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Aug 2, 2016 - ABSTRACT: The formation of a liquid bridge between a cone and a plane is related to dip-pen nanolithography. The meniscus shape and ...
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Meniscus Shape and Wetting Competition of a Drop between a Cone and a Plane Yu-En Liang,† Yu-Hsuan Weng,† Heng-Kwong Tsao,*,‡,§ and Yu-Jane Sheng*,† †

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

‡

S Supporting Information *

ABSTRACT: The formation of a liquid bridge between a cone and a plane is related to dip-pen nanolithography. The meniscus shape and rupture process of a liquid meniscus between a cone and a plane are investigated by Surface Evolver, many-body dissipative particle dynamics, and macroscopic experiments. Dependent on the cone geometry, cone-plane separation, and wetting properties of cone and plane, three types of menisci can be observed before rupture and two types of wetting competition outcomes are seen after breakup. It is interesting to find that after rupture, the bulk of the liquid bridge volume is not necessarily retained by the cone which is more wettable. In fact, a sharp hydrophilic cone often loses wetting competition to a hydrophobic plane. To explain our findings, the “apparent” contact angle of the cone is introduced and the behavior of drop-on-cone/plane system is analogous to that of a liquid bridge between two parallel planes based on this concept. critical distance, the liquid bridge will break.16−19 Despite the rupture of a liquid bridge between two dissimilar planes having been widely investigated,16−19 wetting competition (liquid transfer) between the cone and plane is less understood. Upon the rupture of a liquid bridge, liquid transfer between two dissimilar surfaces relies generally on their contact angle (CA) differences. In the presence of contact angle hysteresis (CAH), it is found that the liquid volume transferred onto the acceptor surface depends strongly on the difference between the receding CAs of the two planes, not their advancing CAs.19,20 When the cone−plane system is considered, not only the CAs of cone and plane are important in controlling the wetting behavior, but also the geometry of the cone (cone angle) may influence wetting competition as the liquid bridge breaks. As the separation distances between two surfaces are below the critical distance, the equilibrium profile (morphology) of a liquid bridge can be determined by solving the Young−Laplace equation,17−19,21 or minimization of the system free energy through the Surface Evolver (SE) simulation.11,22−25 When the separation distance is beyond some critical distance, the liquid bridge becomes unstable and breaks into two separated drops. Since the shape of unstable liquid bridge deforms quickly, the rupture process is rapid and no longer satisfies the quasi-static condition. As a result, the aforementioned thermodynamic

I. INTRODUCTION A drop on a fiber is ubiquitous in nature, such as dew droplets on grass, and in industrial applications, such as wire coating.1,2 Inspired by transporting and collecting dew drops on spider webs, digital microfluidics on fibers3,4 and fibers for water collection in fog filters5−7 have been developed recently. Although fibers are frequently modeled as cylinders, curvature gradients often exist in real fibers and influence the wetting behavior. The simplest case is a conical fiber that possesses a curvature gradient, such as the tip in atomic force microscope (AFM). The wetting behavior of a droplet on a conical fiber has been widely investigated.2,8−11 Owing to the Laplace pressure (curvature) difference along the conical fiber, a barrel-shape drop always moves toward its thick end when the opposed force (e.g., gravity) is absent. However, for a vertical conical fiber, the ascending drop can stop at an equilibrium location stably under gravity. Moreover, both barrel and clam-shell shapes can coexist in a certain range of drop volumes.11 For some applications such as dip-pen nanolithography (DPN), an “ink” (drop) is delivered from an AFM tip (conical fiber) to a planar substrate.12,13 In fact, DPN can be considered as a system of a liquid bridge between a cone and a plane. The capillary forces for a liquid meniscus between a cone and a surface have been studied. The liquid bridge can be developed by an accumulation of adsorbed liquid or by capillary condensation.14,15 It is known that for a liquid bridge between two hydrophilic planes, the neck radius gets smaller with increasing separation distance. As the distance exceeds some © XXXX American Chemical Society

Received: May 25, 2016 Revised: July 18, 2016

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respectively. Two solid−liquid interfaces are involved, including liquidcone (C) and liquid-plane (P). The wettability of them is expressed in terms of liquid-cone CA (θC) and liquid-plane CA (θP), as shown in Figure 1a. The last term denotes the gravitational energy and one has g = 0 in the absence of gravity. Using the Young’s equation, cos θ = (γSG − γSL)/γLG, the dimensionless form of eq 1 becomes

approaches are unable to capture the dynamics of liquid transfer. For two identical planes or spheres, the liquid bridge will break in the middle of the two objects because of symmetry. However, for two dissimilar planes, it is natural to assume that liquid bridge will break at the thinnest point prior to its rupture. On the basis of the liquid bridge profile at the critical distance, one can estimate the amount of liquid retained by each plane after the rupture.17 To understand the rupture process and the outcome of wetting competition accurately, the dynamic approach rather than thermodynamic methods is more appropriate. In this work, the meniscus (liquid bridge) shape and wetting competition of a drop between a cone and a plane are investigated by macroscopic experiments, Surface Evolver (SE) simulations, and many-body dissipative particle dynamics (MDPD) simulations. They are dependent on CAs of cone (θC) and plane (θP), cone angle (α), and the separation distance between the cone tip and the plane (h). For simplicity, those CAs can be regarded as the receding CAs if the CAH exists. The geometry of the cone (α) is found to be an important factor in determining the meniscus shape and outcome of wetting competition. The experimental results agree qualitatively with the simulation results. Three types of meniscus shapes are identified. When h is increased to a critical value, the liquid bridge becomes unstable. The critical distance before rupture (hc) is discussed. After the rupture, it is not necessary for the majority of the drops to stay on the more wettable surface (with smaller CA). Therefore, the apparent CA of the cone (θAC) which is a function of θC and α, is introduced to explain the result of wetting competition.

F * = ALG* − {[cos θA SL *]C + [cos θA SL *]P } +

∭V * (z*) dV *

All lengths are scaled by the capillary length lc and the energy by γLGlc2, where lc = (γLG/ρg)1/2 = 2.71 mm for water. The zero point of the gravitational energy is set at the plane. Once θ and V* are specified, SE iterates to an equilibrium shape corresponding to the global minimum. II.B. Materials and Experimental Method. In addition to theoretical approaches, the morphology of liquid bridge between a cone and a plane and wetting competition are investigated by macroscopic experiments. Various cone angles are studied. Three glass cones (2α = 44°, 65°, and 115°) are ordered from Fuchun Instrument Co., Taiwan. A metal cone with 2α = 137° and a pin tip with 2α = 16° are used as well. To tune the CA of the plane (θP), various materials are used, including the glass slide, polycarbonate sheet, and paraffin film (Parafilm). All of them are obtained from Kwo-Yi Co. (Taiwan). The liquid bridge between a cone and a plane is developed as follows. The volume of the drop used in all of our experiments is 5 μL. Initially, the drop with a specified volume is deposited from a pipet onto the plane. Then the cone is gradually inserted into the drop until it touches the plane. The shape characteristics of the liquid bridge is observed from the Optem 125C optical system directly. For experiments of wetting competition, the plane is moved away from the cone slowly until the breakup of the drop or the detachment of the cone. The average retraction speed in experiments is about 100 μm/s. II.C. Many-Body Dissipative Particle Dynamics (MDPD). MDPD is modified from the dissipative particle dynamics (DPD) which is a coarse-grained particle-based approach and allows simulations for longer time scales and wider length scales than atomistic molecular dynamics. It is able to simulate a solid/liquid/ vapor system.26,27 In MDPD, the conservative force consists of both attractive and repulsive potentials as depicted in eq 3,

II. SIMULATION AND EXPERIMENTAL METHODS Figure 1 shows a schematic diagram of the liquid bridge between (a) a cone and plane and (b) two dissimilar planes. The cone angle (2α)

FijC = aijωc (rij)eij + bij(ρi + ρj )ωd(rij)eij

and the separation (h) between the cone tip and the plane are illustrated in Figure 1a. Because of axisymmetry, the shape of liquid bridge can be characterized by the base diameter (BD) on the plane and apparent BD (diameter of the contact line) on the cone. II.A. Surface Evolver. The equilibrium shape of liquid bridge between a cone and plane can be acquired by the public domain Surface Evolver (SE) software.22 It is a finite element method on the basis of free energy minimization subject to constraints. The total free energy (F) consists of surface energy and gravitational energy,23−26

F = γLGALG + [(γSL − γSG)A SL ]C + [(γSL − γSG)A SL ]P

∭V (ρgz) dV

(3)

where aij < 0 and bij > 0 represent the attractive and repulsive parameters, respectively. rij is the distance between the two particles, and eij is a unit vector to determine the force direction. The weight functions are ωc(rij) = 1 − rij/rc and ωd(rij) = 1 − rij/rd. They vary with rij and decrease to 0 as rij ≥ rc for ωc(rij) and as rij ≥ rd for ωd(rij). Here one has rc = 1.0 and rd = 0.75. Generally, aij and bij are set as −40 and 25, respectively. However, aij between a solid and liquid is set as −34.5 and −32.5 to obtain θC = 40° and θP = 70°, respectively, to study the dynamic process of wetting competition for the case that the cone is more wettable than the plane is simulated. In our simulations, MDPD is implemented by an in-house code and the number densities of liquid and solid particles are 6 and 8, respectively. All units are scaled by the bead mass, bead diameter, and thermal energy. The dimensionless volume of the liquid bridge is 1.81 × 104. Initially, the cone tip is in contact with the plane. During the separation process, the plane is lowered at the very slow rate, 10−5 every time step. The retraction speed 10−3 is small compared to the capillary wave speed O(0.1) (the pinch-off time scale from the Plateau−Rayleigh instability). It is small enough to ensure that the quasi-equilibrium condition is satisfied. The rupture of the drop takes place spontaneously as the critical distance is reached.

Figure 1. Schematic diagram of a liquid bridge between (a) a cone and plane and (b) two planes.

+

(2)

III. RESULTS AND DISCUSSION III.A. Three Types of Meniscus Shapes. The wetting behavior of a drop between a cone and a plane depends on the CAs of plane (θP) and cone (θC), cone angle (2α), and the height (h) of the cone tip to the plane. Dependent on the

(1)

where γLG represents surface tension of the liquid drop and γSL and γSG are solid−liquid and solid−gas interfacial tensions, respectively. ALG and ASL depict the liquid−gas and solid−liquid interfacial areas, B

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Figure 3. Meniscus transformation from truncated cone to neck by changing the cone angle (2α): (a) 2α = 44°, (b) 2α = 65°, and (c) 2α = 115°. Both the cone and plane possess the same wettability (θC = θP = 42°) and drop volume is 5 μL.

Figure 2. Meniscus shapes of a drop between a cone and a plane: (a) with waist, (b) truncated cone, and (c) with neck can be identified based on experiments and SE simulations.

because it requires θP < 90° and θAC > 90°. In Figure 3b with 2α = 65°, one has θAC = 99.5°, and thus the meniscus shape still belongs to type II. However, in Figure 3c with 2α = 115°, θAC is decreased to 74.5° which is smaller than 90°. Therefore, the meniscus shape is transformed to a liquid bridge with neck (type III) since it satisfies θP < 90° and θAC < 90°. SE simulations are performed for all three cases with the cones modeled as truncated cones. The good agreement between SE simulations and experiments reveals that the concept of the apparent CA associated with an imaginary upper plane is valid in determining the meniscus type and its transformation. Since SE simulation is able to capture the experimental result, a systematic study of the influence of the cone angle on the meniscus shape can be accomplished by SE simulations. Consider a system of a drop with Vr = 20 between a cone with θC = 40° and a plane with θP = 70° separated by h = 2. The cone angles vary from 2α = 70° to 180°. Figure 4 shows the simulation results for the system for which the gravity effect is negligible. The apparent base diameter (BD) of the upper plane is smallest when 2α = 70°. As the cone angle is gradually increased, the apparent CA changes from θAC > 90° to θAC < 90°, leading to the meniscus transformation from type II to type III (truncated cone → with neck). As illustrated in Figure 4a, the transition point occurs at θAC = 90°, corresponding to 2α = 80°. The influence of the cone angle on the apparent hydrophilicity associated with the cone can be manifested from the apparent BD on the upper plane. Figure 4b shows the variation of liquid bridge of type III with cone angle. When the cone angle is increased, the apparent BD on the upper plane grows larger, while the BD on the lower plane declines. The apparent BD (cone) and the BD on the plane becomes the same at 2α = 120° because θP = θAC = 70°. When θAC > θP, the apparent BD is smaller than the BD on the lower plane and the cone seems to be less hydropilic than the plane, even though θC < θP. However, as the cone angle is large enough, one has θAC < θP and the cone becomes more hydropilic than the plane eventually. III.C. Critical Distance for Liquid Bridge Stability (Cone Angle Dependence). It is known that for a liquid bridge between two parallel planes, there exists a critical distance beyond which the drop fails to connect both planes. Therefore, it is anticipated that when the distance between the cone and plane exceeds the critical value (hC), the liquid drop becomes thermodynamically unstable. Figure 5 shows the SE simulation

identified for a drop between a cone and a plane based on both experiments and SE simulations. Figure 2a demonstrates a liquid bridge with waist. That is, the widest part of bridge is in the middle, neither on the cone nor the plane. The meniscus shape with the presence of the waist requires that the apparent CA of the top surface (θAC = θC + (90°− α)) and the intrinsic CA of the bottom plane (θP) exceed 90°. As a result, type I has to satisfy the conditions θAC > 90° and θP > 90°. Figure 2b depicts a liquid bridge like a truncated cone. That is, the diameter of the cross section of the truncated cone varies monotonically. It requires that the CA of the bottom plane is less than 90° and the apparent CA of the top surface exceeds 90°, and vice versa. Therefore, type II has to satisfy the conditions θAC > 90° and θP < 90°, or θAC < 90° and θP > 90°. Figure 2c shows a liquid bridge with a neck. That is, the narrowest part of liquid bridge is in the middle, neither on the cone nor the plane. The meniscus shape with a neck requires that the apparent CA of the top surface and the CA of the bottom plane are both smaller than 90°. Consequently, type III must fulfill the conditions θAC < 90° and θP < 90°. III.B. Meniscus Type Transformations by Changing Cone Angles. According to the classification of the meniscus shape, the effects of α and θC can be combined into the apparent CA (θC + (90°− α)), which is taken as the CA associated with an imaginary upper plane containing the contact line on the cone surface as shown in Figure 1b. As a result, the three meniscus types can be easily demonstrated by considering the shape of liquid bridge between two parallel planes with different CAs, θP and θUP = θAC = θC + (90°− α). Evidently, for a given cone material with θC, the apparent CA of the upper plane can be adjusted by varying the cone angle 2α. Therefore, the meniscus type may transform as the cone angle is changed. As shown in Figure 3, three glass cones (θC = 42°) with differenct cone angles (2α) and a glass plane (θP = 42°) are used to observe the cone angle effect on the meniscus shape of the drop (5 μL). Note that all glass cones are not perfect. In fact, they are more or less like truncated cones because of their round tips. Their cone angles are determined by the extrapolation from the truncated cones. In Figure 3a with 2α = 44°, the apparent CA is θAC = 42° + (90°- 22°) = 110°. Consequently, the meniscus shape is a truncated cone (type II) C

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Figure 4. Influence of the cone angle (2α) on the meniscus between a cone and a plane by SE simulations for a system of a drop with Vr = 20, θC = 40°, and θP = 70° separated by h = 2.

planes separated by the distance H is studied. While the CA of the lower plane is θP = 70°, the CA of the upper plane θUP is chosen to equal θAC which varies with the cone angle 2α, as depicted in Figure 1. The inset of Figure 5 shows the comparison of the critical separation (Hc) between the cone− plane and plane−plane systems. Note that in the cone−plane system, H represents the height of the contact line on the cone surface and is different from the tip-to-plane distance (h). Instability occurs at H = Hc corresponding to h = hc. In the plot of Hc versus θUP , the results of both systems agree quite well. Consistent with intuition, the maximum of Hc appears at θUP = θP = 70° (2α = 120°). With respect to the maximum point, the plot of Hc is much more symmetric than that of hc. This result indicates the concept of the apparent CA is useful in interpreting the cone−plane system. To explain the wetting unstability of the system when h is near hc, five cone angles 2α = 60° and 90° (θAC > θP), 120° (θAC = θP), 150°, and 160° (θAC < θP), are discussed for Vr = 20, θC = 40°, and θP = 70°. Figure 6 shows the variation of the total free

Figure 5. SE simulation result in which the critical distance (hc) varies with the cone angle (2α) for liquid bridge of Vr = 20, θC = 40°, and θP = 70° in the absence of gravity.

result in which the critical distance varies with the cone angle for the system of Vr = 20, θC = 40°, and θP = 70° in the absence of gravity. The criterion used to determine the critical distance hc in SE simulation is that the liquid bridge fails to exist as h > hc. That is, all of the drop volume goes to either cone or plane and only a thin filament is left to connect the cone and plane. As α is increased, hc grows accordingly. However, it reaches a maximum value and then decreases with increasing α. The cone angle (2α = 120°) giving the maximum hc corresponds to the apparent CA equal to that of the plane, θAC = θP. In terms of the apparent CA, our result in Figure 5 can be explained by the fact that the liquid bridge between two similar planes can sustain a larger separation, compared to that between two dissimilar planes. When the cone angle is different from 2α = 120°, the system can be considered as a liquid bridge between an apparent plane with θAC = θC + (90° − α) and the plane with θP = 70°. As the cone angle decreases from 120° (θAC > 70°), their difference of the wettability (θAC − θP) rises and therefore the critical distance occurs at smaller values. As the cone angle increases from 120° (θAC < 70°), the wettability difference (θP − θAC) grows and a similar situation occurs. However, compared to the plane with θP = 70°, the apparent plane with θAC is more hydrophobic in the former case and more hydrophilic in the latter case. As a result, the two cases are not symmetric with respect to θAC = θP. The cone−plane system is explained by the plane−plane model in terms of the apparent CA. To verify this model, liquid bridge of the same volume Vr = 20 between two dissimilar

Figure 6. Variation of the total free energy Ft with the separation distance from the cone tip to the plane (h) for Vr = 20, θC = 40°, θP = 70°, and g = 0.

energy Ft with the distance from the cone tip to the plane (h). At a specified cone angle, Ft ascends as h is increased. The maximum of Ft (Fmax t ) occurs at h = hc. This is attributed mainly to the increment of liquid−gas area and the decrease of cone− liquid area. However, at a given h, Ft is lower for larger cone angle because of larger cone−liquid area. This consequence indicates that the reduction of the cone angle weakens the effective solvophilicity of the cone. Note that the critical points D

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Langmuir A (Fmax t ) associated with small cone angles (2α < 120°, θC > θP) are approximately equal to the free energy of a sessile drop with Vr = 20 on the plane with θ = 70°. In contrast, the critical A points (Fmax t ) associated with large cone angles (2α > 120°, θC < θP) must exceed the free energy of a pendant drop on the plane with θ = 40°. D. Wetting Competition: Many-Body Dissipative Particle Dynamics Simulation and Experiment. Although the critical distance for liquid bridge stability can be determined, the result of wetting competition cannot be correctly predicted by SE. Since there always exists a thin filament between the liquid on the upper cone and that on the lower plane, SE fails to simulate the breakup of a liquid bridge. In other words, when h > hc, all of the liquid will be transferred to either the plane or cone because of the artifact in SE. This consequence reveals that the thermodynamic approaches such as free energy minimization and solution of Young−Laplace equation are unable to predict the result of wetting competition without making ad hoc assumptions. As a result, the many-body dissipative particle dynamics (MDPD) simulations, which is a mesoscale particle-based method, is adopted to study wetting competition of a liquid bridge between a cone and plane. We intend to investigate the result of wetting competition for the case that the cone is more wettable than the plane. The system consists of a cone with θC = 40° and a plane with θP = 70°. The separation process starts from h = 0 (the cone tip in contact with the plane). The snapshots of MDPD simulations at various stages are examined by comparing to SE results. They are consistent, indicating the separation process can be regarded as quasi-equilibrium. Two scenarios are considered: (i) large cone angle case (2α = 150°) corresponding to the apparent CA of the upper plane θUP = θAC = 55° < θP and (ii) small cone angle case (2α = 60°) corresponding to the apparent CA of the upper plane θUP = θAC = 100° > θP. According to wetting competition between two dissimilar planes, most of the liquid tends to stay on the more wettable surface after the rupture of liquid bridge. The cone is more wettable than the plane (θP > θC). Therefore, most of the drop is supposed to remain on the cone based on the wettability, as h exceeds hc. Figure 7a shows the MDPD snapshots of the case (i) for a variety of h during the separation process. Note that according to SE, the critical distance is hc ≈ 26. When h is increased, BD on the plane declines gradually. Eventually, the

liquid bridge is ruptured at hc and most of the liquid is pulled off from the plane. As shown in the last snapshot of case (i), the neck breaks up and less than 1% of the drop volume is still left on the less wettable plane. Note that the drop on the tip of the cone surface will continue moving upward because of unbalanced Laplace pressure.2 The dynamic wetting competition process of case (i) from MDPD is demonstrated by Video S1 of the Supporting Information. In the case (ii), the critical distance is hc ≈ 19 according to SE. As demonstrated in Figure 7b, the apparent BD on the cone (diameter of the contact line) decreases gradually when h is increased. It is surprising to find that at h = 19.4, the liquid bridge totally slips off the cone and the whole drop remains on the plane after the separation. This consequence indicates that the drop favors the less wettable plane rather than more wettable cone. This dynamic wetting competition process of case (ii) is demonstrated by Video S2 of the Supporting Information. It seems that the rule of wetting competition between two dissimilar planes cannot apply here. By comparing those two cases, the shape of the surface in terms of the cone angle must come into play in the wetting competition. Our MDPD simulation results can be explained by the concept of the apparent CA θAC between two dissimilar planes. For case (i), the cone angle is large enough and the apparent CA of the cone is θAC = 55° which is less than that of the plane θP = 70°. Therefore, the cone wins the wetting competition. In contrast, for case (ii), the cone angle is too small and the apparent CA is θAC = 100° which is greater than that of the plane. As a result, the cone is “effectively” less wettable than the plane and loses wetting competition. For a comparison between SE and MDPD, the shapes of the meniscus have been plotted in Figure S1 of the Supporting Information. The agreement between them verifies that the quasi-equilibrium condition is satisfied. To verify our theory of apparent CA, we have also performed macroscopic experiments on wetting competition between a cone and a plane. Two systems are considered, (i) water drop (V = 5.0 μL) between a metal cone with 2α = 137° and a polycarbonate sheet and (ii) water drop (V = 5.0 μL) between the tip of a pin with 2α = 16° and a hydrophobic paraffin film (Parafilm). The experiments were conducted several times and similar results were obtained. Since the Bond number is 0.153, the gravitational effect can be ignored. The dynamic wetting competition process is demonstrated by Video S3 and Video S4 of the Supporting Information. Similar to MDPD simulations, the cone is initially in contact with the plane (h = 0), and the distance is gradually increased until the appearance of liquid bridge instability. The process of wetting competition is complicated by the presence of CAH in both cone and plane, in terms of the change of their CAs. However, it is known that the result of wetting competition of a drop between two dissimilar planes with CAH is mainly determined by their receding CAs.19,20 Note that CAH is absent in our model of apparent CA and MDPD simulations and thus θAC and θP correspond to the receding CAs in experiments. In case (i), the advancing and receding CAs of the metal cone are θC,a = 68° and θC,r = 11°, respectively. The advancing and receding CAs of the polycarbonate sheet are θC,a = 85° and θC,r = 74°, respectively. Since the apparent CA of the cone is θAC = 32.5° which is smaller than θP = 74°, it is predicted that most of the water drop tends to adhere to the cone. It agrees with the experiment result, as shown in Figure 8a. In case (ii), the advancing and receding CAs of the pin are θC,a = 76° and θC,r =

Figure 7. Snapshots of MDPD at various stages for θC = 40° and θP = 70°: (a) 2α = 150° corresponding to the upper plane θAC = 55° < θP and (b) 2α = 60° corresponding to the upper plane θAC = 100° > θP. E

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Figure 8. Experimental observations of wetting competition: (a) water drop (V = 5.0 μL) between a metal cone with 2α = 137° and a polycarbonate sheet; (b) water drop (V = 5.0 μL) between the tip of a pin with 2α = 16° and a paraffin film (Parafilm).

well with the critical distances corresponding to the instability of the liquid bridge between two dissimilar planes, indicating the validity of the apparent CA. Furthermore, the consequence of wetting competition between a more wettable cone and less wettable plane (θC < θP) is investigated by MDPD simulations in the absence of CAH. For small cone angles (θAC > θP), the plane wins wetting competition and the liquid bridge totally slipps off the cone. In contrast, for large cone angles (θAC < θP), the cone wins competition and most of the liquid is pulled off from the plane. The experiments of wetting competition between a cone and a plane are also performed. Both results of MDPD simulations and experiments can be explained by the concept of the apparent CA. In spite of CAH in experiments, the receding CA is adopted for our theory.

61°, respectively. The advancing and receding CAs of the hydrophobic plane are θP,a = 110° and θP,r = 80°, respectively. Because the apparent CA of the pin tip is θAC = 143° which is larger than θP = 80°, it is predicted that most of the water drop tends to stay on the plane. Again, it is consistent with the experiment result, as shown in Figure 8b, and slip-off is clearly observed. It should be noted that the changes of the menisci during the separation experiments can be well described by SE simulations taking into account CAH before liquid bridge instability.28 The above results confirm that in addition to wettability (CA), wetting competition relies on the shape of the surface (e.g., α) as well. For wetting competition between two cones with different cone angles, the one with larger α always wins.

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IV. CONCLUSIONS A nanodrop between a cone and a plane is closely related to dip-pen nanolithography. The wetting behavior of a drop between a cone and a plane depends on the CAs of plane (θP) and cone (θC), cone angle (2α), and the separation (h) of the cone tip to the plane. The meniscus shape and rupture process of a liquid bridge are explored by SE simulations, MDPD simulations, and experiments. Three types of meniscus shapes are identified: liquid bridge with waist, with neck, and truncated cone. In analogy to two parallel dissimilar planes, they are useful in determining the outcome of wetting competition. The meniscus type can be realized by the CA of the plane (θP) and the apparent CA (θAC = θC + (90° − α)) of the cone. The latter is defined as the CA associated with an imaginary plane containing the contact line on the cone surface. As a result, one has liquid bridge with waist (θAC > 90° (or θC > α) and θP > 90°), with neck (θAC < 90° (or θC < α) and θP < 90°), and truncated cone otherwise. Since the apparent CA associated with the cone of the same material can be adjusted, meniscus type transformation occurs by changing the cone angle 2α. Even though the cone is more wettable than the plane (θC < θP), the former can possess a higher apparent CA (θAC > θP) and exhibits less wettable behavior. The concept of the apparent CA in the cone-and-plane system is examined by evaluating the critical distance (hc) of liquid bridge stability based on SE simulations. The variation of the system free energy with the separation is calculated as well. The critical distance hc varies with the cone angle and it reaches a maximum value when α = 90° + θC − θP. These results agree

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.6b01990. Comparisons of the shapes of the meniscus between SE and MDPD for 2α = 150° and 60° (PDF) Dynamic wetting competition processes of the large cone angle case (2α = 150°) from MDPD (AVI) Dynamic wetting competition processes of the small cone angle case (2α = 60°) from MDPD (AVI) Dynamic wetting competition processes for large cone angle by experiments (AVI) Dynamic wetting competition processes for small cone angle by experiments (AVI)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. 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. F

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(26) Warren, P. B. Vapor-Liquid Coexistence in Many-Body Dissipative Particle Dynamics. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2003, 68, 066702. (27) Ghoufi, A.; Emile, J.; Malfreyt, P. Recent Advances in Many Body Dissipative Particles Dynamics Simulations of Liquid-Vapor Interfaces. Eur. Phys. J. E: Soft Matter Biol. Phys. 2013, 36, 10. (28) 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.

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DOI: 10.1021/acs.langmuir.6b01990 Langmuir XXXX, XXX, XXX−XXX