Equilibrium Morphological Phase Diagram of Drops in Hydrophilic

Oct 26, 2015 - The equilibrium morphology of a drop in a horizontal tube can provide useful information for two-phase flow in microfluidics devices in...
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Equilibrium Morphological Phase Diagram of Drops in Hydrophilic Cylindrical Channels Yu-En Liang,† Cyuan-Jhang Wu,‡ Heng-Kwong Tsao,*,‡,§ and Yu-Jane Sheng*,† †

Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617, Republic of China Department of Chemical and Materials Engineering and §Department of Physics, National Central University, Jhongli, Taiwan 32001, Republic of China

‡

ABSTRACT: The equilibrium morphology of a drop in a horizontal tube can provide useful information for two-phase flow in microfluidics devices in which the capillary force dominates. A drop-in-tube system is analogous to a drop-on-fiber one and two conformations are obtained, adhered drop and liquid slug, by the approaches of experiments and surface evolver (SE) simulations. The adhered drop conformation tends to exist at small volume, whereas the liquid slug conformation is favored at larger volume. Around the transition volume between the two conformations, both morphologies can coexist. The experimental results are consistent with those of simulation outcomes. The morphological phase diagram of the dropin-tube system is constructed via SE simulations by varying the drop volume and contact angle. Three regimes can be identified through the upper and lower boundary curves: adhered drop only, liquid slug only, and coexistence. Compared to the case with negligible gravity, the adhered drop is more favored than the liquid slug in the presence of gravity. As a result, the coexistence regime expands substantially.

I. INTRODUCTION Gas−liquid two-phase flows are the most common of the twophase flows and frequently encountered in industrial applications such as crude oil transportation. This flow is complicated because it involves a deformable interface. The transport of gas and liquid in tubes can exhibit several topological configurations called flow patterns.1−9 In a slug flow which is inherently intermittent, the gas phase exists as large bubbles separated by liquid slugs.9 The transition from water slug to drop is a main concern in gas-flow channels, such as polymer electrolyte membrane fuel cells. The effects of surface wettability and gravity are important in these applications.10,11 Recently, droplet microfluidics in which drops are created and transported continuously are of great interest for chemical and biological applications.12,13 This microflow deals with a small quantity of liquid in the plug form moving inside a capillary tube and separated by immiscible plugs of another fluid. Certainly, the shape of the liquid plug in a capillary is determined by the capillary forces. The capillary number (Ca) describes the relative importance between viscous forces and capillary forces. When Ca ≪ 1, the capillary force dominates and thus the knowledge of the equilibrium morphology of a drop in a tube can shed some light on flow patterns. From the viewpoint of the morphology of the drop shape, drop-in-tube can be regarded as the reverse situation of dropon-fiber. The morphologies of a liquid drop on a cylinder have been extensively studied.14−19 The conformation varies with the contact angle (θ), drop volume (V), and radius of a cylinder © 2015 American Chemical Society

(a). Two different shapes are observed: axisymmetric barrel and asymmetric clamshell. Generally, the clamshell conformation takes place for high contact angles or small droplets (V/a3) while the barrel conformation exists for low contact angles or large droplets. In the morphological phase diagram (V/a3 vs θ), the crossover between the two conformations reveals the occurrence of the coexistent regime.17 When the gravitational effect becomes important, the axisymmetric barrel drop on a horizontal fiber changes into an asymmetric one.17 For the phase diagram considering gravity, the regime of falling off emerges for large droplets and the regime of the barrel drop dwindles substantially. In this work, the wetting behavior of a drop in a horizontal hydrophilic tube, which resembles the two-phase flow under the condition of very small Ca, is investigated both experimentally and theoretically. Two conformations of the drop-in-tube system are observed. For the experimental observations, their equilibrium shapes are also explored by surface evolver (SE) simulations. The comparison between experimental results and simulation outcomes are made. The drop shape depends on the liquid contact angle (θ) within the tube, tube radius (a), drop volume (V), and gravity (ρg). The morphology phase diagrams in the plot of V/a3 against θ are Received: July 25, 2015 Revised: October 26, 2015 Published: October 26, 2015 25880

DOI: 10.1021/acs.jpcc.5b07212 J. Phys. Chem. C 2015, 119, 25880−25886

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directly. However, because of the significant difference of the refractive index between water and acrylic glass tube and the curvature of the water−air interface, the images have been seriously distorted. As a result, an appropriate correction which will be discussed later is required to recover the true values of the drop shape.

constructed by SE simulations. Under certain conditions, both conformations may coexist.

II. SIMULATION AND EXPERIMENTAL METHODS A. Surface Evolver. In this study, the equilibrium shape of a drop in a horizontal tube can be acquired by the public domain surface evolver software.20 It is a numerical finite element method on the basis of free energy minimization subject to constraints. The total free energy (F) of a drop in a cylindrical tube consists of surface energy and gravitational energy,21,22 F = γLG(ALG − ASL cos θ ) +

∭V (ρgz) dV

III. RESULTS AND DISCUSSION The morphology of a drop on a cylindrical surface can be divided into two scenarios: (i) drop on a fiber and (ii) drop in a tube. In both cases, the drop shape depends on contact angle (θ), cylindrical radius (a), drop volume (V), and gravity (ρg). While the former case is well studied,17 the latter case related closely to two-phase microfluidics is less investigated. The comparison between the two cases can provide us with valuable physical insights. A. Morphologies of Drops in Horizontal Cylindrical Tubes. In the drop-on-fiber system, clamshell and barrel conformations are identified. In some situations, both conformations can coexist at the same condition. Consequently, there are three regimes in the absence of gravity: clamshell only, barrel only, and coexistence of the two shapes. Note that the drop is outside the tube in the drop-on-fiber case. For a drop inside a horizontal cylindrical tube without gravity, the drop morphology of the system may be analogous to that of dropon-fiber. Evidently, when the drop size is large compared to the tube radius, an axisymmetric liquid slug (similar to barrel) is anticipated. In contrast, when the drop size is small compared to the tube radius, an asymmetric adhered drop (similar to clamshell) is expected. When the effect of gravity is not negligible, the liquid slug becomes axis-asymmetric as well. The typical shapes of both adhered drop and liquid slug acquired from experiment and SE simulation are demonstrated in Figure 1. Note that only the visualizations of the air−water interface

(1)

where γLG is the surface tension and ALG and ASL denote the liquid−gas and solid−liquid interfacial areas, respectively. The second term on the right-hand side of eq 1 is associated with the gravitational energy and the case without gravity corresponds to g = 0. The contact angle θ is related to the interfacial tensions by the Young’s equation, cos θ = (γSG − γSL)/γLG, where γSG and γSL represent the solid−gas and solid− liquid interfacial tensions, respectively.21,22 Note that the minimization of the functional in eq 1 yields the Young− Laplace equation. In this work, all lengths are scaled by the tube radius a, and the energy by γLGa2. The dimensionless form of eq 1 is then given by F * = ALG* − cos θASL * + G

∭V* (z*) dV *

(2)

where the dimensionless gravitational constant is defined as G = (a/lc)2 with the capillary length lc = (γLG/ρg)1/2. For pure water used in our experiment, one has lc = 0.271 cm. The zero point of the gravitational energy is set at the axial axis of the horizontal tube. Note that, in this work, the exact (analytical) solution of the drop shape is generally not available except for an axisymmetric liquid slug in the absence of gravity. In SE simulations, surfaces are modeled as unions of triangles with vertices and evolved from an initial shape until a conformation with the local minimum energy is reached. The droplet morphology with a local minimum of the total energy F* corresponds to the equilibrium state. Once cos θ, G, and V* are given in eq 2, the equilibrium shape of the droplet (liquid− gas interface) in a cylindrical tube can be acquired, including the surface areas (ALG* and ASL*) and the altitude of each triangular element z*. In general, it takes about 1 h on Intel Core i7-3770 M processor to reach an equilibrium droplet shape in this study. B. Experimental Observation. The tube used in the dropin-tube system is purchased from Kwo-Yi Co. (Taiwan) and is made of acrylic glass with the inner diameter of 5 mm and the outer diameter of 8.15 mm. The water contact angle (θ) on the tube surface is determined by placing a tiny drop on the exterior wall of the tube and it is about 70°. The surface tension is determined by the drop shape analysis system DSA10-MK2 (Krüss, Germany) with the pendant drop method.23 The drop with a specified volume inside a horizontal cylindrical tube is developed by placing drops carefully from a pipet onto the inner surface of an acrylic glass tube. Small vibrations are always applied after liquid is added inside the tube to make sure that the observed shape is truly an equilibrium one. Different drop morphologies can be seen as the volume of the liquid drop is increased. The characteristics of the equilibrium drop shape in the tube can be observed from the Optem 125C optical system

Figure 1. Typical shapes of both the adhered drop and liquid slug acquired from experiment and SE simulation.

are shown for the results obtained from SE simulations. An oblique view is also provided for three-dimensional demonstration purpose. Since contact angle hysteresis is absent in this study, all contact angles along the contact lines for adhered drops and slugs have to be the same. The shape of an adhered drop with V* = 3 in a cylindrical tube with θ = 40° for G = 0 and 1 obtained from SE simulations are illustrated in Figure 2, including side view (radial direction), front view (axial direction), top view (radial direction), and oblique view. The case of negligible gravity, i.e., G = 0, is illustrated in Figure 2a. When the gravity effect is considered, i.e., G = 1, the interface of an adhered drop is pulled down and flattened by the gravitational force, as shown in Figure 2b. As a 25881

DOI: 10.1021/acs.jpcc.5b07212 J. Phys. Chem. C 2015, 119, 25880−25886

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case of negligible gravity, i.e., G = 0, is depicted in Figure 3a. Similar to a barrel drop on a fiber without gravity,17 the liquid slug in a tube for G = 0 is also axisymmetric, as can be seen from side and top views. From side view, the contact line of the slug is just like a pair of parallel lines. However, when the gravity effect is considered, i.e., G = 1, the interface of a liquid slug becomes axis-asymmetric, similar to a barrel drop on a horizontal fiber. Because of the gravitational force, the width of the bottom (WB) is greater than that of the top (WT), as shown in Figure 3b. The waist corresponding to the shortest distance between two menisci is shifted upward. For specified volume and contact angles, two different initial shapes (liquid−gas interfaces) of drops in a cylindrical tube are used for SE simulations: (i) two parallel planes and (ii) tetrahedron. The former initial shape tends to evolve to the liquid slug conformation, while the latter evolves to the adhered drop conformation. However, as shown in Figure 4, if the drop

Figure 2. Adhered drop conformation for V* = 3 and θ = 40° acquired from SE simulations. (a) G = 0. (b) G = 1.

result, the height of the contact line (HE) decreases and the width of the drop (W) increases. For a large enough drop, a liquid slug forms. The shape of a liquid slug with V* = 6 in a cylindrical tube with θ = 40° can also be acquired from SE simulations. Several views for visualization of air−water interfaces of a liquid slug are illustrated in Figure 3. Different from the adhered drop, liquid slug wets around the perimeter of the horizontal tube, as observed from both front and oblique views. Because the tube is hydrophilic, the air−water interface is a concave surface. The

Figure 4. Unstable conditions in SE simulations for (a) liquid slug and (b) adhered drop.

volume is too small, the initial shape (i) corresponding to a liquid slug fails to evolve to a physically existing shape. The final outcome seems like a liquid slug but the two menisci overlap. As result, the liquid slug cannot be stably formed and the adhered drop is preferred. In contrast, if the drop volume is too large, the initial shape (ii) corresponding to an adhered drop cannot develop a physically existing conformation either. The final result looks like a liquid slug but two menisci are connected by a thread along the cylindrical tube wall. Consequently, the adhered drop cannot be stably developed and the liquid slug is favored. These simulation observations reveal that, for a given tube radius, there exist two critical volumes, VA and VS, and VA > VS. The conformation of an adhered drop is observed as V < VA. In contrast, the conformation of a liquid slug is seen as V > VS. Evidently, both conformations can be acquired when VS < V < VA. B. Comparison between Experimental and Simulation Results of Drops in Hydrophilic Cylindrical Tubes. A typical experimental image of a liquid slug inside a horizontal tube (side view) is shown in Figure 5. The front view of the tube is shown as well and it can be regarded as the reference because of its absence of optical distortion. The differences of the thickness of the tube wall (t) and the inner tube diameter (Di) between the side and front views demonstrate the substantial optical distortion due to the refractive index mismatch. Apparently, Di determined from the side view of

Figure 3. Liquid slug conformation for V* = 6 and θ = 40° obtained from SE simulations. (a) G = 0. (b) G = 1. 25882

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water, as illustrated in Figure 5c. The distortion in the horizontal direction is negligible and thus the horizontal calibration is not necessary. Figure 6 shows our experimental results of water drops inside a cylindrical tube. The inner diameter of the tube is 5 mm and

Figure 6. Experimental observation of water drops inside an acrylic glass tube. Three regimes are observed: adhered drop only, liquid slug only, and coexistence of both shapes. The inner diameter is 5 mm and the contact angle θ = 70°.

the water contact angle is θ = 70°. Three regimes are observed, including adhered drop only, liquid slug only, and coexistence of adhered drop and liquid slug. Consistent with the qualitative prediction in section IIIA, the state of adhered drop is present as V < 90 μL. On the other hand, the state of liquid slug appears as V > 50 μL. Both states can be seen in the region of 50 < V < 90 μL. When the drop volume is increased, the height and width (W) of the adhered drop grow. Note that the maximum height of the water−air surface is located at the center of this surface for small drops V < 20 μL. However, the maximum height is shifted to the edge of the air−water surface (contact line) for large drops V > 20 μL. Both cases are clearly demonstrated in the front views of Figure 7. For convenience, the edge height of the adhered drop (HE) is observed. For a liquid slug, the width of the bottom (WB) is greater than that of the top (WT) because of the gravitational force. Both WB and WT rise as the drop volume is increased. For comparisons, SE simulation results are presented in Figure 7 for all the cases of

Figure 5. (a) Experimental observation of a liquid slug inside an acrylic glass tube. Di is the inner diameter and t denotes the thickness of the tube wall. (b) Image of a PTFE rod with diameter 2 mm located at the meniscus of a liquid slug inside a tube. The rod diameter measured from the water side is larger than the virtual diameter. (c) The calibration curve between the observed diameter and the virtual one for four rod sizes.

the liquid slug is greater than that from the front view. In contrast, t decided from the side view is smaller than that from the front view. Note that the outer diameter of the tube is the same for both side and front views. To make a calibration curve to describe the characteristics of the drop shape correctly, polytetrafluoroethylene (PTFE) rods with four diameters (about 1, 2, 3, and 4 mm) are used. They have a refractive index of 1.315, which is close to that of water. The image correction is not needed when the PTFE rod is placed in the tube without water. In the presence of water, nonetheless a significant distortion is distinctly seen in Figure 5 for a rod with a diameter of 2 mm. Apparently, the diameter of the rod in the liquid slug is greater than that outside the liquid slug. Therefore, the calibration curve for the vertical direction can be acquired in the plot of the rod diameter in air against that in

Figure 7. SE simulation results of water drops inside a cylindrical tube. Three regimes are identified. The inner diameter is 5 mm and the contact angle θ = 70°. 25883

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Figure 8. (a) The variation of the geometrical characteristics of the adhered drop with the drop volume. Here, W stands for the drop width and HE for the edge height. (b) The variation of the geometrical characteristics of the liquid slug with the drop volume. Here, WB stands for the width of the bottom and WT for the width of the top. The experimental observations (symbols) agree well with SE simulation outcomes (lines).

Figure 6. They agree with each other qualitatively but the quantitative comparison requires the correction of the optical distortion. Note that contact angle hysteresis is not considered in our SE simulations. Therefore, the coexistence always occurs, whether the contact angle hysteresis is present or not. Figure 8a shows the variation of the geometrical characteristics of the adhered drop (W and HE) with the drop volume which varies from 5 to 90 μL. The lines represent the simulation outcomes while the symbols denote the experimental results. Note that the experimental results for HE were corrected by the calibration curve in Figure 5c. Evidently, our experimental observations agree reasonably well with SE simulation outcomes. Figure 8b depicts the variation of the geometrical characteristics of the liquid slug (WT and WB) with the drop volume which varies from 50 to 200 μL. Again, the experimental results are consistent with SE simulations. The lower limit of the liquid slug volume determined from experiments (VS = 50 μL) is greater than that from simulations (VS = 30 μL). This difference may be due to the instability of the thin film between two menisci caused by experimental noises. Moreover, as illustrated in Figure 12, which will be discussed later, the free energy of the liquid slug is much higher than that of the adhered drop near the lower boundary of the coexistent regime. Small disturbances from the surroundings will destabilize the metastable state of the liquid slug. As a result, the predicted value of the drop volume (without disturbance) can be lower than the experimental value (with disturbance). C. Phase Diagram of Drops in Hydrophilic Cylindrical Tubes. The equilibrium conformation of a drop inside a horizontal tube depends on contact angle (θ), drop volume (V), tube radius (a), and gravity (ρg). For specified V/a 3 and θ, the equilibrium morphology can be obtained by SE simulations. The phase diagram is constructed by a collection of those results. The cases with and without gravity are considered. The gravity effect is negligible if G = (a/lc)2 ≪ 1. That is, the system can be considered as G = 0 if the tube radius is small compared to the capillary length. As shown in Figure 9 for G = 0, the phase diagram consists of three regimes: adhered drop only, liquid slug only, and coexistence of adhered drop and liquid slug. When the liquid volume is increased at a given

Figure 9. Morphological phase diagram of a drop in a tube without gravity (G = 0). The symbols ■ and □ represent adhered drop only and liquid slug only, respectively. In the coexistence regime, the symbol depicted as a red open circle denotes adhered drops with lower free energy and the red solid circle depicts liquid slug with lower free energy.

contact angle, the conformation changes from the regime of adhered drop only to the regime of coexistence, and further to the regime of liquid slug only. The regime of the coexistence is in the range of VS < V < VA, where the lower and upper limits vary with the contact angle. In general, both VS and VA decline with increasing θ. The boundary line in the regime of coexistence depicts the condition that the surface free energy of the adhered drop is equal to that of the liquid slug. Beyond the boundary line, the surface free energy of the adhered drop exceeds that of the liquid slug. Below the boundary line, the situation is reversed. Evidently, the phase diagram of a drop in a tube is distinctively different from that of drop-on-fiber. As long as the drop volume is 2.5 < V* < 3.5, both adhered drop and liquid slug conformations can exist for all contact angles. For a given volume V* = 5 (in the vicinity of VA*(θ)), the drop can exhibit both conformations for small contact angle but prefers the liquid slug shape for large contact angle (θ > 32°). As the contact angle is increased, the liquid drop tends to lower the solid−liquid interfacial area, and thus the characteristic length of the wetting area on the tube wall is reduced under the 25884

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disappearance of the adhered drop, which can be characterized by HE and HS. As illustrated in the inset of Figure 10, both HE and HS of the case G = 1 are significantly smaller than those of G = 0 for V* = 5 and θ < 32°, because the gravity force pulls down the adhered drop. As a result, compared to the case G = 0, the adhered drop can exist at larger drop volume and contact angle and the upper boundary of the coexistent regime moves upward substantially. The lower boundary of the coexistent regime is manifested by the disappearance of the liquid slug, which can be characterized by Wm. As illustrated in the inset of Figure 11, Wm of the case G = 1 is significantly smaller than that of G = 0 for V* = 2 and θ > 50° because the gravity force causes the axis-asymmetric menisci, reducing the minimum separation between them. As a result, compared to the case G = 0, the liquid slug exists only at larger drop volume so that the separation between two menisci is increased. Consequently, the lower boundary of the coexistent regime moves upward as well. Since the gravity effect is more significant for larger drop volume, the displacement of the upper boundary is more than that of the lower one. In all, gravity prefers the adhered drop shape rather than the liquid slug conformation. In polymer electrolyte membrane fuel cells, water formation and transport through the porous electrode, particularly gas diffusion layer, often involves drops in hydrophobic channels. Figure 12 also shows the morphological phase diagram of a

condition of constant drop volume. Therefore, the height of the saddle point (HS) associated with the liquid−gas interface of the adhered drop grows with increasing θ, as illustrated in Figure 10. The edge height (HE) is close to the limiting value

Figure 10. Variation of the geometrical characteristics of an adhered drop in a tube with the contact angle θ for V* = 5 without gravity (G = 0) and with gravity (G = 1).

2a = 2 for all contact angles. When θ > 32°, the edges become in contact and the adhered drop fails to exist. Only the liquid slug is present. That is, for θ = 32°, the upper boundary of the coexistent regime is VA*(θ) = 5. For a given volume V* = 2 (in the vicinity of VS(θ)), the drop can show both conformations for large contact angle but favors the adhered drop shape for small contact angle (θ < 20°). As the contact angle is decreased, the liquid drop tends to increase the solid−liquid interfacial area, and thus the characteristic length on the tube wall is increased. For a liquid slug, the slug width (W) grows and the curvature of menisci increases accordingly. As a result, the minimum width between the menisci (Wm) descends with decreasing θ, as shown in Figure 11. When the contact angle is small enough (θ < 20°),

Figure 12. Morphological phase diagram of a drop in a tube in the presence of gravity (G = 1). The symbols ■ and □ represent adhered drop only and liquid slug only, respectively. In the coexistence regime, the symbol depicted as a red open circle denotes adhered drops with lower free energy and the red solid circle depicts liquid slug with lower free energy.

drop in a hydrophobic tube for 90° ≤ θ ≤ 150° at G = 1. The result is qualitatively similar to that in a hydrophilic tube and there exist three regimes as well. Nonetheless, as illustrated in Figure 12, the air−water interface of an adhered drop is like a spherical cap rather than the saddle shape and the air−water interface of a liquid slug is a convex surface rather than a concave one for hydrophilic tubes. Moreover, in contrast to hydrophilic tubes, VS(θ) and VA(θ) associated with the coexistent regime rise with increasing θ for θ > 90°. That is, the regime of adhered drop only expands as the tube wall becomes more hydrophobic. According to our phase diagram for very small capillary number, gas diffusion will be blocked by the water slug as long as the drop volume exceeds 140 μL in a Teflon-coated tube with the radius of 2.71 mm and the contact angle 120°. To avoid gas blockage, a hydrophilic tube with low contact angle has to be used to develop adhered drops.

Figure 11. Variation of the geometrical characteristics of a liquid slug in a tube with the contact angle θ for V* = 2 without gravity (G = 0) and with gravity (G = 1).

the menisci will contact and the liquid slug fails to exist. Only the adhered drop occurs. Since the condition of two overlapped menisci will not happen for a cylindrical plug, the liquid slug can exist for very small volume if θ approaches 90°. When the gravity effect is important (G = 1), the characteristics of the phase diagram are still similar to G = 0 but the coexistent regime expands, as shown in Figure 12. The upper boundary of the coexistent regime is manifested by the 25885

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(3) Sugiura, S.; Nakajima, M.; Tong, J. H.; Nabetani, H.; Seki, M. Preparation of Monodispersed Solid Lipid Microspheres Using a Microchannel Emulsification Technique. J. Colloid Interface Sci. 2000, 227, 95−103. (4) Sugiura, S.; Nakajima, M.; Seki, M. Effect of Channel Structure on Microchannel Emulsification. Langmuir 2002, 18, 5708−5712. (5) Garstecki, P.; Gitlin, I.; DiLuzio, W.; Whitesides, G. M.; Kumacheva, E.; Stone, H. A. Formation of Monodisperse Bubbles in a Microfluidic Flow-focusing Device. Appl. Phys. Lett. 2004, 85, 2649− 2651. (6) Link, D. R.; Anna, S. L.; Weitz, D. A.; Stone, H. A. Geometrically Mediated Breakup of Drops in Microfluidic Devices. Phys. Rev. Lett. 2004, 92, 054503(1)−054503(4). (7) Anna, S. L.; Bontoux, N.; Stone, H. A. Formation of Dispersions Using ‘‘flow focusing’’ in Microchannels. Appl. Phys. Lett. 2003, 82, 364−366. (8) Thorsen, T.; Roberts, R. W.; Arnold, F. H.; Quake, S. R. Dynamic Pattern Formation in a Vesicle-Generating Microfluidic Device. Phys. Rev. Lett. 2001, 86, 4163−4166. (9) Zheng, B.; Ismagilov, R. F. A Microfluidic Approach for Screening Submicroliter Volumes against Multiple Reagentsby Using Preformed Arrays of Nanoliter Plugs in a Three-Phase Liquid/Liquid/ Gas Flow. Angew. Chem., Int. Ed. 2005, 44, 2520−2523. (10) Cheah, M. J.; Kevrekidis, I. G.; Benziger, J. B. Water slug formation and motion in gas flow channels: the effects of geometry, surface wettability, and gravity. Langmuir 2013, 29, 9918−9934. (11) Cheah, M. J.; Kevrekidis, I. G.; Benziger, J. B. Water slug to drop and film transitions in gas-flow channels. Langmuir 2013, 29, 15122− 15136. (12) Hsieh, S. F.; Chang, C. P.; Juang, Y. J.; Wei, H. H. Stretching DNA with Electric Fields Beneath Submicron Interfacial Constriction Created by a Closely Fitting Microdroplet in a Microchannel. Appl. Phys. Lett. 2008, 93, 084103(1)−084103(3). (13) Juarez, G.; Arratia, P. E. Extensional Rheology of DNA Suspensions in Microfluidic Devices. Soft Matter 2011, 7, 9444−9452. (14) 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. (15) 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. (16) McHale, G.; Newton, M. I. Global Geometry and the Equilibrium Shapes of Liquid Drops on Fibers. Colloids Surf., A 2002, 206, 79−86. (17) 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. (18) 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. (19) 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. (20) Brakke, K. The Surface Evolver. Exp. Math. 1992, 1, 141−165. (21) 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. (22) Liang, Y. E.; Tsao, H. K.; Sheng, Y. J. Drops on Hydrophilic Conical Fibers: Gravity Effect and Coexistent States. Langmuir 2015, 31, 1704−1710. (23) Hong, S. J.; Chang, F. M.; Chou, T. H.; Chan, S. H.; Sheng, Y. J.; Tsao, H. K. Anomalous Contact Angle Hysteresis of a Captive Bubble: Advancing Contact Line Pinning. Langmuir 2011, 27, 6890− 6896.

IV. CONCLUSIONS The equilibrium morphology of a drop in a horizontal hydrophilic tube can provide some useful information for two-phase flow in microfluidic devices in which the capillary force dominates over the viscous force. That is, in the limit of small capillary number (Ca → 0), the surface tension is important while the dynamic viscosity is negligible. The morphology of drop-in-tube is analogous to that of drop-onfiber, depending on the contact angle (θ), drop volume (V), tube radius (a), and gravity (ρg). When the gravity effect is negligible, an axisymmetric liquid slug (analogous to barrel) can be developed for the drop with the size large compared to the tube radius. In contrast, an asymmetric adhered drop (analogous to clamshell) occurs as the drop size is small compared to the tube radius. Both conformations can coexist in a certain range of liquid volume. When the effect of gravity is considered, the liquid slug becomes axis-asymmetric. Moreover, the interface of an adhered drop is pulled down and flattened by the gravitational force. The drop-in-tube experiments have been conducted and three regimes are identified: adhered drop only, liquid slug only, and coexistence of both conformations. By correction of the optical distortion, the geometrical characteristics of the equilibrium shape are acquired for various drop volumes. SE simulations are also performed for those systems. The comparison between experimental results and simulation outcomes shows good agreement. The morphological phase diagrams of drop-in-tube systems with and without gravity are constructed via SE simulations by varying V/a 3 and θ. They are distinctly different from those of drop-on-fiber systems. The upper and lower boundary curves, VA(θ) and VS(θ), separate the diagram into three regimes. Both VS(θ) and VA(θ) decline with increasing θ. At a given contact angle θ, the liquid slug is formed for V > VS(θ) and the adhered drop is developed for V < VA(θ). Both conformations can coexist as VS(θ) < V < VA(θ), where the drop size is comparable to the tube radius. In the presence of gravity, the adhered drop is favored because the gravitational energy comes into play. That is, even though the drop is in the regime of liquid slug only at G = 0, it can form a stable adhered drop as well at G = 1 in addition to liquid slug. As a result, the upper boundary of the coexistent regime (VA) moves upward substantially compared to the case of G = 0. Generally, gravity increases the stability of the adhered drop shape.

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AUTHOR INFORMATION

Corresponding Authors

*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 Y.-J.S and H.-K.T. thank the Ministry of Science and Technology of Taiwan for financial support.

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REFERENCES

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DOI: 10.1021/acs.jpcc.5b07212 J. Phys. Chem. C 2015, 119, 25880−25886