Directional Transport of a Liquid Drop between ParallelâNonparallel

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Directional Transport of a Liquid Drop between Parallel-Nonparallel Combinative Plates Yao Huang, Liang Hu, Wenyu Chen, Xin Fu, Xiaodong Ruan, and Haibo Xie Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b00172 • Publication Date (Web): 25 Mar 2018 Downloaded from http://pubs.acs.org on March 25, 2018

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Directional Transport of a Liquid Drop between Parallel-Nonparallel Combinative Plates Yao Huang, Liang Hu*, Wenyu Chen, Xin Fu, Xiaodong Ruan, Haibo Xie State Key Laboratory of Fluid Power & Mechatronic Systems, Zhejiang University, 38 Zheda Road, Hangzhou 310027, China

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ABSTRACT. Liquids confined between two parallel plates can perform the function of transmission, support or lubrication in many practical applications, due to which to maintain liquids stable within their working area is very important. However, instabilities may lead to the formation of leaking drops outside the bulk liquid, thus it is necessary to transport the detached drops back without overstepping the working area and causing destructive leakage to the system. In this study, we report a novel and facile method to solve this problem by introducing the wedgelike geometry into the parallel gap to form a parallel-nonparallel combinative construction. Transport performances of this structure were investigated. The criterion for self-propelled motion was established, which seemed more difficult to meet than that in the nonparallel gap. Then, we performed a more detailed investigation into the drop dynamics under squeezing and relaxing mode, since the drops can surely return in hydrophilic combinative gaps, while uncertainties arose in gaps with weak hydrophobic character. Therefore, through exploration of the transition mechanism of drop motion state, a crucial factor named turning point was discovered and supposed to be directly related to the final state of the drops. On the basis of the theoretical model of turning point, the criterion to identify whether a liquid drop returns to the parallel part under squeezing and relaxing mode was achieved. These criteria can provide guidance on parameter selection and structural optimization for the combinative gap, so that the destructive leakage in practical production can be avoided.


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INTRODUCTION Liquids trapped between two parallel plates are ubiquitous both in daily life and in industrial processes. They play an important role in a wide range of applications including printing, microfluidics, hydraulics and integrated circuit manufacture, since the liquid drops or puddles 1-7

may act as a part of the optical path, or serve as hydrodynamic support, hydrodynamic lubrication and so on. Under these circumstances, in order to guarantee the normal operation of the system, it is of great importance to confine the liquid within its working area and maintain the liquid stable. However, the stability may be destroyed under external forces. For example, in immersion lithography, the liquid confined between the lens and wafer will be deformed or even ruptured into drops under strong shear stress due to the high-speed scanning motion of wafer. Once the liquid 1, 8

drops break through the confinement of the lens-wafer gap, they will be left on the wafer and cause exposure defects, which should be avoided in practical production. Therefore, it is vital to find a method to transport the detached drops back to the bulk liquid without overstepping the working area, that is, directional drop transport is desired to prevent the leakage of the liquid confined in a parallel gap from bringing damages to the normal operation of the system. It is well known that chemical gradient and micro/nano structure designed on a surface can be used to propel liquid drops in preferential directions, which can also be applied to the parallel 9-12

gap to realize a directional transport of the confined liquid drop. This movement can further be 13

strengthened by repeated squeezing and relaxing the drop when the plates possessing a significant wetting hysteresis. However, the disadvantages of these surfaces like mechanical strength, 14

durability and high production cost still restrict their practical application.

15-16

Therefore, a simpler

method is desired in our cases. It is also known that, a nonparallel (wedgelike) geometry can lead to instability of a confined drop due to the differential Laplace pressure along the length of the


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drop.

12, 17-21

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Without consideration of contact angle hysteresis (CAH), a liquid drop confined between

nonparallel hydrophilic plates will fill the corner of these two plates spontaneously.

22-23

However,

when CAH is taken into account, this self-propelled motion may be hindered, hence the drop should meet certain requirements to restart and then fill the corner.

24-25

Furthermore, the drop can

also move toward the corner by squeezing and relaxing one of the plates periodically.

23-24, 26-28

Due to

its simple structure and good adaptability, nonparallel gaps have practical applications in, for instance, water collection, liquid-liquid microextraction and oil recovery.

29-32

Nevertheless, parallel gap cannot be replaced by nonparallel geometry since the liquids confined between two parallel plates play an important role in many practical applications, as we mentioned above. Therefore, we come up with a novel and simple method for directional drop transport by introducing the wedgelike geometry to the border of the working area in parallel gap to form a parallel-nonparallel combinative structure (or p-n gap for short). In normal conditions when the confined drops remain steady or quasi-steady, only the parallel part of the p-n gap works, same as usual. Once the liquid stability damaged, the nonparallel part will come into play as a rescue, aiming at shipping the detached drops back to the parallel part. When this situation arises, the different parts of the drop’s air-liquid interface will be bounded by different gap geometry, thus the influence of the parallel part on drop motion should be taken into consideration. The parallel gap may facilitate or suppress the directional drop transport, resulting in the dynamics of the drop quite different from the traditional one in a nonparallel gap. Accordingly, it is very important to know under what situations a directional drop transport can be realized, so that an effectively leakage prevention can be achieved. In this paper, we carried out a study on the dynamic behavior of a drop confined in a parallelnonparallel combinative gap. The conditions for self-propelled drop motion were identified by the


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comparison of liquid pressure on two opposite edges of a drop, which seemed more difficult to be satisfied than that in the nonparallel gap. Then, we performed the experiments with mechanical actuation and found that things became more complicated under squeezing and relaxing mode. The transition mechanism of drop motion state was explored by systematic comparative analysis and a crucial factor named turning point was discovered in squeezing phase, which is supposed to be the reason preventing the drop from returning. Subsequently, we built a theoretical model of turning point based on squeeze flow theory and Young-Laplace equation, by which, the criterion was put forward to identify whether a drop returns to the parallel part under certain conditions in squeezing and relaxing mode. These criteria allow us to effectively design and optimize the combinative gap structure to achieve a leakage prevention function in practical production. Furthermore, with a multistage arrangement or a combination of surface energy gradient, the performance of the p-n gap can be further strengthened. LIQUID DROPS TRAPPED IN A FIXED P-N GAP In this section, the dynamic behavior of a liquid drop trapped in a fixed p-n gap is investigated. As illustrated in Figure 1 is a p-n gap, the bottom surface is a plane plate horizontally arranged, while the top surface is a combination of two parts: the parallel part is located on the left and the nonparallel part on the right. A liquid drop is confined in the middle of the gap, which represents the case that a detached drop exceeds the working area and needs to be moved back to the parallel part. The left and right edges of the drop are named leading edge and trailing edge, respectively. We use ℎ" and 𝛼, separately, to denote the height of the parallel gap and the opening angle of the wedgelike gap. The points that the leading edge intersects with the bottom and top plates are 𝑎" and 𝑏" , the ones which trailing edge intersects with the bottom and top plates are 𝑎& and 𝑏& . Let 𝑜 represent the inflexion of the top plate, which connect the parallel part and the nonparallel part,


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and set 𝑆 to be the distance between 𝑏& and 𝑜 point. Then, 𝜃" and 𝜃& are used to denote the equilibrium contact angle at the contact point on leading and tailing edge respectively.

Figure 1. Cross-sectional schematic of a liquid drop confined in a fixed p-n gap. We define 𝑃+ and 𝑃, as the atmospheric pressure and pressure inside the liquid drop, separately. Let 𝑃," and 𝑃,& denote the liquid pressure inside the leading and trailing edge respectively. Here, we made two assumptions to conduct the research. First, the half heights of the liquid drops are less than the capillary length of the liquid, so gravity effect can be neglected.24-25 Second, the radius of curvature perpendicular to the edges is considered to be infinite. Therefore, in this section, a 2D model is employed to describe the drop morphology, and the two edges of the drop are considered as a part of a circle. Based on the assumptions and Young-Laplace equation,33-34 𝑃," and 𝑃,& can be written as: 𝑃," =

−2𝛾 cos(𝜃" ) + 𝑃+ ℎ" (1)

𝑃,& =

𝛼 𝛼 −2𝛾 cos 9 2 : ∙ cos(𝜃& + 2 ) ℎ" + sin(𝛼) ∙ 𝑆

+ 𝑃+ (2)

where 𝛾 denotes the surface tension of the liquid and 0 < 𝛼 < 𝜋⁄2.


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Since the surfaces we used in reality are not ideally smooth and homogeneous, the advancing contact angle (𝜃+ ) and receding contact angle (𝜃B ) are different from each other. Therefore, the leading and trailing edges remain pinned when the corresponding contact angles haven’t increased to 𝜃+ or decreased to 𝜃B . If the leading edge want to advance towards the parallel part, 𝜃" should reach 𝜃+ , together with a trailing-to-leading orientation flow driven by the pressure difference (𝑃,& > 𝑃," ). Similarly, the trailing edge retreats only when both 𝜃& = 𝜃B and 𝑃,& > 𝑃," are satisfied. If the whole drop moves back to the parallel part without any external assistance, the directional self-propelled motion of the drop is accomplished. Accordingly, with the help of eqs 1 and 2, the dynamic behavior of the drop can be divided into four regions according to the opening angle and the surface wettability, as shown in Figure 2. Drop motion is free from the influence of opening angle when 𝜃+ ≤

E &

E

E

or 𝜃+ > 𝜋 − & . If 𝜃+ ≤ & ,

E

the drop surely can move back to the parallel part, while if 𝜃+ > 𝜋 − & , the drop cannot return, it may either stay still or move outward to the nonparallel part. By contrast, in the remaining two regions, the dynamic behavior of the drop is affected by the opening angle obviously. Whether the opening angle α can reach the threshold 2(𝜃+ − 𝜃B ) or not, determines the final state of the confined drop. It can be seen from the figure that, if satisfied, the drop certainly returns; while if

G &

E &

< 𝜃+
0 , while the other 𝑃,& −𝑃," < 0 . Since ℎ" ⁄𝑆 is constant on this occasion, this watershed is a straight line with positive slope. Therefore, the zone above the divide (zone 1) tends to possess a smaller ℎ" and larger 𝑆, which corresponds to the compression of the top plate. While the other one (zone 2) corresponds to the lifting of the top plate. If we can change operating conditions to adjust the parameters of a drop initially at rest into the zone with positive pressure difference, then the drop can return spontaneously. Point a (ℎ"+ ,𝑆+ ) and b (ℎ"I ,𝑆I ) located in zone 1 and 2, respectively, was used to judge the sign of pressure difference in two zones. Then we have (𝑃,& − 𝑃," )J − (𝑃,& − 𝑃," )K 𝛼 𝛼 ~ 𝐶 ∙ cos 9 : ∙ cos 9𝜃B + : 2 2 (3) E

E

where C is positive. It’s obvious that the sign totally determined by cos 9 : ∙ cos 9𝜃B + :. For & & E

hydrophilic uncertain region (& < 𝜃+

0, that is, 𝑃,& −𝑃," > 0 in zone 1. Similarly, we can have 𝑃,& −𝑃," < 0 in zone 2. Therefore, if we want to drive the stationary drop to return, we need to compress the top plate. On the contrary, G

E

for hydrophobic uncertain region (& < 𝜃+ < 𝜋 − & and 𝛼 ≥ 2(𝜃+ − 𝜃B )), the drop can return to the parallel part by lifting the top plate. This new method to propel the drop back to the parallel part was validated by two experiments. In each experiment, ultra-thin glass sheets with thickness of about 100 µm were used to form the gaps. One sheet was placed horizontally on the operation table as a bottom plate, and the top plate was constructed by splicing two glass sheets closely and sticking them together with polyimide tape. The left sheet was horizontally arranged and positioned at a specified height above the bottom plate using a high precision translation stage. Then the right sheet was lifted or lowered down by a micromanipulator, thus a specific angle with the bottom plate can be realized. The influence of


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splicing gap on drop motion can be ignored since the gap size is dependent on the thickness of the sheets, which is very thin in our cases. Water drops were used in all the experiments. We provided two kinds of plates to verify the aforementioned two cases. One was coated with normal SiO

2

coatings, exhibiting a hydrophilic character with the values of 𝜃+ and 𝜃B for water to be 74° and 53° respectively, with an error of 2°. Another kind of plates was coated with hydrophobic treated SiO coatings, and the values of 𝜃+ and 𝜃B were 99° and 87° separately, expressing a weak 2

hydrophobic character. Moreover, since the parallel gap is involved in our structure, the impact of drop volume on drop dynamics can be described by the trailing edge position 𝑆. Thus, the results of this work is generalized for various drop volumes, as long as the initial sizes of the liquids confined in parallel and nonparallel part are of the same order. However, for the occasions where the difference between these two sizes is too large, our results may no longer be valid. Water drops were put in the middle of p-n gaps at the beginning to ensure experimental repeatability. In the first experiment for hydrophilic uncertain region (Figure 3), the value of α was 30.4°, it was obvious that 𝛼 < 2(𝜃+ − 𝜃B ) was satisfied. The confined water drop was initially at rest. Then compressed the top plate to decrease the gap height from 1.5 mm to 1 mm, a new balance between 𝑃," and 𝑃,& was achieved. The water drop began to move towards left until it totally returned to the parallel part. On the contrary, to validate the hydrophobic uncertain region (Figure 4), weak hydrophobic plates with smaller CAH were used to ensure 𝛼 ≥ 2(𝜃+ − 𝜃B ). The drop that initially stationary returned to the parallel part through increasing the gap height, which was in good agreement with theoretical analysis.


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Figure 3. Propel a drop initially at rest to the parallel part by decreasing the gap height when the plates are hydrophilic and 𝛼 < 2(𝜃+ − 𝜃B ). (a) A water drop was initially at rest. (b) When the top plate was squeezed to decease the gap height from 1.5 mm to 1 mm, the drop started to move again toward the parallel part and (c) The drop finally returned to the parallel part. The scale bars represent 1 mm.


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Figure 4. Propel a drop initially at rest to the parallel part by increasing the gap height when the plates are weak hydrophobic and 𝛼 ≥ 2(𝜃+ − 𝜃B ). (a) A water drop was initially at rest. (b) When the top plate was lifted to increase the gap height from 1 mm to 1.5 mm, the drop started to move again toward the parallel part and (c) The drop finally returned to the parallel part. The scale bars represent 1 mm. SQUEEZING AND RELAXING OF A DROP IN A P-N GAP Preliminaries For the cases whose plates possessing a large CAH, self-propelled motion of the confined drop may be hindered. Then using mechanical actuation, i.e. squeezing and relaxing the top plate periodically, provides an alternative way to realize the directional transport of the drop. Under this circumstance, the leading and trailing edge may behave differently due to the asymmetry of the structure. If the leading edge advances while the trailing edge remains pinned during the squeezing phase or the trailing edge retreats while the leading edge remains pinned during the relaxing phase, it is referred to as asymmetric depinning (A.D.). While if both leading and trailing edges advance and retreat, it will be labeled with symmetric depinning (S.D.).

23, 27

The

most efficient way for drop transport is to let the drop move in a pure A.D. mode. Once S.D. is involved, the drop advances in a slipping state, and the corresponding speed will be slowed down. And worst of all, if S.D. dominates the whole squeezing and relaxing phases, a net inward motion of the drop may still be achieved due to a horizontal flow from the trailing edge to the leading edge during the relaxing process, albeit inefficiently. Accordingly, the criterion for directional drop 24

transport in our p-n gap is provided only for the motion in a pure S.D. mode, so the other two cases


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will be guaranteed. The criterion we proposed is a lower bound and must be satisfied to transport the drop back to the parallel part. Since both leading and trailing edge are not pinned, the pressure difference between the two edges can easily be derived as the contact angles on both sides are 𝜃+ in squeezing phase and 𝜃B in relaxing phase. According to eqs 1 and 2 we have: cos(𝜃) ℎ" 𝛼 𝛼 cos(𝜃) − sin 9𝜃 + 2 : ∙ sin(2 ) −2𝛾 ∙ ℎ" + (sin 𝛼 ∙ 𝑆)

𝑃,& − 𝑃," = 2𝛾 ∙

(4) where 𝜃 can be replaced by 𝜃+ when the drop is being squeezed, or 𝜃B when it’s being relaxed. It follows from this equation that 𝑃,& − 𝑃," > 0 during the whole squeezing and relaxing G

process if 𝜃+ < &, so the drop surely can return to the parallel part when the plates are hydrophilic. However, drop behavior in the region 𝜃+ >

G &

is still unknown: returning, staying still or even

leaving away are all possible. Since this kind of surfaces are commonly used in industrial production, it is necessary to carry out a systematically investigation of the dynamic behavior of a G

drop confined in a p-n gap with 𝜃+ > & , and come up with a criterion to identify whether the drop returns to the parallel part under certain conditions. Experimental Results and Discussions In consideration of the composite structure of a p-n gap, the governing parameters affecting the drop behavior in squeezing and relaxing mode can be categorized into four classes. They are mechanical parameters, i.e. the amount of squeezing and relaxing of the drop (∆ℎ); geometrical parameters, i.e., 𝛼; wettability parameters, i.e. 𝜃+ , 𝜃B and CAH of the plates; and initial position


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parameters, i.e. ℎ"T and 𝑆 T . As the two edges are located in different parts of the gap, the dynamic behavior of the leading edge can only be affected by gap height and surface wettability, but the dynamics of the trailing edge can also be influenced by opening angle and the position of the trailing edge. Hence, the two edges will go through a very different evolution process during the whole trial. These characters are first explored though experimental investigation. The experimental setup used in this section was similar with the one aforementioned. The water drop was placed in the middle of the p-n gap at first and the volume of the drop was fixed at 50 µl in all experiments. The wettability of the plates was arranged to fit the situation to be explored where 𝜃+ need to be hydrophobic. And the complexity of the analysis was also taken into consideration, so we set 𝜃B hydrophilic, under which condition the drop always tended to return in the relaxing phase. Surfaces with similar wettability are very common and widely used in actual industrial production. Consequently, the weak hydrophobic plates aforementioned were chosen for experiments and the corresponding 𝜃+ and 𝜃B were 99° and 87°, respectively, with an error of 2°. We realized squeezing and relaxing motion by height variation, considering height variation is very common in many industrial occasions, which may due to system vibration or pressure fluctuation. Finally, the opening angle was fixed at α = 10°, which was insufficient to propel the drop toward the parallel part spontaneously. During each trial, the combinative top plate was first set to a total parallel state and adjusted to the desired height, then held the left part of the top plate still and made the right part slowly rotate from 0° to 10° to reach the final position. This can help us to avoid the undesired drop motion during adjusting the gap height. Afterwards, the combinative top plate was squeezed for a distance ∆ℎ, and then relaxed for the same amount. The squeezing and relaxing process were repeated for no more than 20 cycles. Several combinations of ℎ"T and ∆ℎ were tested. We grouped trials that


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have same ℎ"T but different ∆h, and changed the tested plates after each group was done. Every group should be repeated on at least five sets of plates. During each group, the operating sequence of different trials should be rearranged to avoid the influence of wettability diversity along the surface and aging over time on the results. After experiments, the numbers of cycles needed to transport the drop back to the parallel part were recorded and summarized in a diagram, as shown in Figure 5.

Figure 5. Regime diagram for droplet transport within the p-n gap under squeezing and relaxing mode. The black numbers denote the number of cycles required to transport the drop back to the parallel part and the crosses mean the drop cannot return to the parallel part within the limited cycles. The experiments with certain results and uncertain results are represented by solid circles and half-hollow circles, respectively. This figure illustrates the various regimes of droplet transport observed in our experiments when ℎ"T and ∆ℎ are varied respectively. Here, we use ℎ"W to denote the gap height after squeezing, then we have ℎ"W = ℎ"T − ∆ℎ. During the experiments, ℎ"T varied from 1.8 mm to 0.8 mm, with ℎ"W no less than 0.3 mm, under which parallelism of the plates will have a great impact on the result. The


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unphysical region refers to the cases cannot be achieved in the actual situations. While the drops in outrange region may be too large to ignore the gravity or even break up, which make the assumption untenable. The region left, which is encircled by red dashed line, is the functional region. According to the final state of the confined drop, this region can be divided into three parts. The first part refers to an initialization regime, which illustrates a minimum amount of ∆ℎ needed to initiate the bulk motion (∆ℎ"∗ ). This phenomenon has already been indicated in nonparallel (wedgelike) gap studies, and come into a conclusion that ∆ℎ"∗ is only related to the opening angle and the surface wettability, which can also be applied to our studies. Although ∆ℎ"∗ here is less than 1 mm 27

according to the experimental conditions, it ruined the transport efficiency of the drop when ∆ℎ = 0.1 mm, leading to an unfinished returning within the limited cycles. Only with a higher ∆ℎ, a complete directional transport can be achieved, as shown in the other two parts. However, the drop behaves quite differently with the variation of ℎ"T and ℎ"W . When ℎ"T is large, decrease ℎ"W (increase ∆ℎ) seems a good way to accelerate drop’s returning. While as ℎ"T getting smaller, a threshold of ℎ"W appears at which the drop possesses the highest efficiency. Once the threshold is passed, i.e. keep on decreasing ℎ"W , the drop motion will slow down again. What is even worse is that, when ℎ"T is around 0.8 mm, the experimental results seems uncertain: in some cases the drop returned to the parallel part, while in other cases the drop stayed still without any movement, as the third part shown in Figure 5. This unexpected phenomenon poses a serious barrier for directional drop transport, and a more detailed study on drop dynamics in squeezing and relaxing process is needed. The impact of ℎ"T and ∆ℎ on dynamic behavior of the leading and trailing edges was first investigated, which is shown in Figure 6 and Figure 7. It is obvious that drop motion under two cases are quite different. When ℎ"T = 1.5 mm, the behavior patterns of the two edges seemed like


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stairs, which are referred to as A.D. mode. By contrast, the patterns shown when ℎ"T = 1 mm were more like saw tooth, meaning in each trial, both leading and trailing edges advance and retreat, known as S.D. mode. Furthermore, A.D. mode possesses a much higher transport efficiency than S.D., which is in good agreement with the analysis we carried out above. Both cases exhibit faster movement with the increase of ∆ℎ, which means a larger inward displacement during each cycle and less cycles needed for drop returning.

Figure 6. Comparison of the associated motion of leading (red circles) and trailing (blue squares) edges under squeezing and relaxing mode in one trial with different ∆ℎ. (a) ℎ"T = 1.5 mm and ∆ℎ = 0.1 mm. (b) ℎ"T = 1.5 mm and ∆ℎ = 0.3 mm.


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Figure 7. Comparison of the associated motion of leading (red circles) and trailing (blue squares) edges under squeezing and relaxing mode in one trial with different ∆ℎ. (a) ℎ"T = 1 mm and ∆ℎ = 0.1 mm. (b) ℎ"T = 1 mm and ∆ℎ = 0.25 mm. It also can be seen in Figure 7 that, compared with the trailing edges, the movements of the leading edges are relatively small, and show no remarkable change with the increase of ∆ℎ. As the drop is in a quasi-static state, the movements of two edges are caused by the inner flow driven by pressure difference. 𝑃T , the pressure in the middle of the drop due to squeezing, should be higher than 𝑃," and 𝑃,& to make leading and trailing edge move left and right respectively. The shorter displacement of leading edge shown in Figure 7 means a smaller pressure difference within the parallel part, in other words, a higher 𝑃," at the leading edge. This phenomenon is mainly due to the specificity of our structure. Since the geometry is an assemble of a parallel gap and a nonparallel gap, the pressure increase more rapidly at the leading edge than the trailing edge when the top plate is getting lower and lower. Therefore, the leading edge is much harder to push when the gap height is small. Nevertheless, it should be noted that all comparisons introduced here based


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on the premise that the drop is initially in the middle of the p-n gap, which means 𝑆 T was not taken into consideration. A different 𝑆 T may alleviate or aggravate the difficulty in drop transportation. In the following experiments, ℎ"T and ∆ℎ were further fixed, so 𝑆 T became the only variable in all trials. 𝑆 T was changed by a deviation of initial drop position from the middle of the gap. Comparative experiments were carried out with different drop locations and a set of experimental results are given in Figure 8. Both of the experiments were carried out with ℎ"T = 1 mm, and ∆ℎ = 0.4 mm, but different 𝑆 T . It is shown that the results are totally different. In Figure 8a, only three cycles are needed to transport the drop back to the parallel part, while the drop in Figure 8b doesn’t move at all. Besides, the behavior pattern shown in Figure 8a seems like a transition from A.D. to S.D., and there is no obvious difference in displacement between leading and trailing edge. Whereas the drop in Figure 8b experienced an extreme situation in which its leading edge cannot be pushed while its trailing edge behaved symmetrically in the squeezing and relaxing process. These phenomena demonstrate that 𝑆 T do change the drop behavior: a smaller 𝑆 T can improve the transport, whereas a larger 𝑆 T may suppress the drop motion, which should be avoided in practical application. However, how these parameters affect the dynamic behavior of the drop is still unknown.


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Figure 8. The impact of 𝑆 T on the associated motion of leading (red circles) and trailing (blue squares) edges under squeezing and relaxing mode. For both trials, ℎ"T = 1 mm and ∆ℎ = 0.4 mm. (a) Initial position of the liquid drop is closer to the parallel part, i.e. with a smaller 𝑆 T . (b) Initial position of the liquid drop is closer to the nonparallel part, i.e. with a larger 𝑆 T . After a systematic analysis of the experimental results, a new phenomenon is discovered in the squeezing phase, and a crucial factor directly related to drop dynamics is proposed, which is supposed to be the reason for the transition of drop motion state. Figure 9 shows a complete squeezing and relaxing cycle when ℎ"T = 1.4 mm. Among them, Figure 9a-d represent a squeezing phase, and Figure 9e represents a relaxing phase. After one cycle is finished, a net inward movement can be achieved. During the cycle, we can see that both leading and trailing edge advance in Figure 9b-c. However, after that, the trailing edge keeps advancing while the leading edge gets pinned. This phenomenon has never been mentioned in previous studies before. Therefore, we name the special moment when the pinning of the leading edge occurs in squeezing phase the turning point.


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In a cycle which contains the turning point, once the turning point is reached, water drained by the compression of the top plate totally flow outwards to the nonparallel part, leading to a large extending movement of the trailing edge. This situation is a disaster to drop returning. Under normal circumstances, the drop can realize an inefficient transport to the parallel part even if it moves in a pure S.D. mode, due to the horizontal flow in the relaxing phase. However, after the 24

turning point, the outward displacement of the drop in squeezing phase cannot be surpassed by relaxing phase anymore, that is, the net transport cannot be achieved under this condition. Hence, whether and when the turning point occurs have a serious impact on the final state of the confined drop. If there is no turning point during the squeezing phase, the drop can move back to the parallel part after several cycles. Or if the turning point occurs during the squeezing process, the transport efficiency will be reduced. Furthermore, if the whole squeezing phase is in a state with pinned leading edge, then the drop cannot return, it will stay in original position no matter how many cycles are repeated.


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Figure 9. One squeezing and relaxing cycle during which the turning point occurs. (a-d) The top plate was being squeezed. (a) both the leading and trailing edge were pinned. (b-c) both the leading and trailing edge advanced and the turning point occurred when (c) was over. (d) the trailing edge advanced while the leading edge got pinned. (e) A net movement was achieved after the whole cycle. Arrows indicate the direction of the edge movement. ℎ"T = 1.4 mm. The scale bars represent 1 mm. It follows that turning point is supposed be the key factor that responsible for the diversity of the experimental results when initial gap height is small, as shown in Figure 5. Therefore, a further investigation on the turning point should be carried out to obtain the criterion for predicting whether a drop returns to the parallel part under specific conditions by the squeezing and relaxing method. Theoretical Model According to the foregoing analysis, the occurrence of turning point is the result of comprehensive effect of multiple parameters, which increase the difficulty of turning point prospecting. Therefore, if we want to make a precise prediction of the turning point to identify the drop behavior, the internal mechanism should be explored. Based on the definition of turning point we introduced above, a force balance appears at this moment within the parallel part, i.e., 𝑃T = 𝑃," , which paves a way to establish the theoretical model of turning point. As the whole process can be treated as quasi-static, 𝑃," can be solved by eq 1. Then, the pressure in the middle of the drop, 𝑃T , which is still unknown, become the only obstacle to establish the turning point model. During the squeezing phase, once the turning point is passed, the flow field within the gap is greatly simplified since the water drained by the compression totally flow towards the trailing


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edge. Therefore, a theoretical model of 𝑃T can be achieved within the nonparallel part based on squeeze flow theory. Under this circumstance, the calculation of volume flow rate is involved, thus a 3D drop model is needed to ensure the volume conservation. Furthermore, two additional assumptions should be proposed in the modeling of 𝑃T . First, in order to simplify the integral process in squeeze flow calculation, the curvature of air-liquid interface is further neglected, since the size of interface is much smaller than drop radius, making the curvature unimportant in integral. Second, the opening angle of nonparallel part cannot be too large, thus the velocity of the drop along vertical direction can be ignored. Accordingly, the drop can be simplified as a 3D composite hexahedron whose horizontally-projected shape is a quadrate. Schematic of the squeeze flow model after simplification is shown in Figure 10. According to the assumptions mentioned above, the leading edge and trailing edge are replaced by leading and trailing interface here. For easy viewing, the planes on which the leading and trailing interfaces located are named leading plane and trailing plane, as shown in the figure. We use ℎ" and ℎ& , respectively, to denote the height of the leading interface (also the height of the parallel gap) and trailing interface. Let aa and bb denote the two contact lines that leading interface intersects with the bottom and top plates, separately. Set cc and dd to be the two contact lines that the trailing interface forms with the bottom and top plates, respectively. Then, oo is the joint line of the parallel and nonparallel part of the top plate. Use K to stand for the distance between bb and oo, while S denotes the distance between dd and oo. Let B represent the width of the drop. Use V and 𝛼, respectively, to denote volume of the drop and opening angle of the nonparallel part. It follows from geometric analysis that


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Figure 10. Schematic of the squeeze flow model after simplification.

𝐵 =

−

sin(2𝛼) ∙ 𝑆 & \sin& (2𝛼) ∙ 𝑆 ] + + 4𝑉ℎ" 4 16 2ℎ" (5)

To investigate the squeeze flow properties, the Reynolds number Re should be first estimated.

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The Reynolds number is defined as Re = 𝜌ℎ𝑈/𝜇, where 𝜌 and 𝜇 are the density and dynamic viscosity of the liquid, while U represents the squeezing speed of the top surface. Therefore, U should be small to make Re much less than 1. In this case, the flow in the gap is laminar, so the problem can be solved under the creeping flow approximation, i.e., neglecting inertial and gravitational terms in the Navier–Stokes equations.

36-38

Only the inner flow within the nonparallel part is taken into account in squeeze flow model, so the driving pressure difference is 𝑃T − 𝑃,& . Consider a slice vertical to the flow direction in the nonparallel part, whose thickness is dx, height is h, and have a distance of x from the vertical plane though joint line oo. Since squeeze flow is a special case of pressure-driven flow, we have d𝑃 12𝜇𝑞i =− d𝑥 𝐵ℎj (6)


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where 𝑞i represents the volume flow rate though the slice. As the model is built at the moment of the turning point, any further compression on the top plate will lead to a drainage though the slice. Thus, we obtain 𝑞i = 𝑈𝐵 (𝑥 + 𝐾 ) (7) Assumptions mentioned above indicate that 𝐾 = 𝐵 − 𝑆 ∙ cos(𝛼) . According to geometric analysis, we can have 𝑥 = (ℎ − ℎ" )⁄ tan(𝛼) and d𝑥 = dℎ⁄tan(𝛼). Then it follows from eqs 6 and 7 that 1 1 ( ( ) ( ) ) + tan 𝛼 ∙ 𝐵 − sin 𝛼 ∙ 𝑆 − ℎ ∙ p " ℎ& ℎj −12𝜇𝑈 q & r ∙ dℎ tan (𝛼 )

d𝑃 = o

(8) Pressure distribution in the nonparallel gap can be achieve by integrating eq 8 as 𝑃=

6𝜇𝑈 [𝐵 − 𝑆 ∙ cos (𝛼)] tan(𝛼 ) ∙ ℎ& 6𝜇𝑈 (2ℎ − ℎ" ) + 𝐶 + tan& (𝛼 ) ∙ ℎ& (9)

Two boundary conditions can help us get the final relationship between 𝑃T and P,& : 𝑃 = 𝑃T when ℎ = ℎ" (i.e. at the leading interface); while 𝑃 = 𝑃,& when ℎ = ℎ& (i.e. at the trailing interface). Thus the final model in the nonparallel gap can be written as 𝑃T − 𝑃,& =

𝐶" 𝑆[𝐵ℎ" + (𝐵 − 𝑆 cos(𝛼 ))ℎ& ] + 𝐶& ℎ"& ∙ ℎ&& (10)


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where 𝐶" and 𝐶& are empirical coefficient obtained from a fitting to the experimental results. Here, 𝐶" ~6𝜇𝑈 ∙ cos (𝛼) and ℎ& = ℎ" + sin (𝛼) ∙ 𝑆. It can be seen that, once the opening angle and the drop volume are determined, the pressure difference within the nonparallel part is only related to ℎ" and 𝑆. Besides, it is obvious that there is a linear relationship between the two sides of the equation, which provides an approach to experimental verification. Although 𝑃T is unobtainable from experimental results during verification, we have 𝑃x" = 𝑃T at the moment of turning point. Thus, it can be derived that 𝑃T − 𝑃,& = 𝑃," − 𝑃,& = −∆𝑃 (11) With the help of eqs 10 and 11, the experimental data is depicted in the Figure 11 and shows a good linearity, which in an excellent agreement with the theoretical prediction. Thus, it can be demonstrated that our simplified squeeze flow model can give a good description of the pressure distribution within a drop confined in a p-n gap at the moment of turning point.

Figure 11. Experimental verification of the squeeze flow model. Experimental results are represented by blue circle, and the linear fit of experimental results is indicated by the dotted line.


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Criterion and Validation According to the analysis aforementioned, once the opening angle and the drop volume are determined, the pressure difference in the nonparallel part is only related to ℎ" and 𝑆. Therefore, we can use a two-dimensional coordinate (ℎ" , 𝑆) to represent the turning point. As the validity of the simplified model has been verified, theoretical solution of the turning point can be achieved based on the combination of squeeze flow model and Young-Laplace equation. Since the drop is moving in a pure S.D. mode during the squeezing phase, we have −∆𝑃 = 2𝛾 ∙

𝛼 𝛼 cos(𝜃+ ) − sin 9𝜃+ + 2 : ∙ sin 9 2 : ℎ&

cos(𝜃+ ) −2𝛾 ∙ ℎ"

(12) With the aid of eqs 10-12, we can obtain the theoretical values of turning points by iterative computing in Matlab, as shown in Figure 12, represented by the red dotted line. There is an inflexion point on the curve when h1 is around 0.6 mm, which divided the curve into two parts. The variation trend of discrepancy between the pressure difference calculated by squeeze flow model and by Young-Laplace equation is quite different in these two parts. On the left, two pressure differences differ significantly with each other, so the results possess a good unicity. Whereas these two values are nearly the same on the right, making any deviation being amplified and having larger influence on the final results, thus precise turning point is unavailable. However, considering that the flow mechanism within the gap remains unchanged, thus we use the extrapolation to acquire the turning points at high h1 , as the black line shown in Figure 12. Domain under the corrected turning point curve represents the cases 𝑃T > 𝑃," , where the turning points have not been reached. On the contrary, cases in the domain above have already passed the turning


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point. From the figure, we can observe a monotonicity of the turning point in (ℎ" , 𝑆) coordinate system: the lower turning point occurs, the further trailing interface lies. Moreover, Figure 12 shows a good agreement between the turning points detected in experiments and the theoretical prediction values.

Figure 12. Comparison between the experimental results of turning point (circles) and the corresponding theoretical values. The red dotted line represents the original theoretical prediction curve, while the one after corrected is indicated by the solid black line. The inserts denote the dynamic behavior of a liquid drop in one squeezing and relaxing cycle before/after the turning point. The scale bars represent 1 mm. This curve is of great significance as it can offer a criterion of whether a drop confined in a p-n gap returns or not. Here, the position (ℎ"∗ , 𝑆 T ) when the drop begins to extend under the compression is needed for prediction. It should be noted here, this position is different from the initial position (ℎ"T , 𝑆 T ) since the squeezing height should reach a minimum ∆ℎ"∗ to initiate the bulk motion (i.e. ℎ"∗ = ℎ"T − ∆ℎ"∗ ).

25, 27

Therefore, (ℎ"∗ , 𝑆 T ) is more appropriate for the prediction,

and ∆ℎ"∗ can be estimated using the method in the reference mentioned above. According to the


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previous analysis, once (ℎ"∗ , 𝑆 T ) is located above the curve, drop cannot return to the parallel part. On the contrary, drop can return by means of repeated squeezing and relaxing. In our cases, ∆ℎ"∗ is less than 0.1 mm, so the initial position can be used for prediction directly. Experimental verification of this criterion is shown in Figure 13. It can be seen that this criterion can make an accurate prediction of the final state of the drop.

Figure 13. Experimental verification of the criterion based on turning point model. Theoretical turning point curve is represented by black solid line, and the initial positions of both the drops who can return (solid circle) and not returns (hollow circle) are also plotted. In addition, the initial positions of ideally located drops with the same volume are calculated and indicated by the red dotted line, which is used to explain the uncertain results arose when the gap height is 0.8 mm. Besides, the ideal initial position of the trailing interface when the drop (also 50 µl) lies in the middle of the gap is calculated and indicated by the red dotted line in Figure 13. It is obvious that the ideal initial position locates below the turning point curve with a large distance between each other when the gap height is around 1 mm or above, which means it is more difficult for a drop confined in a p-n gap with a large initial gap height to reach the turning point in squeezing phase.


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Therefore, a drop can always return to the parallel part under this circumstance. However, when the gap height is around 0.8 mm, the ideal initial position is very close to the turning point curve, thus the impact of trailing interface location 𝑆 T on the dynamic behavior and the final state of the drop becomes more severely. This explains why the experimental results become uncertain when the gap height is 0.8 mm. If the gap height continues to decrease, a drop of 50 µl with an ideal initial position becomes unable to return to the parallel part. With this method, the prediction for the final state of the drop is accomplished, which seems quite simple and clear. SUMMARY AND CONCLUSIONS In summary, directional transport of a liquid drop has been realized between parallel-nonparallel combinative plates, which provides a new route to prevent the liquid in a parallel gap from stepping out the working area and causing a damage to the whole system. It has been shown that the dynamics of the liquid drop confined in this combinative gap differs from both the traditional parallel and nonparallel gaps. The criterion for self-propelled motion was established through a comparison of the liquid pressure at leading and trailing edge, which seemed more difficult to meet than that in the nonparallel gap. Besides, the dynamic behavior of the drop under squeezing and relaxing mode was also investigated. It was shown that the drop in the hydrophilic combinative gap could surely return to the parallel part, while uncertainties arose in weak hydrophobic cases. Systematic comparative analysis of the impact of experiment parameters on drop behavior revealed a new phenomenon that the trailing interface advances while the leading interface gets pinned in squeezing phase (whose beginning named turning point), which is supposed to be the reason for transition of drop motion state. Finally, the criterion to determine whether a drop returns under squeezing and relaxing mode can be achieved with the help of the theoretical model of turning point we proposed in this paper. Although not all wetting properties were taken into


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account, the cases involved in this report covered the majority of surfaces used in actual industrial production. Therefore, the designed parallel-nonparallel combinative gap can play an important role in diverse practical applications such as fluid manipulation and leakage prevention in microfluidics, hydraulics and semiconductor manufacturing. AUTHOR INFORMATION Corresponding Author E-mail: [email protected] Notes The authors declare no competing financial interest. ACKNOWLEDGMENT 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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(18) Heng, X.; Xiang, M.; Lu, Z.; Luo, C. Branched ZnO Wire Structures for Water Collection Inspired by Cacti. ACS Appl. Mater. Interfaces 2014, 6, 8032-8041. (19) Chen, H.; Zhang, P.; Zhang, L.; Liu, H.; Jiang, Y.; Zhang, D.; Han, Z.; Jiang, L. Continuous Directional Water Transport on the Peristome Surface of Nepenthes Alata. Nature 2016, 532, 8589. (20) Rajaram, M.; Heng, X.; Oza, M.; Luo, C. Enhancement of Fog-Collection Efficiency of a Raschel Mesh Using Surface Coatings and Local Geometric Changes. Colloids Surf., A 2016, 508, 218-229. (21) Heng, X.; Luo, C. Liquid Drop Runs Upward between Two Nonparallel Plates. Langmuir 2015, 31, 2743-2748. (22) Concus, P.; Finn, R. Discontinuous Behavior of Liquids between Parallel and Tilted Plates. Phys. Fluids 1998, 10, 39-43. (23) Prakash, M.; Quéré, D.; Bush, J. W. Surface Tension Transport of Prey by Feeding Shorebirds: the Capillary Ratchet. Science 2008, 320, 931-934. (24) Luo, C.; Heng, X.; Xiang, M. Behavior of a Liquid Drop Between Two Nonparallel Plates. Langmuir 2014, 30, 8373-8380. (25) Ataei, M.; Chen, H.; Tang, T.; Amirfazli, A. Stability of a Liquid Bridge between Nonparallel Hydrophilic Surfaces. J. Colloid Interface Sci. 2017, 492, 207-217. (26) Bush, J. W.; Peaudecerf, F.; Prakash, M.; Quéré, D. On a Tweezer for Droplets. Adv. Colloid Interface Sci. 2010, 161, 10-14.


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(37) Neto, C.; Evans, D. R.; Bonaccurso, E.; Butt, H.-J.; Craig, V. S. Boundary Slip in Newtonian Liquids: a Review of Experimental Studies. Rep. Prog. Phys. 2005, 68, 2859-2897. (38) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics: with Special Applications to Particulate Media, Springer, 2012.


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Directional Transport of a Liquid Drop between ParallelâNonparallel

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