Pinning Effects of Wettability Contrast on Pendant Drops on

Oct 18, 2016 - The morphology and dynamics of the pendant drops attached to chemically patterned surfaces (pattern-pinned pendant drops) with differen...
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Pinning Effects of Wettability Contrast on Pendant Drops on Chemically Patterned Surfaces Liang Hu, Yao Huang, Wenyu Chen,* Xin Fu, and Haibo Xie State Key Laboratory of Fluid Power & Mechatronic Systems, Zhejiang University, Hangzhou 310058, China ABSTRACT: The morphology and dynamics of the pendant drops attached to chemically patterned surfaces (pattern-pinned pendant drops) with different hydrophilic/hydrophobic contrasts were investigated experimentally and numerically. During the experiments, the evolution of the contact angle and the maximum drop volume were found to be different from those of traditional pendant drops, whose contact line is pinned on the edge of the tips (tip-pinned pendant drops), and the deviation is related to both the pattern radius and the wettability contrast. Then, a hypothesis was proposed to illustrate the behavior of the contact line after it reached the pattern boundary, based on the premise that the pattern boundary possessed a certain width or fuzziness. It was concluded that the special phenomena in this case were due to the movement of the contact line, and the maximum contact radius was presented as a key parameter for the pattern-pinned drops, which is directly related to the stability and the maximum volume of the drops. Furthermore, through a simulation study on patternpinned pendant drops, the vibration performance of the meniscus was revealed as a superposition of two vibration behaviors including a low-frequency vibration due to the inertia effects and a high-frequency vibration due to the surface tension gradient within the boundary region. In addition, the hypothesis proposed above was also verified. Finally, a forecasting model to predict the maximum contact radius for the pattern-pinned pendant drops was built for different liquids and pattern wettabilities. This allows us to effectively design and optimize chemically patterned surfaces to achieve a desired pinning function or a pendant drop with desired properties.

1. INTRODUCTION Pendant drops are ubiquitous both in industrial processes and in daily life. They are important in a wide range of applications including printing, tensiometry, water harvesting, and DNA microarrays.1−5 Pendant drops can be formed at the lower end of an orifice plate or the tip of a nozzle, and both of these cases have been extensively investigated theoretically and experimentally.6−9 The body of this study on pendant drops mainly concentrates on the computation of the profile and the stability analysis as well as on the determination of the weight or volume of the detached drops.10−13 The factors affecting the shape and stability of a pendant drop come from both the liquid properties such as viscosity or surface tension and the interactions with the solid surface it is attached to. For a pendant drop hanging under a tip, the three-phase contact line is fixed, whereas the contact angle and the height vary with respect to the volume during the whole dropformation process. This can be called a constant contact line (CCL) mode, which is based on a term used in the study on the evaporation of sessile droplets.14,15 The profile of a tippinned pendant drop can be obtained by solving the Young− Laplace equation numerically: the apex of the drop is set as the origin and the fixed contact radius as the boundary condition.16,17 Because the wettability of the tip will make the problem complicated, capillary dripping tips are usually treated to be completely wetting or nonwetting, so that the contact radius is chosen to be either the inner radius or the outer radius of the tip.18 © 2016 American Chemical Society

When compared with those of tip-pinned drops, which have attracted a lot of attention from researchers, there are far fewer studies on pendant drops hanging from an infinite solid surface.19 A drop hanging from an infinite surface behaves quite differently from the tip-pinned drop as the contact area between the liquid and the solid surface is changing with respect to the volume and the contact angle depends on the property of the solid surface. In most cases, the solid surface is supposed to be smooth and homogeneous, so that the contact angle remains unchanged during the whole process, which can be referred to as a constant contact angle (CCA) mode.14,15 Our team has carried out a systematic study on the flow field in immersion lithography, where a high-index liquid is inserted into the gap between the lens and the wafer surface to improve the resolution.20 We are trying to use the energy barrier of the virtual boundary formed by a switch between hydrophilic and hydrophobic surfaces (smooth but heterogeneous) to confine the liquid and to prevent the liquid from leakage, to improve the yield of the products. Recent advances in surface modification techniques have allowed surfaces to be fabricated with well-defined chemically heterogeneous patterns.21−23 These patterns with wettability contrast can change the wetting properties of a surface; therefore, they are widely used in liquid manipulation techniques.24−26 During our experiments on Received: September 8, 2016 Revised: October 15, 2016 Published: October 18, 2016 11780

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Figure 1. Experimental setup for studying the dynamics of the pattern-pinned pendant drop formation. The inset on the top left is the top view of the glass plate we used in the experiments. The inset on the top right demonstrates the side view of a pendant drop.

also used as a verification to the hypothesis. Finally, a model to predict the maximum contact radius for a pattern-pinned drop is built, which is useful for designing a specific chemically heterogeneous pattern with certain distribution characteristics to realize an efficient pinning function and can also be used to form a pendant drop as the tip-pinned case.

pendant drops, it was found that after the meniscus of the drop reaches the pattern boundary, although the contact line seems to be pinned on the boundary, the contact angle and the profile behave in a different way with respect to those of the traditional tip-pinned drops. As there has been a lack of attempts to introduce chemically heterogeneous patterns in the study of the pendant drop so far, how a pendant drop interacts with the patterned surface is still unknown. Therefore, it is necessary to investigate the behavior and dynamics of a pendant drop attached to chemically patterned surfaces, which can provide guidance for the design in immersion lithography and also provide a new method to generate pendant drops in many cases. Chemically heterogeneous surfaces have been investigated extensively in sessile drop cases, which mainly concentrate on the equilibrium contact angle, contact angle hysteresis, and the equilibrium morphology of the contact line.27−30 It is found that the droplets covering a single pattern domain exhibit contact angles that do not satisfy the usual Young equation. This phenomenon can be explained as the pattern boundaries possess a certain width or fuzziness in reality, and only the position-dependent generalized Young equation is fitted here.31,32 This position-dependent characteristic will also affect the formation of a pendant drop, whose three-phase contact line is located within the pattern boundary region. However, as pendant drops can be elongated and broken under gravity, the dynamic behavior that they exhibit will be quite different from that of sessile drops. In this paper, we carry out a study on the evolution of the pendant drops attached to surfaces with chemically heterogeneous patterns, aimed at uncovering the mechanism of the interaction between the contact line of the pendant drop and the patterns of the solid surface. We report on the evolution of the contact angle and the maximum drop volume in the experiments and find that the dynamic behavior of a patternpinned drop is affected more dramatically by the contact radius than that of the tip-pinned drops. Then, a hypothesis is proposed to explain these phenomena based on the movement of the three-phase contact line and the maximum contact radius. A simulation study based on the volume of fluid (VOF) method is followed by which a detailed and more targeted investigation can be achieved as the dimension and the wettability contrast of the pattern boundary region can be altered randomly in this way. The simulation consequences are

2. EXPERIMENTAL SECTION 2.1. Experimental Details. The experiment apparatus consisted of an orifice plate through which the liquid was delivered at a constant volumetric flow rate and a pendant drop was formed and grown under the plate, as seen from Figure 1. Ultrapure deionized water (surface tension γ = 72.8 mJ/m2; viscosity η = 0.997 mPa·s) was used as the liquid in all experiments, with the flow rate being controlled by a microsyringe pump (Shenchen SPLab02) at 1 mL/min (the drop formation process at this flow rate can achieve a reliable picture of drop evolution and a good repeatability6). A quartz glass plate was chosen as the solid substrate, and the diameter of the orifice in the middle of the plate was 0.8 mm, which is smaller than the size of the plate. Chemically heterogeneous patterns with a single circular shape in the millimeter scale were constructed with sol−gel coatings. A hydrophobic pattern was first formed by spray-coating, meanwhile protecting the hydrophilic area with a mask. Then, the hydrophilic coating could be printed on the inner surface as this area could accept the sol−gel coating, but the outer hydrophobic area would repel the sol−gel coating. This realized two kinds of patterns with the same dimension but with different wetting characteristics. In both cases, the diameters of the patterns varied from 5 to 9 mm. For the patterns with a larger wettability contrast (here named pattern A), the equilibrium contact angle of the solid surface in the inner region was 27 ± 2°, which was surrounded by the outer region whose equilibrium contact angle was 106 ± 2°. On the other hand, patterns with a smaller wettability contrast (pattern B) possessed an equilibrium contact angle of 53 ± 3° inside of the boundary, and the angle outside of the boundary was the same with pattern A. All contact angles in the experiments were measured using a contact angle measuring device (Theta optical tensiometer T200-Auto1B). The side view of the pendant drop was captured using a high-speed video camera (Phantom M320S with a Sigma 105 mm micro lens). Together with a high-brightness LED (SHIBUYA LHP-40W) as backlight, the images of the drops were captured at a time resolution of 2000 fps. These results were analyzed using an image-processing algorithm in MATLAB. The edge of the drop was extracted using a gray-level fit near the border of the drop. Contact angles were determined by a third-order polynomial fitting applied to the lower part of the droplet contour (approximately 15−20% of the droplet height) close to the three-phase contact line. 11781

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Figure 2. Time sequence of shapes of a water drop forming under a glass surface with a pattern radius of 2.5 mm at the liquid flow rate of 1 mL/min. The relative times for each image are (a) t = 0, (b) 300, (c) 900, (d) 2000, (e) 4000, (f) 5500, (g) 5610, (h) 5630, (i) 5637, and (j) 5639 ms. 2.2. Experimental Results. The experiments were performed on patterns A and B for all radii in the range. During the experiments, the pendant drop underwent three periods with different morphologies and dynamic characteristics; the entire process is shown in Figure 2. It first grew with a CCA before touching the pattern boundary, as shown in Figure 2a−c. Then, a “fake CCL” motion similar to the tip-pinned drops appeared after the meniscus of the pendant drop reached the pattern boundary, as shown in Figure 2d−f. Finally, when the drop achieved a critical size, a tiny increment in volume would make the drop unstable and break into the fragments, as shown in Figure 2g−j. We focused on the second period whose contact line seemed pinned on the pattern boundary: this is the period that gives rise to the specificity that distinguishes the pattern-pinned drop from the traditional pendant drop in a CCA or CCL mode. The contact angle was used to describe the behavior of the pendant drop in the description of the experimental results. The morphology parameters of the tip-pinned drops, which can be calculated numerically from the Young−Laplace equation, were used for comparison with the experimental results as the theoretical values. It is well known that in the tip-pinned case, the contact angle increases to a maximum and then decreases as the volume is progressively increased, and the maximum value of the contact angle is related to the tip radius. However, the contact angle in our experiments shows a different growing trend when compared with the theoretical values. In Figure 3, the experimental results of the contact angle versus volume on pattern B are presented as an example with the corresponding theoretical values. It can be seen from the figure that the contact angle fits well

with the theoretical values far before reaching the maximum. As the contact angle getting close to the maximum, the deviation between the experimental and theoretical results becomes obvious. Then, after crossing the peak, the contact angle decreases with a different rate when compared with the traditional tip-pinned drops. This rate is found to be related to both the pattern radius and the wettability contrast. With regard to the pattern radius, we can see that the contact angle decreases faster than the theoretical values when the pattern radius is 2.5 mm. However, the decreasing rate slows down with the growth of the radius, and when the radius is 4.5 mm, the decreasing rate is much slower than that of the theoretical values. Besides, the characteristic of the decreasing rate changing with the pattern radius is affected by the wettability contrast, which can be seen in Figure 4. The

Figure 4. Comparison of contact angles under different pattern/tip radii before losing stability. Results from experiments on pattern A and pattern B are shown by triangles and circles, respectively. Squares represent the theoretical values. (Lines are present only to show the trends for each situation). three angles used for comparison at the same radius were chosen at the same volume, which was close to but did not reach the critical volume in all three cases. We can see from the image that a pendant drop on pattern A shows a more moderate variation in the contact angle when changing the pattern radius than the drop on pattern B, and both of them behave gently when compared with the theoretical values in the tip-pinned drop situation. It can be seen from Figure 3 that the maximum volume of a drop attached to the pattern also differs from the corresponding theoretical values as the descending rate of the contact angle varies. The maximum volume in the experiments was determined using a concise method as follows. A nondimensional parameterthe ratio of height and contact radiuswhich could give a better description of the morphology changes was used. Its gradient approached zero when the drop was stable but increased suddenly with a sharp turn on the curve once the drop lost its stability, which made it easier to distinguish the maximum volume at the onset of instability. The comparison of the maximum volume between the experimental results on pattern B and the corresponding theoretical results is shown in Figure 5a, which shows that the maximum drop volume that a heterogeneous pattern can hold is slightly different from the traditional tip-pinned drops; the volumes of pattern-pinned drops are affected by the contact radius more severely. Besides, the maximum volume held by different wettability contrasts is also different, as shown in Figure 5b. Here, a

Figure 3. Evolution of contact angle vs drop volume obtained from the experimental results on pattern B (squares) and the theoretical values (solid line) after the contact line reaches the pattern boundary. 11782

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boundary region in our cases represents a series of hystereses, so that the above behavior is repeated incessantly; the schematic diagram is shown in Figure 6. It can be seen from the figure that the CCL and the

Figure 6. Schematic diagram of the movement of the three-phase contact line within a pattern boundary region. The CCA and CCL behaviors are indicated by pink and blue arrows, respectively. CCA behaviors alternated after the contact line reached the pattern boundary region. The CCA behavior is responsible for the expansion of the contact area, and the CCL mode is responsible for conquering the energy barrier that is caused by the surface tension gradient within the pattern boundary region. However, as the true pattern boundaries can be treated as a continuous transition region, the alternation of these two modes becomes quite smooth. In our experiments, all pendant drops seemed to be pinned on the pattern boundaries on a macroscale. However, on a microscale, the position of the contact line may vary in this region. The maximum contact radius (Rmax) was determined by the competition between the meniscus contact angle and the intrinsic contact angle at the local position. Because the intrinsic contact angle is supposed to increase with the contact radius, when the maximum meniscus contact angle can not cope with the local intrinsic contact angle anymore, the contact line will stop at this position and the pendant drop gets its Rmax. Therefore, we can estimate that the actual contact line position of the drop with a 4.5 mm pattern radius is smaller than 4.5 mm, whereas the actual contact line position of the drop with a 2.5 mm pattern radius is larger than 2.5 mm. The deviation between the actual Rmax and the ideal boundary location will lead to a great influence on the macrodynamics of the pendant drop, which was introduced in Experimental Results Section. Therefore, it can be concluded that Rmax is an important parameter in the pattern-pinned drop formation. The characteristics described above bring about some unique features that belong only to the pattern-pinned pendant drops and can be used to explain the special phenomena observed during our experiments. As mentioned, Rmax might deviate from the ideal pattern radius under different radii since the pattern boundary region possesses a finite width in reality. Additionally, according to the calculation results of the Young−Laplace equation, the maximum meniscus contact angle is inversely proportional to the contact radius, which is why the maximum meniscus contact angles obtained in experiments appear different from the theoretical values and exhibit a radius-related behavior. Besides, as shown in Figure 3, the descending speed of the contact angle varies under different pattern radii, which is quite similar to that observed in the traditional CCL mode. In the CCL mode, when increasing the tip radius, the contact angle will go through a more placid evolutional process, with the descending speed slowing down at the same time. However, it can be seen that the decline curves of the contact angle in our experiments sweep across a larger area than the area swept by the theoretical values when changing the pattern/tip radius, which illustrates that the dynamic behavior of the patternpinned pendant drop is affected by the contact radius more dramatically. In addition, as the distribution of the intrinsic contact angle on pattern A is more compact than that on pattern B because of a larger wettability contrast, the locations that achieved the maximum meniscus contact angle under different radii lie closer to each other within the boundary region on pattern A, which means that the values of Rmax under different pattern radii vary in a smaller range on pattern

Figure 5. (a) Comparison of maximum drop volume between the experimental results on pattern B (circles) and the corresponding theoretical values (squares) under different pattern/tip radii. (b) Comparison of maximum drop volume, which is represented by a dimensionless parameter on pattern A (triangles) and pattern B (circles) under different pattern radii. ΔV is the difference in the maximum drop volume between the experimental results and the corresponding theoretical values under the same radius. dimensionless parameter ΔV/R3 was used to measure the maximum drop volume because the variation trends were more clearly shown in this way. From the figure, we can see that under a small radius, the maximum drop volume held on pattern A is smaller than the one on pattern B. However, the difference in the maximum volume between these two wettability contrasts weakens with the increase in the radius. When the pattern radius is 4.5 mm, the maximum drop volumes on pattern A and pattern B are nearly the same. As concluded above that the volumes of pattern-pinned drops are affected by the contact radius more severely than those of tip-pinned drops, it seems that a larger wettability contrast of the pattern will strengthen this characteristic. 2.3. Hypothesis and Discussion. The aim of this section is to advance a hypothesis of how the contact line of the pendant drop interacts with the pattern boundary region and to explain the special phenomena found during the experiments. We first introduce the premise of the hypothesis: all chemically heterogeneous pattern boundaries possess a finite width or fuzziness. The reason why the contact angle can change with volume when the drop seemed pinned on the pattern boundaries is due to the microscopic movement of the contact line within the pattern boundary region. A familiar assumption has been adopted in the generalized position-dependent Young equation for sessile drops,32,33 but there is still a lack of deeper cognition with the mechanism of how the contact line behaves within the pattern boundary region, which makes it inapplicable in the pendant drop case, where the contact line behavior is more complicated. Besides, we distinguish two kinds of contact angles: intrinsic contact angle and meniscus contact angle. The former represents the inherent property of the solid surface at the local position and the latter refers to the angle formed by the gas−liquid interface and the horizontal solid surface at the three-phase contact line, as shown in the top right inset in Figure 1. Because both the contact radius and the contact angle change with the volume, the behavior of the contact line within the boundary region is a combination of the CCA and CCL modes. If we assume that the intrinsic contact angle increases linearly throughout the whole boundary region, at each point in the boundary region, the contact line will encounter an energy barrier when it tries to extend. This situation is similar to the contact angle hysteresis on the usual surfaces, where the contact line will not move until the meniscus contact angle reaches the advancing contact angle of the solid surface. However, the 11783

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Langmuir A. As Rmax is the key parameter in determining the dynamic behavior of the pattern-pinned drops, a smoother change in Rmax can make the dynamic behavior of the drops vary in a more gentle way. That is why the pendant drop on pattern A shows a more moderate variation in the contact angle than the drop on pattern B when changing the pattern radius, as shown in Figure 4. The maximum drop volume is also affected by the special characteristics of the contact angle of pattern-pinned drops, which can be seen in Figure 5. This parameter is mainly determined by the latter half of the formation process after the contact angle reached the maximum, as the dynamic behavior of the pendant drop begins to be different from each other for the tip-pinned case and the patternpinned case at this moment. The total variation in the contact angle during this period is nearly the same in all three cases because the final contact angle at the dripping moment is independent of the surface properties. Thus, the key parameter determining the maximum drop volume is the descending speed of the contact angle. A slower descending speed will lead to a larger drop volume and vice versa, as shown in Figure 3. This feature further proves the conclusion that the dynamic behavior of the pattern-pinned drop is affected by the contact radius more severely.

the contact line through the comparison with case 1. Case 6 was a homogeneous surface without the contrast wettability pattern, and case 4 is prepared for the error analysis in the next section. In cases 3 and 4, the flow rate of 1 mL/min was replaced by a larger rate to shorten the calculation time. As a larger pattern radius could weaken the effects that the flow rate had on the pendant drop,6 this flow rate of 3.015 mL/min could be used in the simulation when the pattern radius was 4.5 mm. Here, we supposed that the distribution of wettability in the boundary region was a smooth and continuous transition. The change in the contact angle on the solid surface was realized using user-defined functions and was assigned to the specified location in Fluent. Before simulation, the liquid drop was initialized as a segment of a sphere, and the volumetric flow rate in the experiment was realized as a velocity inlet in the boundary settings.

3.2. RESULTS AND DISCUSSION Figure 7 shows the simulation results of the contact radius and the height of the pendant drop versus drop volume in cases 1

3. SIMULATION SECTION 3.1. Simulation Details. A numerical study was conducted to get a deeper and more targeted insight into the dynamics of the contact line as the dimension and wettability contrast of the heterogeneous pattern can be altered randomly in the simulation. The shape of a three-dimensional axisymmetric pendant drop was computed numerically using the CFD software Fluent, and the evolution of the gas−liquid interface was tracked by the VOF method. Gravity was incorporated into the simulation; the surface tension of the liquid and the interaction with the substrate were also considered. The simulation here was used to give a more targeted and detailed investigation of the discrepancy between the tip-pinned drops and the pattern-pinned drops and to cultivate a deeper understanding of the impact of the pattern boundaries on the pendant drop formation. Therefore, the boundary region between the two areas with opposing wetting properties was well-defined in the simulation. We built six simulation models; the wetting properties, dimensions, and flow rate of each model are given in Table 1. All surfaces in the simulation models were

Figure 7. Simulation results of the contact radius and height of the pendant drop vs drop volume in case 1 (the blue line and the black line) and in case 2 (the pink line and the red line), respectively.

Table 1. Wetting Properties, Dimensions, and Flow Rate of the Simulation Modelsa case

θin (°)

Δθ (°)

w (mm)

R0 (mm)

Q (μL/min)

1 2 3 4 5 6

55 55 55 30 55 55

50 100 50 75 50 no

1 1 1 1 1 no

2.5 2.5 4.5 4.5 2.5 no

1000 1000 3015 3015 3015 1000

and 2. The growth of the contact radius shows an obvious inflection point after the contact line reaches the boundary region, after which the growth rate slows down. The maximum values that the contact radius can achieve are quite different between these two cases, which is in accordance with the experimental results. Meanwhile, the height of the drop keeps increasing with the volume. Here, the wettability gradient through the boundary region is the reason for these discrepancies. It acts like a damping force to the movement of the contact line, which becomes stronger with a tighter distribution of the wettability, as can be seen from the comparison between case 1 and case 2. To investigate the evolution of the contact angle, the maximum contact angle was taken as the separatrix that divided the whole formation process of a pendant drop into two parts. An interesting phenomenon was observed during the ascent stage of the evolution of contact angle during the simulation, which can be seen from Figure 8. The black line represents a drop in case 6, a homogeneous surface, whereas the blue line

a θin is the contact angle of the area inside of the boundary, Δθ is the contact angle difference throughout the whole boundary region, w is the width of the boundary region, R0 is the ideal pattern radius with no width, and Q is the flow rate of the liquid.

supposed to be smooth. The shape of the heterogeneous patterns used in the simulation was circular, which is the same as the shape used in the experiments. However, the pattern boundary region is magnified in dimension, forming a circular ring of width 1 mm. The wettability contrast, dimension, and flow rate were changed, respectively, in cases 2, 3, and 5 to estimate the effects that these factors have on the dynamics of 11784

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Figure 10. Evolution of the contact angle vs volume of two patternpinned drops with different wettability contrasts; case 1 is shown by the blue line and case 2 is shown by the black line.

Figure 8. Ascent stage of the evolution of contact angle in the simulation of case 1 (the blue line) and case 6 (the black line).

theoretical values here were calculated using the maximum contact radius obtained by simulation as a boundary condition. It can be seen that the contact angle decreases with a rate different from that of the theoretical values, faster in case 1 and slower in case 3, which is in good agreement with the experimental results shown in Figure 3. The simulation results of the two pattern-pinned drops with different wettability contrasts are shown in Figure 10. The descending rate of the contact angle in case 2 is larger than that of the other one, and both of these two cases are faster than that exhibited by the theoretical values. This can be explained by the hypothesis introduced above: the maximum contact radius in case 1 is larger than that in case 2, which can be seen in Figure 7, so that the descending rate of the contact angle will get slower when the maximum contact radius increases. This phenomenon can also be verified by the experimental results shown in Figure 4, where case 1 corresponds to pattern B and case 2 corresponds to pattern A. On the basis of the above results, one can draw the conclusion that the contact radius has a more significant influence on the pattern-pinned pendant drops than on the traditional tip-pinned drops. Therefore, a little change in the maximum contact radius may have a great impact on the macroscopic morphology and stability of a pattern-pinned pendant drop. This characteristic is directly related to the maximum drop volume that a heterogeneous pattern can hold, which can be used to achieve a wider range of drop volumes within a limited space. In practice, one can obtain a desired maximum contact radius through an optimization of the parameter design of the pattern boundary region. A forecasting model of the maximum contact radius used in the pendant drop hanging under the chemically heterogeneous pattern is introduced in the following section.

represents a drop in case 1. It can be seen that the dynamic behaviors of the contact angle under these two cases are quite different. For the former, the meniscus vibrates during the expanding process and the vibrational frequency decreases as the volume increases. The source of the vibration mainly comes from the inertia effect caused by liquid injection and the region initialization in the simulation. However, the vibration of the meniscus in case 1 shows a very different mode as the volume increases, especially when the volume exceeded 30 μL, the frequency seems a superposition of two vibration behaviors. This is because there are two vibration sources in this situation: one is the low-frequency vibration due to the inertia effect as the former case, and the other one is a high-frequency vibration due to the surface tension gradient within the boundary region, which is unique in the pattern-pinned situation. This phenomenon occurs mainly in the ascent stage of the dropformation process and can be weakened by a higher liquid flow rate (compared with the simulation results in case 5). Later, the descent stage of the evolution of contact angle was explored, and the results are shown in Figures 9 and 10. Figure 9 demonstrates a comparison between the simulation results of cases 1 and 3 against the corresponding theoretical values. The

4. FORECASTING MODEL Generally speaking, the pattern-pinned pendant drops exhibit a behavior very much similar to the traditional tip-pinned drops. However, a drop of this kind seems more sensitive to the maximum contact radius, which will lead to a different performance on the morphology, stability, and the final volume. This characteristic can be used to realize an efficient pinning function and can also be used to form a pendant drop as in the tip-pinned case. However, with the advantages of a wider application including for the cases where it is not suitable for introducing a topology boundary into the structure and a wider range of drop volumes within a limited space, it is necessary to

Figure 9. Descent stage of the evolution of contact angle in experiments (the blue line) compared with that of the corresponding theoretical values (the black line) in simulation cases 1 and 3. 11785

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Langmuir perform a parameter design for the chemically patterned surfaces. With the help of highly advanced fabrication techniques, a chemically patterned surface with sophisticated wettability distribution on a micrometer, or even a nanometer, scale can be realized. It may be possible to achieve pendant drops with desired shapes, properties, and behaviors with a chemically pattern-pinned method. To achieve this goal, a forecasting model is established to predict the maximum contact radius, which is very important in determining the dynamic behavior of the contact line. This model is based on the balance between the maximum meniscus contact angle and the intrinsic contact angle. The evolution of the meniscus contact angle with volume can be calculated as the CCL mode using the Young−Laplace equation. The Young−Laplace equation describes the mechanical equilibrium of the interface between two homogeneous fluids, which relates the Laplace pressure across an interface with the curvature of the interface and the interfacial tension34 ⎛1 1 ⎞ Δp = Δp0 − ρgz = γ ⎜ + ⎟ R2 ⎠ ⎝ R1

Figure 11. Evolution of meniscus contact angle vs volume during the formation of the pendant drops as a function of the pattern radius. The top right inset shows the relationship between the maximum meniscus contact angle and the Bond number.

Once the contact radius is decided, the maximum contact angle can be calculated using eq 7. The wettability distribution of the pattern boundary region may vary in different occasions, but in most cases, the contact angle is thought to transit linearly within the boundary region,35 which is given by

(1)

where Δp is the Laplace pressure across the interface; Δp0 is a reference pressure at z = 0 (namely, the apex of the drop); and R1 and R2 are the principal radii of curvature. This equation can be solved numerically by a set of coupled first-order differential equations in terms of the arc length s measured from the drop apex dϕ = (H − Z) − (sin ϕ)/X dS

(2)

dX = cos ϕ dS

(3)

dZ = sin ϕ dS

(4)

X(0) = Z(0) = ϕ(0) = 0

(5)

⎛ R − R0 + θi(R ) = θin + ⎜⎜ w ⎝

R=

(6)

(8)

(J +

J 2 − 4MK )2 4M2

(9)

⎛ ρg ⎞1/4 1 J = −1.756·⎜ ⎟ = −1.756· a ⎝γ ⎠

(10)

⎛ 2R − w ⎞ ⎟ − 3.256 K = θin − Δθ·⎜ 0 ⎝ 2w ⎠

(11)

M=

measures the importance of surface tension forces compared with gravity, both of which dominate the formation process of the pendant drops. Here, R in the Bo equation represents the contact radius, where the meniscus and the solid surface interact. The relationship between θm_max and Bo shown in the top right inset in Figure 11 can be approximated by the following equation θm_max(R ) = −1.756·Bo0.25 + 3.256

⎞ ⎟⎟ ·Δθ ⎠

where R is the abscissa of the contact radius; θin is the equilibrium contact angle of the inner area of the solid; Δθ represents the difference in the equilibrium contact angle across the whole boundary domain; w is the width of the boundary region, which is usually in a micrometer scale; and R0 is the radius of an ideal pattern with no width. It should be noted that the contact angle hysteresis is not involved in this model at the moment. If the contact angle hysteresis cannot be ignored, one should replace θin and Δθ with the advancing contact angle of the solid surface. As the meniscus contact angle is equal to the intrinsic contact angle when the contact line expanding to its maximum position, the maximum contact radius of a pattern-pinned pendant drop can be deduced with the use of eqs 6−8

The parameters indicated in capital letters in the above equations represent dimensionless variables that are nondimensionalized using the capillary length a = γ /ρg . Besides, φ is the angular inclination of the drop interface to the x axis; H is the dimensionless hydrostatic heights at the drop apex. The relationship between the contact angle and the pendant drop volume under different pattern radii is shown in Figure 11. The black line in the figure represents the maximum contact angle at each pattern radius. This curve cannot be calculated directly but can be determined by interpolation from the computed θ and V values. The top right inset in Figure 11 shows the relationship between the maximum meniscus contact angle (θm_max) and the Bond number (Bo). The Bond number Bo = ρgR2/γ

w 2

Δθ w

(12)

From eqs 9−12, it can be seen that once the liquid used in the experiment is determined as well as the geometrical and wetting parameters of the chemically patterned surface are known, the maximum contact radius can be located precisely. This can be proved by a comparison between the simulation results and the calculated results obtained through the forecasting model introduced above, which is shown in Figure

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patterned surfaces to achieve a desired pinning function or a pendant drop with desired properties.

12. It can be seen that the calculated values are in good agreement with the simulation results, and the maximum error

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (No. 51575476), the Science Funding for Creative Research Groups of the National Natural Science Foundation of China (No. 51521064), and the Fundamental Research Funds for the Central Universities (No. 2016XZZX002-08).

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Figure 12. Comparison of the maximum contact radius obtained using the simulation (circles) and the forecasting models (squares). The maximum error (the black line) is less than 1.2%.

REFERENCES

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is less than 1.2%. If the simulation condition is taken into consideration, then it can be concluded that the wettability distribution and the pattern boundary dimension nearly have no effect on the accuracy of the model. The only factor that does matter is the fluid flow rate: when it becomes larger, the formation process of the pendant drop tends to deviate a little from the steady state, which will reduce the accuracy of the model. Therefore, our forecasting model for the maximum contact radius is suitable for the steady state and the quasisteady state.

5. CONCLUSIONS We performed experiments and numerical simulations to characterize the morphology and dynamics of the pendant drop pinned on a surface with a single circular chemically heterogeneous pattern. The calculated values of the traditional tip-pinned drops were chosen as the theoretical values and reference for comparison. We showed that the evolution of the contact angle and the maximum drop volume in experiments deviated from the theoretical values, and the deviation was found to be related to both the pattern radius and the wettability distribution. Then, a simple hypothesis based on the movement of the three-phase contact line was proposed to illustrate the dynamics of the contact line when it stepped into the pattern boundary region. The behavior of the contact line within the boundary region is a combination of the CCA and CCL modes, and the experimental observations were satisfactorily explained by this hypothesis. Most of all, the maximum contact radius was emphasized as a key parameter for the pattern-pinned drops because of the characteristic that the dynamic behaviors of the pattern-pinned drops are affected more dramatically by the contact radius when compared with the traditional tip-pinned drops, which will lead to an influence on the stability and the maximum volume of the drops. Besides, with the help of the numerical simulation, we carried out a detailed study of the interaction between the contact line and the solid surface. The vibration performance of the meniscus was discussed, and the hypothesis we proposed was verified here. Finally, the forecasting model built in this study enables us to predict the maximum contact radius of a pattern-pinned pendant drop, for different liquids and pattern wettabilities. This allows us to effectively design and optimize the chemically 11787

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DOI: 10.1021/acs.langmuir.6b03318 Langmuir 2016, 32, 11780−11788