop. In a capillary valve, the pinned contact line can move across the edge as the pressure difference exceeds the valving pressure, which depends on the geometrical characteristic and wetting property of the valve opening. When CAH is considered, the valving pressure is elevated because the capillary force is enhanced by the receding contact angle. For a pendant drop under gravity, the maximal capillary force is achieved as the lower contact angle reaches 180° in the absence of CAH. However, in the presence of CAH, four regimes can be identified by three critical drop volumes. The lower contact angle can exceed 180°, and therefore the drop takes on the shape of a light bulb, which does not exist in the absence of CAH. The comparisons between Surface Evolver simulations and experiments are quite well.
I. INTRODUCTION The phenomenon of liquid driven by an external force but trapped at the end of a microchannel has many practical applications. For instance, capillary passive valves are used to regulate liquid flow in microfluidic systems for the control of the sequence of bio/chemical analyses. The liquid can be trapped at the edge of the valve opening due to the pinning of a fluid interface on geometrical edges, as can be seen in Figure 1a.
The simplest case of capturing liquid drops is a liquid column trapped in a vertical capillary tube of radius R. The equilibrium condition requires the balance of the capillary force against the downward force F acting on the liquid column.9,10 2πRγlg(cos θu − cos θl) = F
The external force F may originate from pressure difference between the upper and lower ends (Δp) or gravitational force associated with the liquid volume (V). The capillary force is the product of γlg cos θ and the perimeter of the contact line (2πR), where γlg depicts the liquid−gas interfacial tension and θ the contact angle. In order to resist F in a hydrophilic tube, the upper contact angle (θu) must be smaller than the lower one (θl), i.e., cos θu > cos θl. If the trapped drop is not located at the capillary end, the static equilibrium has to be attributed to contact angle hysteresis (CAH). However, the drop may be halted at the end of the capillary without resorting to CAH. In this case, it is often observed that the lower contact angle is greater than the intrinsic contact angle (θ∗), i.e. θl > θ∗. This consequence implies that the lower contact angle does not necessarily follow the intrinsic contact angle depicted by the Young’s equation9,11,12 even in the absence of CAH. When the liquid drop is captured at the capillary end, the lower contact line is pinned at the edge of the capillary. The downward external force (F) is equal to (πR2)Δp for the capillary valve and ρgV for the pendant drop. As F is increased, θl is generally observed to grow so that the upward capillary
Figure 1. Skematics of a drop trapped at the end of a micochannel. (a) A cylindrical capillary valve under the pressure difference. (b) A cylindrical tube under the gravitational force.
The capillary valve is generally achieved on a sudden enlargement of flow channel. Capillary valves are essentially “passive” and attractive because it is insensitive to physicochemical properties of the liquid and easy to fabricate.1−3 A pending drop, as shown in Figure 1b, defined as a drop suspended from the end of a capillary can also be regarded as liquid trapped at the tube end. Since the shape of such a drop is determined by the gravitational force and surface tension, the pendant drop method has been generally employed to extract the surface tension from the image of the drop shape.4−8 © 2013 American Chemical Society
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Received: July 15, 2013 Revised: September 4, 2013 Published: September 4, 2013 12154
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force is elevated to balance the external force. As θl > 90°, a pendant drop can be seen and the lower meniscus also provides upward capillary force. Obviously, the maximum supporting capillary force is expected to occur at θl = 180°. However, it is quite often to see the pendant drop with θl > 180° in experiments. In fact, surface tension measurement is often performed with such a pendant drop profile based on Young− Laplace equation.4−8 One naturally arising question is that, from the point of view of the force balance, how the pendant drop with θl exceeding 180° is formed as F is gradually raised. For example, the falloff of the pendant drop from the capillary end often takes place at the critical value of θl greater than 180°. Unfortunately, the corresponding mechanism is still unclear. Despite of the fact that the phenomena of liquid trapped at the end of a capillary has been applied to practice, its physical understanding is still far from being complete, particularly as CAH is involved. In this paper, the liquid drop captured at the end of a capillary will be explored systematically by Surface Evolver simulations (SE) and experiments. We focus on capillary valve and pendant drop with and without CAH. In section II, the methods of SE simulations and pendant drop experiments are briefly described. In section III, the liquid trapped at the edge of a capillary valve is investigated. In section IV, the pendant liquid drop suspended from the end of the vertical cylindrical tube is studied. At last, a conclusion is given in section V.
macroscopic origin of CAH (roughness, chemical heterogeneity). However, the current work uses two values of interfacial energy during the simulation (at the changing solid−liquid interface and the changing solid−gas interface) based on the microscopic origin of CAH: molecular interlocking. B. Materials and Experimental Methods. The experiments about the pendant drop located at the end of a vertical tube are performed in the absence/presence of CAH. The transparent tube is made of poly(methyl methacrylate), and its radius is R = 1.7 mm. The initial pendant drop is formed by injecting the liquid from the lower end of the tube. The volume of the pendant drop is gradually increased by consecutive dropwise addition. In order to prevent the pendant drop spreading outward along the tube wall, the surface of the tube end is modified to become nearly superhydrophobic by the spray (PINOLE, Japan), which contains alcohol, isohexane, organopolysiloxane mixture, and petroleum distillates. The side view of the liquid column and pendant drop is observed by the lens of Optem 125C, and an enlarged image of the menisci is acquired. All data are analyzed by the measuring software, MultiCam Easy 2007 (Shengtek, Taiwan). As the liquid diiodomethane (CH2I2) is used, CAH is negligible and the capillary length is lc = (γlg/ρg)1/2 = 1.25 mm. The intrinsic contact angle is θ∗ = 77°. The initial volume of pendant drop is 2 mm3, and the dropwise addition of CH2I2 is 2 mm3. In contrast, when water is employed, CAH is present, and the advancing and receding contact angles are θa = 89° and θr = 77°, respectively. The capillary length is lc = 2.71 mm. The initial volume is 30 mm3, and the dropwise addition of water is 2 mm3.
III. LIQUID DROP CAPTURED AT CAPILLARY VALVE A drop can be trapped at the capillary valve in a microfluidic system because of the pinning of the fluid interface on geometrical edges. When the pressure difference across the drop is increased, the curvatures of menisci are adjusted accordingly. As the pressure difference is increased and reaches the valving pressure, the drop becomes able to exit the valve. The bulging of the meniscus corresponding to a Laplace pressure is given by the Young−Laplace equation3,9
II. SIMULATION AND EXPERIMENTAL METHODS A. Surface Evolver Simulation. Numerical simulations by public domain finite element Surface Evolver (SE) package developed by Brakke are performed.13,14 The basic concept of SE is to minimize the energy of a surface of a drop subject to various forces and constraints. The minimization is achieved by evolving the surface down the energy gradient. SE models surfaces as unions of triangles with vertices. They are iteratively moved from an initial shape until a minimum energy configuration is obtained. It has been applied for the study of various wetting phenomena.10,15−25 In our simulation, the system consisting of a liquid drop located at the end of a microchannel is represented by 4226 vertices, 12 416 edges, and 8192 facets. The total liquid volume in the system is kept identical during iteration. The calculation is done until the total free energy difference is converged within the acceptable tolerance of 10−5. Note that in SE calculations all lengths are scaled by the capillary length (lc) and free energy is scaled by γlglc2. To ensure that our SE simulation results are stable, the final drop shape (menisci) is always subject to small perturbations by “jiggling” through which vertices are randomly moved.13 Once the system reaches the equilibrium state, shaking the grid points will not change the simulation results at all. To confirm that the simulation result has approached the minimum state, the jiggling process has been repeated three to five times. In reality, mechanical vibration can be used as jiggling to achieve the most stable contact angle.26 In general, it takes about 3 h of CPU time on a PC Intel Core i7-2600 3.4 GHz running Windows 7 to obtain an equilibrium data point. CAH is incorporated in our SE simulations. Upon wetting, the solid−liquid interface is restructured over a short period of time, and thus the solid−liquid interfacial tension decrease from γsl to γsl′ .11,18,22,23 That is, during simulations, the solid−liquid tension within the wetted region changes to γ′sl while that in the exterior region remains unchanged, γsl. Since there exist two solid−liquid interfacial tensions, one can define two contact angles associated with droplet wetting: the advancing angle θa with cos θa = (γsg − γsl)/γlg and the receding angle θr with cos θr = (γsg − γsl)/γlg according to Young’s equation,9,11,12 where γsg denotes the solid−gas interfacial tension. Recently, an algorithm for including contact angle hysteresis in Surface Evolver was also developed.20,24 The retention forces were introduced as local constraints in the variation approach. Such a macroscopic model is based on heuristic arguments but without including the
2γlgC = Δpi
(2)
where C denotes the mean curvature of the meniscus and Δpi the pressure difference across the liquid interface. For a small drop with the size less than the capillary length, the gravity effect can be neglected. Because Δpi does not change along the interface, C is a constant everywhere, and thus the meniscus takes the shape of spherical cap. During the capturing process of the drop by the valve, the characteristics of menisci, including meniscus positions and contact angles, vary with increasing pressure difference across the drop. They can be determined by the force balance, eq 1, under the constraint of constant volume. The valving pressure beyond which the fluid will invade the larger channel depends on the surface tension of the fluid, the wetting properties of the channel wall, and the geometrical characteristics of the capillary valve. It can be analytically decided at the point when the trapped drop tends to expand over the wall connecting the valve to the larger channel. SE simulations are also performed and the outcomes are compared with the analytical results. The ideal case corresponding to the absence of CAH is studied first, and the presence of CAH is then taken into account to demonstrate its effect. The capillary valve is generally used in the microfluidic system in which the characteristic diameter is small compared to the capillary length. For a horizontal microchannel, the gravity effect may affect the meniscus only when the diameter is comparable to the capillary length. The 12155
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liquid volume (V 1/3 ) is irrelevant in determining the importance of gravity. Therefore, gravity is neglected in both the analytical model and SE simulation. A. In the Absence of CAH. As shown in Figure 1a, consider a liquid drop captured at the end of a tube of radius R (capillary valve) under the pressure difference Δp = pu − pl > 0, where pu and pl denote the air pressures above and below the liquid drop, respectively. According to eq 1, the capillary force that traps the liquid drop must be adjusted to resist the driving force F = πR2Δp. In the absence of CAH, the upper contact line of θu lies within the tube and must equal θ∗ to satisfy Young’s equation.9,11,12 On the other hand, since the lower contact line of θl is pinned at the edge of the valve, θl has to be altered from θ∗. In order to satisfy the force balance, one has ⎛ R̃ Δp ̃ ⎞ θl = cos−1⎜cos θ∗ − ⎟ > θ∗ 2 ⎠ ⎝
(3)
where Δp̃ = Δplc/γlg and R̃ = R/lc. Without knowing the geometrical characteristics of the valve, the ideal maximum of the upward capillary force can be determined by cos θl = −1 in eq 1. That is, the maximal capillary force provided by the lower meniscus is 2πRγlg with θl = 180°. Therefore, the ideal maximum of the capillary force that the liquid drop can afford is max
Fc̃
= 2πR̃(cos θ∗ + 1)
F̃max c
(4)
Fmax c /(γlglc).
where = As a result, the ideal maximal pressure difference that the trapped liquid can sustain is given by Δpmax ̃
2 = (cos θ∗ + 1) R̅
Figure 2. (a) θu and θl of a trapped drop as a function of the pressure difference Δp̃ in the absence of CAH for V = 20 mm3, lc = 2.71 mm, R = 0.44 mm, and θ∗ = 60°. Lines represent the theoretical results, and symbols depict the outcomes of SE simulations. z̃u and z̃l as a function of Δp̃ are plotted in the inset. (b) The divergent process of a trapped drop along the edge of the capillary valve in the absence of CAH by SE simulations for α = 90° at Δp̃ = 18.
(5)
To examine our foregoing analysis, SE simulations are performed without any constraint on the movement of the contact line. In other words, contact line pinning along the edge occurs naturally. Consider a liquid drop with V = 20 mm3 captured at the valve with R = 0.44 mm and θ∗ = 60°. As demonstrated in Figure 2a, one has θu = θ∗ and θl grows with increasing Δp̃. The simulation outcome can be depicted eq 3 quite well. The positions of the upper contact line and the center of the lower meniscus scaled by lc, z̃u, and z̃l, are also obtained, as shown in the inset of Figure 2a. Note that the end of the tube is set as z̃ = 0. As Δp̃ is increased, z̃u and z̃l descend gradually. When Δp̃ = Δp̃max, they reach the minima, z̃min and u z̃min l . Again, the simulation results agree quite well with the force balance subject to the condition of constant volume. While z̃l varies with Δp̃, one has z̃min = R̃ . l The critical pressure difference, beyond which the valve is turned on and the fluid starts to pass through the valve, is defined as the valving pressure (Δp̃v). If the lower contact line is not allowed to wet the expanded region, one has Δp̃v = Δp̃max with θl = 180°. In reality, the geometrical characteristic and wetting property of the valve opening will affect the valving pressure Δp̃v, which is generally less than Δp̃max with θl = 180°. As the pressure difference is increased to a certain value, the apparent contact angle at the edge reaches the condition of wetting invasion across the corner boundary, φ = θl − (180° − α) = θ∗.27−30 Here α denotes angle spanned between the capillary wall and the expanded wall, as shown in Figure 1a. As φ < θ∗, wetting invasion does not take place and contact line pinning occurs because of the edge effect,10,25,31 instead of CAH. At this bifurcation point, further increment of Δp̃ may either push the more part of the drop out of the capillary just like the ideal case or force the pinned contact line moving
across the corner to wet the expanded region. In the former case, the radius of curvature of the lower meniscus descends, and the lower contact angle rises. In the latter case, on the contrary, the radius of curvature of the lower meniscus ascends, but the lower contact angle remains at θ∗. In comparison between the two possibilities, one finds that the former case satisfies the Young−Laplace equation but the latter case has lower interfacial free energy. Because of the satisfaction of the Young−Laplace equation, the former state corresponds to the extremum of the free energy. However, wetting invasion across the edge of the valve can lower the system energy, and thereby the former (ideal) state is not stable. Figure 2b demonstrates the evolving process for Δp̃ > Δp̃v. The change from step 2 to 3 indicates that the trapped drop proceeds toward the latter scenario before it reaches the former scenario. According to the principle of free energy minimization, the first case is ruled out and the second case is anticipated. That is, as the pressure difference exceeds Δp̃v corresponding to φ = θ∗, the pinned contact line at the edge will move across the corner to wet the expanded surface connected to the larger channel. Moreover, as the solid−liquid area is increased after wetting invasion, the system free energy declines monotonically and the radius of curvature of the lower meniscus grows. This consequence indicates that the local minimum can not be reached along the expanded surface, and the Young−Laplace equation will never be satisfied. Although the second case is not stable as well, we believe that once Δp̃ > Δp̃v, the evolving 12156
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well with eq 6. According to eqs 6 and 7, the valving pressure depends on the maximal lower contact angle (θmax l ), which is in turn determined by the wetting property and geometrical characteristic of the valve (θ∗ and α). Therefore, one is able to manipulate the valving pressure by the design of the capillary valve. Basically, the ideal valving pressure (Δp̃max) can be reached by large θ∗ or small α. B. In the Presence of CAH. The microchannel without CAH can provide the asymptotic behavior of the capillary valve. In reality, CAH often exists.23,28,32 Here, the influence of CAH on the behavior of capillary valve is investigated. In the presence of CAH, the contact angle is no longer kept at an unique angle (θ = θ∗), but can vary in a range θr ≤ θ ≤ θa, where θa and θr denote the advancing and receding contact angle, respectively. In this study, we have θa = θ∗. Initially, consider a drop located at the end of the tube. At equilibrium, the pressure difference is absent (Δp̃ = 0), and thus the upper and lower contact angles are equal (θu = θl = θa). As the pressure difference is increased, the contact line of the upper meniscus is pinned due to CAH and that of the lower meniscus is pinned as well due to the edge effect corresponding to the boundary minimum of free energy.10,25,27,29−31 However, the increased pressure Δp̃u pushes the center of the upper meniscus downward, and thus the upper contact angle θu is lowered. Under the condition of constant drop volume, the center of the lower meniscus must descend, and thereby the lower contact angle θl is elevated. When Δp̃u is large enough, θu is reduced to θr, and the upper contact line is able to move down according to CAH. Consequently, one has θa ≥ θu ≥ θr and θl ≥ θa. The variations of the positions and contact angles of menisci with the pressure difference are depicted in Figures 4a,b. As Δp̃ is increased, three regimes separated by two critical pressure differences, Δp̃c1 and Δp̃c2, can be identified. In regime I where Δp̃ < Δp̃c1, z̃u remains unchanged but θu declines. The conservation of drop volume requires the descending of z̃l. In order to balance the external force, θl must be increased to afford enough capillary force. When Δp̃ = Δp̃c1, θu decreases to θr. In regime II where Δp̃c1 < Δp̃ < Δp̃c2, the upper contact line pinning vanishes and z̃u declines gradually with θu = θr. Similar to regime I, z̃l descends and θl continues to increase. Eventually, θl reaches 180° when Δp̃ = Δp̃c2. Here the lower contact line is always pinned at the edge, and the ideal maximal pressure difference is considered.
course of the drop will follow this scenario. Based on the above argument, the maximal lower contact angle is θlmax = min(θ∗ + 180° − α , 180°)
(6)
where min(a,b) denotes the smaller one of a and b. The valving pressure Δp̃v is then given by 2 Δpṽ = (cos θ∗ + cos θlmax ) (7) R̃ In order to examine our foregoing analysis, SE is performed to simulate the condition of Δp̃ > Δp̃v. Figure 2b illustrates the divergent process of the liquid under Δp̃ = 18 in the valve with α = 90°. Note that Δp̃ is greater than Δp̃v = 16.8. It is clearly shown that the trapped liquid can not sustain such a pressure difference and invades the expanded wall. Moreover, the lower contact line continues expanding until the simulation diverges. The validity of eq 6 can also be verified by SE simulations. Figure 3a shows the variation of the maximal lower contact
Δpṽ = Δpc̃2 = Δpmax ̃ =
2 (cos θr + 1) R̃
(8)
In regime III where Δp̃ > Δp̃c2, the capillary force is unable to resist the driving force, and the liquid drop cannot be trapped at the valve anymore. SE simulations are also conducted to verify the effect of CAH for V = 20 mm3, R = 0.44 mm, θa = 60°, and θr = 30°. According to the force balance (eq 1) under the constant volume constraint, one has Δp̃c1 = 12.93 and Δp̃c2 = 22.98. Results from SE simulations are consistent with the theoretical analyses based on eq 1. As mentioned before, the valving pressure can be altered by the wetting property and geometrical characteristic of the valve, which affect the unpinning behavior of the lower contact line. Since the movement of the lower contact line corresponds to wetting invasion, only the advancing contact angle is involved for the crossover from regime II to regime III. Therefore, eq 6 for the maximal lower contact angle is still applicable with θa = θ∗. However, the upper contact meniscus always provides the capillary force with 2πRγlg
Figure 3. (a) The maximum of the lower contact angle θmax varies with l valve wettability θ∗ in the absence of CAH for V = 20 mm3, lc = 2.71 mm, R = 0.44 mm, and α = 90°. Lines represent the theoretical results, as a and symbols depict the outcomes of SE simulations. (b) θmax l function of valve geometry α at θ∗ = 60°.
angle with the intrinsic contact angle for V = 20 mm3, lc = 2.71 mm, R = 0.44 mm, and α = 90°. For hydrophobic surfaces (θ∗ > 90°), one has θmax = 180°, and the valving pressure equals the l ideal maximal pressure difference Δp̃max. In contrast, for hydrophilic surfaces (θ∗ < 90°), θmax decays linearly and the l deviation from Δp̃max grows, as θ∗ is decreased. Figure 3b depicts the variation of θmax with α for θ∗ = 60°. When the valve l is designed with α < θ∗, θmax can reach 180° and Δp̃v = Δp̃max. l However, θmax decays linearly and the deviation from Δp̃max l rises as α is increased. The results of SE simulations agree quite 12157
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determined. The ideal case corresponding to the absence of CAH is studied first, and the presence of CAH is then taken into account to demonstrate its effect. A. In the Absence of CAH. A liquid drop with volume V suspended at the end of the capillary can be regarded as a trapped drop due to contact line pinning of the lower meniscus. In the absence of CAH, the contact angle of the upper meniscus is kept at the intrinsic contact angle, θu = θ∗. In order to sustain the weight of the drop, the contact angle of the lower meniscus must be raised to fulfill the force balance (eq 1) ⎛ Ṽ ⎞ θl = cos−1⎜cos θ∗ − ⎟ ⎝ 2πR̃ ⎠
(10)
where Ṽ = The upper limit of the maximal drop volume Ṽ max can be determined from the maximal capillary force corresponding to θl = 180° V/lc3.
̃ = 2πR̃(cos θ∗ + 1) Vmax
(11)
As the gravity effect becomes dominant, the equilibrium shape of the lower meniscus satisfying the Young−Laplace equation may cease to exist and the drop falls off before θl reaches 180°. That is, θl = 180° can only be attained for R̃ ≪ 1. In general, the critical value of θl before falling off declines with increasing R̃ . In order to validate our theory in the absence of CAH, both experiments and SE simulations are performed for a capillary with R = 1.7 mm, lc = 1.25 mm, and θ∗ = 77°. Figure 5a illustrates the shape of the pendant drop at three different volumes: Ṽ = 2.06, 6.17, and 9.25. Obviously, the extent of bulging out of the lower meniscus rises with increasing Ṽ . The comparison between experiment and simulation shows that they agree quite well. The quantitative features of the trapped drop are also shown in Figure 5b. As the drop volume is increased, the location of the upper contact line (z̃u) grows but the upper contact angle (θu) remains at θ∗. At the same time, the center of the lower meniscus (z̃l) declines while the lower contact angle (θl) grows. Again, the simulation outcome is quantitatively consistent with the experimental result. Both data points can be well depicted by eq 10. However, our simple theory fails to predict the critical values of θl and Ṽ before falling off. It is found that the liquid drop falls off as θl ≈ 146° in experiment and 156° in SE simulation. Accordingly, the critical volume is Ṽ = 9.3 in experiment and 10.2 in SE simulation. The divergent process associated with a falling drop observed in SE simulation is demonstrated in Figure 5c for Ṽ = 10.3. The typical process involves drop stretching and neck forming and getting distended. B. In the Presence of CAH. The behavior of a pendant drop at the end of the cylindrical tube becomes more complicated in the presence of CAH. In fact, the contact line of the upper meniscus can be pinned due to CAH, and thereby the upper contact angle can be adjusted as well. While θl can be acquired from the force balance in the absence of CAH, there are two unknowns associated with eq 1, θu and θl, in the presence of CAH. As a result, they have to be simultaneously determined by SE simulations. As the drop volume is increased by injection of liquid from below, the positions of menisci and contact angles vary, and four regimes separated by three critical volumes, Ṽ c1, Ṽ c2, and Ṽ c3, can be identified as shown in Figure 6. In regime I where Ṽ ≤ Ṽ c1, the upper meniscus rises and the upper contact angle remains at the advancing contact angle (θu = θa) because
Figure 4. (a) θu and θl of a trapped drop as a function of the pressure difference Δp̃ in the presence of CAH for V = 20 mm3, lc = 2.71 mm, R = 0.44 mm, θa = 60°, and θr = 30°. Lines represent the theoretical results, and symbols depict the outcomes of SE simulations. (b) z̃u and z̃l are plotted against Δp̃.
cos θr in regime II. As a consequence, the valving pressure corresponding to Δp̃c2 is given by 2 Δpṽ = (cos θr + cos θlmax ) (9) R̃ Evidently, the valving pressure is elevated by CAH. The influence of the advancing contact angle is through θmax in l accord with eq 6. Note that for a given pressure difference there may exist multiple solutions due to CAH.10,23,32 In our case, different drop geometries may be observed if one reduces the pressure difference from Δp̃v. However, in this study, we focus only on the process associated with pressure increment.
IV. PENDANT DROP AT THE END OF A VERTICAL CAPILLARY In the capillary valve with the negligible gravity effect, the lower meniscus bulges out due to pressure difference between the two menisci. In contrast, the drop without pressure difference will be suspended at the end of vertical capillary because of downward gravitational force. In the former case, the shapes of the menisci can be described by spherical cap because of uniform pressures. In the latter case, however, the shapes of menisci deviate from spherical cap due to position-dependent hydrostatic pressure. As a result, they have to be determined by solving the Young−Laplace equation.33−35 This difficulty can be circumvented by resorting to the balance of the forces acting on the pendant drop.9,10 The upper limit of the maximal drop volume before falling off can be decided. By taking into account gravity, SE is also employed to investigate the characteristics of the pendant drop. The maximal drop volume can be 12158
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Figure 5. (a) In the absence of CAH, the equilibrium shapes of pendant drops from experiments and SE simulations at lc = 1.25 mm, R = 1.7 mm, and θ∗ = 77° for Ṽ = 2.06, 6.17, and 9.25. (b) θu and θl of a pendant drop as a function of the drop volume Ṽ in the absence of CAH. The dashed line represents eq 1. Hollow symbols depict the outcomes of SE simulations, and solid symbols represent the experimental results. z̃u and z̃l as a function of Ṽ are plotted in the inset. The error bars are smaller than the symbol sizes. (c) The divergent process of a pendant drop in the absence of CAH by SE simulations at Ṽ = 10.3. The process involves drop stretching and neck forming and getting distended.
Figure 6. (a) In the presence of CAH, the equilibrium shapes of pendant drops from experiments and SE simulations at lc = 2.71 mm, R = 1.7 mm, θa = 89°, and θr = 77°. (b) θu and θl of a pendant drop as a function of the drop volume Ṽ in the presence of CAH. Hollow symbols depict the outcomes of SE simulations, and solid symbols represent the experimental results. The error bars are smaller than the symbol sizes. (c) The variations of z̃u, z̃l, and R̃ max with Ṽ are plotted in the presence of CAH (Δθ = θa − θr = 12°). The error bars are smaller than the symbol sizes.
injection from below leads to wetting invasion of the upper meniscus. Nonetheless, the lower contact angle has to grow in order to provide additional capillary force as depicted in eq 10. In regime II where Ṽ c1 ≤ Ṽ ≤ Ṽ c2, since the upper contact line reaches its highest position, it is pinned with increasing Ṽ . Consequently, the upper contact angle declines toward the receding contact angle (θr) due to contact line pinning. The decrease of θu furnishes significant upward capillary force. It is interesting to note that the lower contact angle still grows monotonically and even exceeds 180°. That is, the drop takes on the shape of a light bulb (R̃ max > R̃ ), which does not exist in the absence of CAH. As θl > 180°, the upward capillary force afforded by the lower meniscus decreases but is compensated by the upper meniscus. In regime III where Ṽ c1 < Ṽ ≤ Ṽ c3, because the upper contact angle has arrived at the receding contact angle (θu = θr), the capillary force offered by the upper meniscus reaches its maximum. Moreover, the upper contact
line is able to descend as Ṽ = 9.3 is increased. At the same time, the lower contact angle must decline to provide more upward capillary force. In regime IV where Ṽ > Ṽ c3, the equilibrium shape of the lower meniscus cannot satisfy the Young−Laplace equation anymore and thus falls off, just like regime III in the absence of CAH. The quantitative behavior of a pendant drop in the presence of CAH is demonstrated through both experiments and SE simulations for a capillary with R = 1.7 mm, lc = 2.71 mm, θa = θ∗ = 89°, and θr = 77°. Figure 6a shows the shape of the pendant drop for three different volumes: Ṽ = 2.01, 3.52, and 12159
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5.12. As Ṽ is increased, z̃u, z̃l, θu, θl, and R̃ max vary. For Ṽ = 5.12, the lower meniscus looks like a light bulb and θl exceeds 180°. Note that in the absence of CAH θl is always less than 180° because the capillary force provided by the upper meniscus is always unchanged. The variations of θu and θl with the drop volume Ṽ are shown in Figure 6b. According to SE simulation, the three critical volumes are acquired: Ṽ c1 = 3.77, Ṽ c2 = 4.82, and Ṽ c3 = 5.22. In regime II, θl becomes 180° at Ṽ = 4.1 and reaches its maximum, θl ≅ 198°, at Ṽ = Ṽ c2. The maximal value of θl agrees well with that obtained from experiment (θl ≅ 196.4°). Note that the maximal θl corresponds to the largest volume of the drop whose upper contact line can still be pinned. In regime III, the drop falls off at θl = 188.6°, which is close to the experiment value 190.9°. The variations of z̃u, z̃l, and R̃ max with Ṽ are given in Figure 6c. In regime I, while z̃l decreases monotonically, z̃u grows and is able to reach its maximum z̃max = 2.76 at Ṽ = Ṽ c1. The maximal radius of the u pendant drop equals the radius of the capillary R̃ max = R̃ . In regime II, z̃l descends fast and z̃u remains unchanged because of contact line pinning. At Ṽ ≅ 4.1 where θl = 180°, R̃ max begins to exceed R̃ . At Ṽ ≅ 4.52, z̃l drops and R̃ max grows suddenly. That is, the lower meniscus starts to have a shape of the light bulb. In regime III, z̃l continues decreasing and z̃u begins to decline. The lowest stable position of z̃u is at z̃u = 2.15, corresponding to Ṽ = Ṽ c3. However, Rmax seems to be kept at about 1.22 R. When Ṽ > Ṽ c3, the pendant drop falls down. As can be seen from Figures 6b,c, the results from SE simulations and experiments agree well. Our analysis clearly shows that the surface tension determination based on the force balance such as Tate’s law9 can be systematically improved by taking into account the effect of CAH.
and the upper contact angle declines toward the receding contact angle. In regime II where Δp̃c1 < Δp̃ ≤ Δp̃c2, the upper contact line descends with θu = θr. Since Δp̃c2 corresponds to the valving pressure, the drop cannot be trapped anymore in regime III where Δp̃ > Δp̃c2. Owing to the increment of the upper capillary force, the valving pressure is elevated by CAH, as shown in eq 9. For a pendant drop suspended at the capillary end, the gravitational force leads to the deviation of the shape of menisci from spherical cap. As the drop volume is increased, the lower contact angle is adjusted from θ∗. In the absence of CAH, θl grows monotonically. As gravity becomes dominant, the shape of the lower meniscus fails to satisfy the Young−Laplace equation, and thus the drop falls off before θl reaches 180°. In general, θmax decays with increasing R̃ . In the presence of CAH, l four regimes are separated by three critical volumes. In regime I where the upper contact angle remains at θa, the lower contact angle is adjusted to resist the gravitational pull. In regime II where the upper contact angle decays from θa to θr, significant upward capillary force is provided by the upper meniscus. The upper contact line is pinned with increasing Ṽ . The lower contact angle continues growing and is allowed to exceed 180°. As a result, the drop has the shape of a light bulb (R̃ max > R̃ ), which does not exist in the absence of CAH. In regime III where the upper contact angle is kept at θr, the upper contact line is able to descend as Ṽ is increased. The lower contact angle has to decline to provide more upward capillary force. In regime IV, the captured drop cannot satisfy the Young−Laplace equation and thus falls off.
V. CONCLUSION In this paper, the liquid drop captured at the capillary end, which is observed in capillary valve and pendant drop technique, is investigated theoretically and experimentally. It is a special case of the trapped liquid drop in the microchannel.10 Because of contact line pinning of the lower meniscus, the external force acting on the liquid drop due to pressure difference or gravity can be resisted by the capillary force. Owing to the edge effect,10,25,27,29−31 the lower contact angle is able to rise from the intrinsic contact angle so that the force balance is fulfilled. In the absence of CAH, the upper contact angle remains at θ∗. However, in the presence of CAH, the upper contact angle can descend to contribute more capillary force. The coupling between the lower and upper contact angles determines the equilibrium shape of the captured drop. In a capillary valve, the drop movement is stopped by the edge effect. By increasing the pressure difference over the valving pressure, the pinned contact line can move across the edge and the drop starts to move again. As Δp̃ is increased, the lower contact angle grows from θ∗. Under the ideal condition where the lower contact line is always pinned regardless of the valve characteristics, the maximal capillary force can be provided at θl = 180° and the ideal maximal valving pressure is given in eq 5. When the geometrical characteristic (α) and wetting property (θ∗) of the valve opening are considered, the movement of the lower contact line takes place at θl = 180° + θ∗ − α. The valving pressure is reduced from Δp̃max as expressed in eq 7. When CAH is considered, three regimes can be identified by two critical pressure difference Δp̃c1 and Δp̃c2. In regime I where Δp̃ ≤ Δp̃c1, the upper contact line is pinned
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AUTHOR INFORMATION
*E-mail
[email protected] (Y.-J.S.). *E-mail
[email protected] (H.-K.T.). Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS
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REFERENCES
This research work is supported by National Science Council of Taiwan.
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