Behavior of a Liquid Bridge between Nonparallel Hydrophobic

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Behavior of a Liquid Bridge between Nonparallel Hydrophobic Surfaces Mohammadmehdi Ataei, Huanchen CHEN, and Alidad Amirfazli Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b03508 • Publication Date (Web): 17 Nov 2017 Downloaded from http://pubs.acs.org on November 21, 2017

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Behavior of a Liquid Bridge between Nonparallel Hydrophobic Surfaces

Mohammadmehdi Ataei, Huanchen Chen, and Alidad Amirfazli*

Department of Mechanical Engineering, York University, Toronto, ON, M3J 1P3, Canada * Corresponding author: [email protected]

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Abstract When a liquid bridge is formed between two nonparallel identical surfaces, it can move along the surfaces. Literature indicates that the direction of bridge movement is governed by the wettability of surfaces. When the surfaces are hydrophilic, the motion of the bridge is always towards the cusp (intersection of the plane of the two bounding surfaces). While, the movement is hitherto thought to be always pointing away from the cusp, when the surfaces are hydrophobic. In this study, through experiments, numerical simulations, and analytical reasoning, we demonstrate that for hydrophobic surfaces, wettability is not the only factor determining the direction of the motion. A new geometrical parameter, i.e. confinement () was defined as the ratio of the distance of the farthest contact point of the bridge to the cusp, and that of the closest contact point to the cusp. The direction of the motion depends on the amount of confinement (). When the distance between the surfaces is large (resulting in a small ), the bridge tends to move towards the cusp through a pinning/depinning mechanism of contact lines. When the distance between the surfaces is small (large ), the bridge tends to move away the cusp. For a specific system, a maximum value ( ) exists. A sliding behavior (i.e. simultaneous advancing on the wider side, and receding on the narrower side) can also be seen, when a liquid bridge is compressed such that the cf exceeds the . Contact angle hysteresis ( ) is identified as an underpinning phenomenon that together with fundamentally explains the movement of a trapped liquid between two hydrophobic surfaces. If there is no , however, i.e. the case of ideal hydrophobic surfaces, the will be a constant; and we show that the bridge slides towards the cusp when it is stretched, while it slides away from the cusp when it is compressed (note sliding motion is different form motion due to pinning/depinning mechanism


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of contact lines). As such, the displacement is only related to geometrical parameters such as the amount of compression (or stretching) and the dihedral angle between the surfaces.

1. Introduction It has been well reported that a capillary bridge between two nonparallel solid surfaces can have a bulk motion along the surfaces under certain conditions, e.g. during compressing and stretching [1-14]. This process is seen in nature; e.g. shorebirds trap preys inside a liquid bridge formed between their nonparallel beaks, and move it mouthwards by compressing and stretching the bridge [7]. Such phenomenon can potentially be used in different practical applications to transport small amounts of liquids e.g. in lab-on-a-chip devices, or fog collectors [3, 4]. Figure 1 shows a schematic of a liquid bridge between two identical nonparallel surfaces with a dihedral angle ( ) between them. In Fig. 1, and are the values of the contact angle between the bridge and the surfaces on the narrow and wide sides, respectively. The values of and are limited to the advancing ( ) and the receding ( ) contact angles of the surfaces; this is due to Contact Angle Hysteresis ( = − ) phenomenon. The distance from the rightmost contact point of the bridge to where the plane of two surfaces intersect (o), and the width of the contact area of the bridge are denoted as and , respectively.


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Fig. 1 Schematic of a liquid bridge between two nonparallel hydrophobic surfaces. and are the apparent contact angles. A stable bridge (i.e. one in the mechanical equilibrium) between two nonparallel surfaces has a constant Laplace pressure across the liquid-air interface. This condition requires the geometrical relationship given by Eqn (1) to be satisfied [1, 7, 8]:

2 =

− 2

(1)

Equation (1) was derived with the assumption that the in-plane geometry of the leftmost and the rightmost menisci of the bridge are approximately truncated circles (this does not mean that the interface shape is spherical); and neglecting the effect of gravity. For the first assumption to hold, the contact angle on the top and bottom surfaces on each side of the bridge should be equal [1]. Although Eqn. (1) was derived in in Ref. 1 in the context of hydrophilic surfaces, but its utility is not limited to such surfaces i.e. there are no limiting assumptions for the range of contact angles. The difference when applying Eqn. (1) to hydrophobic surfaces rather than hydrophilic surfaces is that the sign of the curvature will change from negative to positive. Since , Eqn. (1) leads to the inequality:


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2 1

− 2

(2)

Equation (2) is a necessary condition for a stable bridge to exist between two nonparallel surfaces. The cosine functions in Eqn. (2) changes from positive to negative when the variables (

or − ) become larger than . As such, the conditions for bridge stability can be

different, if the surface wettability changes. In the range of < −

, surfaces are hydrophilic

and the in-plane curvatures of the leftmost and the rightmost menisci of the bridge are concave [1]. If the bridge is between two hydrophilic surfaces, to satisfy < −

inequality (denoted

as an ‘I bridge’), both cos and cos − should be positive; so, Eqn. (2) leads to: −

(3)

According to Eqn. (3), for a stable ‘I bridge’, is always larger than . If a bridge is compressed and stretched under a quasi-static condition (i.e. small Capillary and Weber numbers), the pressure inside the bridge can be assumed to be static at all times i.e. the additional pressure time derivatives such as the transient pressures due to the motion of the surface can be neglected. Thus, Eqns. (1-3) remain valid during the compressing and stretching processes. Because for an ‘I bridge’, during the quasi-static compression, contact angles (CAs) increase to compensate the change in the bridge’s height until reaches first, and consequently the contact line on the narrow side moves towards to surfaces’ cusp [2, 7, 8]. During the quasi-static stretching, CAs decrease until reaches first, and consequently the contact line on the wide side move towards to surfaces’ cusp. Such movements of the contact


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lines eventually lead to a bulk motion of the bridge that is always towards to the cusp of the surfaces in both compressing and stretching processes (i.e. in a loading cycle) [2, 7, 8].

However, if the bridge is formed between two surfaces satisfying inequality (denoted as an ‘O bridge’ hereafter), surfaces are hydrophobic and the in-plane curvatures of the leftmost

and the rightmost menisci of the bridge are convex [1]. Hence, both cos and cos −

are negative and Eqn. (2) leads to: − <

(4)

Different from an ‘I bridge’ where the only acceptable answer for Eqn. (3) is , Eqn. (4) can have three different solutions: (when − is still less than ), = , or . Therefore, in a quasi-static compressing and stretching cycle, the motion of contact lines for an ‘O bridge’ can be different. In literature, for an ‘O bridge’ experiencing loading cycles, only an outwards motion, moving away from the cusp has been reported [8, 10]. However, given the argument above, this should not be the only possible motion. In this study, we represent various possible motions of an ‘O bridge’ in which the bridge may move towards, or away from the cusp. The motion can be enabled either by a pinning/depinning mechanism of contact lines, or sliding of the bridge, as it will be explained later. We will use both analytical and experimental approaches to study the effect of different parameters (e.g. wettability) for each type of the motion discussed. At the end, a discussion on the behavior of ‘O bridges’ between ideal surfaces will also be presented.


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2. Experimental methods In the experimental process, the bridge was formed by approaching a tilted solid surface (upper surface) onto a drop sitting on another solid surface (lower surface) (see Fig. 2). The shape as well as the movement of the bridge was recorded by one camera (Basler A321, with the resolution of 760×522) facing the plane shown in Fig. 2. The lower surface was fixed on a stage. The bridge was compressed and stretched by moving the top surface (at 0.01&&/) with a motion controller system (ILS100CC and XPS-C6 Motion Controllers from Newport). In each test, a 2 () distilled water drop was used, and all experiments were conducted at approximately 21 °C. Two different types of hydrophobic surfaces, OTS ( = 115.7° ± 1°, = 101.3° ± 1°) and Teflon AF ( = 126.7° ± 1°, = 116.7° ± 1°) were used in the experiments. These surfaces have a similar , but different i.e. Δ ≈ 10°. The contact angle values of these surfaces were measured using the Sessile Drop Method with a resolution of 0.1° and accuracy of ±1°. Detailed information about the fabrication procedure and CA measurements are given in the Supporting Information (SI) Section S1. It should be noted that due to small Ca 4

= µU/γ ∼ O (10−6), Weber number We = ρU2 √3 /γ ∼ O (10−8), and Bond number (Bo = gρR2/γ ∼ O (10−2) of the systems, the gravity, viscous and inertia effects were considered negligible i.e. all the experiments were treated as quasi-static; g is the gravitational acceleration; R is the radius of the best fitted circle to the contact line of the bridge on the lower surface; U is 4

the velocity of the motion of the top surface; √3 is the cubic root of the liquid volume; and γ, µ, and ρ are the surface tension, viscosity and density of the liquid, respectively. The Weber number will be of the same order of magnitude mentioned above, if the characteristic velocity, U, is taken as the maximum velocity of the contact lines (instead of the surface's velocity).


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The height of the upper surface with respect to the lower surface ( ) was measured from the lower surface at the location of the sessile drop apex (see Fig. 2 (a)). The reference point for the horizontal motion of the bridge was considered as the center of the contact line on the bottom surface with positive direction towards the cusp (see 5 in Fig. 2 (b)). In the experiments, the amount of compressing and stretching were the same and equal to ∆ . 4

For the system studied here √3 is the characteristic length [1]; consequently, all length 4

4

4

4

parameters can be normalized with √3 , e.g., ∆ ∗ = ∆ / √3 , ∗ = / √3 , 5 ∗ = 5/ √3 , 4

4

∗ = / √3, and ∗ = / √3. So, the volume (V) of the bridge does not need to be considered as one of the system variables.

Fig. 2 (a) The distance between the surfaces was measured from the lower surface at the drop apex. (b) The origin for the horizontal motion of the bridge was considered to be from the center of the contact line on the lower surface.

3. Results and Discussions An example of the motion of an ‘O bridge’ (between Teflon AF surfaces and = 2° ) undergoing compressing and stretching is shown in Fig. 3. At ∗ = 0.86 the bridge has just 8 ACS Paragon Plus Environment

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formed (Fig. 3 (a)); the value of and are 119° and 117°, respectively. When the bridge was compressed, and increased to compensate the change in ∗ until reached to the advancing contact angle value first. Further compression moved the contact line on the narrow side towards the cusp (Fig. 3 (b)). During this stage (‘a’ to ‘b’), although was increasing, its value remained in the range of < < , so the wider side of the bridge was pinned. At ∗ = 0.56, the value of also reached (i.e. = = ). At this point, by compressing the bridge to ∗ = 0.36, the wider side of the bridge moved away from the cusp while started to decrease causing the narrower side to remain pinned instead (Fig. 3 (c)). After the stretching began, from ∗ = 0.36 to ∗ = 0.51, both and decreased until reached first, and the narrower side receded away from the cusp with the wider side pinned. However, from ∗ = 0.51 to ∗ = 0.81, where = = , the direction of motion was reversed again and the wider side receded towards the cusp with the narrower side pinned (i.e. increased). From the above example, the value of ∗ emerges as a key element that governs the direction of the horizontal movement of the liquid bridge in a compressing and stretching cycle. Specifically, the direction of motion changes at an intermediate ∗ value. From ∗ = 0.86 to ∗ = 0.56 in compressing, and from ∗ = 0.51 to ∗ = 0.81 in stretching, and the bridge moves towards the cusp. Whereas from ∗ = 0.56 to ∗ = 0.36 in compressing, and from ∗ = 0.36 to ∗ = 0.51 in stretching, and the bridge moves in the opposite direction. It appears that at large ∗ , the bridge tends to move towards the cusp. And at smaller ∗ , the bridge moves in the opposite direction. In literature [8, 10], only the motion of bridge away from the cusp corresponding to small ∗ has been reported. While by having a larger range of ∗ , a diversity of the motions can be observed.


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Several questions arise here: Why the direction of motion changes at an intermediate ∗ ? What is the governing mechanism behind the effect of ∗ on the direction of motion of an ‘O bridge’? Can ∗ be adjusted only to move the bridge towards the cusp, or away from the cusp? And what are the effects of CAs and on the direction of motion of the bridge? We start by addressing how ∗ affects the direction of motion of an ‘O bridge’. Then, the mechanism for movement of the bridge towards, or away from the cusp, will be discussed. At the end, the effect of CAs and

on the motion of a bridge will be provided.


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Fig. 3 Motion of a liquid bridge between two Teflon AF surfaces during a compression and stretching cycle. Motion of a meniscus (i.e. on the narrower side or the wider side of the bridge) towards the cusp and away from the cusp are shown with blue and green arrows, respectively. (a) Initial shape of a bridge between two Teflon AF surfaces with = 2° after formation of the bridge. (b) The bridge is compressed and the narrower side of the bridge advances towards the cusp. (c) The compressing continues and the wider side of the bridge starts to advance away from the cusp. (d) Stretching began and the narrower side of the bridge recedes away from the cusp. (e) As stretching was continued the wider side of the bridge starts to recede. 11 ACS Paragon Plus Environment

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3.1. The effect of :∗ on direction of motion To understand the effect of ∗ on the direction of motion of a bridge, one should understand how changing ∗ affects CAs, ∗ , and ∗ . By compressing a bridge (i.e. decreasing ∗ ), due to conservation of liquid volume, the reduction in ∗ should be compensated either by an increase of and , or depinning of contact line and spreading of the liquid on surfaces i.e. an increase of ∗ . In either of these scenarios, the reduction in ∗ shifts the cusp (o) to a location closer to the bridge which reduces the value of ∗ (see Fig. 1); this is because is fixed, so even if all the contact lines are pinned, the reduction in ∗ still changes in the location of the cusp. Therefore, the ratio ∗ ∗ /∗ increases, as ∗ decreases. The ratio ∗ ∗ /∗ represents the amount of confinement (cf) for a bridge between two surfaces i.e. a large means a large confinement for a bridge between two surfaces, and vice versa. Taking the non-dimensional form of Eqn. (1), one has:

cos 2 =

cos − 2

(5)

For a fixed , is only governed by and . Note that Eqn. (5) is independent of the surface tension of the liquid, ;, explicitly. However, the surface tension implicitly affects the value of CAs. In Fig. 4, the contour plot of Eqn. (5) (i.e. versus and ) for the range of 92° < , ≤ 150° ( = 2°) is plotted. The values shown on the contour lines are values. The blue-dotted line is the line of = , therefore, it divides the plot into two regions: one with (Region I) and the other with (Region II); see Fig. 4. Region I denotes a small confinement (i.e. small ), whereas Region II represents a large confinement (i.e. large ). The 12 ACS Paragon Plus Environment

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minimum value of (denoted as >? ) occurs when the distance between the surfaces is so

large that ∗ ≫ ∗ , which implies >? ≈ 1. Therefore, according to Eqn. (5), cos − ≈

cos , which reduces to:

− ≈

(6)

Since , any = / smaller than ≈ 1 is not representing a physical system. For a given surface, and are limited by and . For systems studied here (Teflon and OTS) the dashed squares in Fig. 4 show the possible theoretical range for . In Fig. 5, the contour plot of Eqn. (5) for the bridge discussed in Fig. 3 is plotted (Teflon AF surface at = 2° ). The black circles correspond to values of the bridge during the compressing and stretching (some circles are labeled with ‘a’ to ‘e’ corresponding to each of the snapshots in Fig. (4)). The values were calculated, both by measuring and and using RHS of Eqn. (5), and by using and in the definition of (i.e. = / ). The calculated by either way were in good agreement with each other; this shows the validity of Eqn. (1). All data points between processes ‘a’ to ‘b’ in compressing, and between ‘d’ to ‘e’ in stretching, fall within the Region I. This can be used to explain the behavior of the bridge from ‘a’ to ‘b’, and ‘d’ to ‘e’ (yellow dashed-lines in Fig. 5) as follows: since in this region, would reach earlier than in compression (process ‘a’ to ‘b’), and in stretching, would reach earlier than (process ‘d’ to ‘e’). Thus, the bridge moves towards the cusp in these two processes. As such, an ‘O bridge’ behaves like an ‘I bridge’ as long as is sufficiently small to fall in the Region I (e.g. at large ∗ ). On the other hand, when increases and falls in the Region II, needs to be smaller than to uphold the Eqn. (5). The values of from processes 13 ACS Paragon Plus Environment

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‘b’ to ‘c’ and ‘c’ to ‘d’ fall in the Region II (red dashed-lines in Fig. 5). Since in this region, the bridge moves away from the cusp. Considering above, Eqn. (5) can be used to explain the change in the direction of the motion at different ∗ by correlating ∗ to . In general, an ‘O bridge’ tends to move away from the cusp when the cf value is large. And it tends to move towards the cusp when the distance between the surfaces are large (small cf). Such distinction between these two types of motion has never been studied before. The sensitivity of the motion to can be used in practice; in next section, we will demonstrate that only by adjusting ∗ one can unidirectionally move an ‘O bridge’ towards the cusp, or away from it using three distinct approaches.


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Fig. 4 Contour plot of versus and for the range of 92° < , ≤ 150° for = 2°.


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Fig. 5 Contour plot of versus and for the liquid bridge in Fig. 3 undergoing the same compressing and stretching depicted in Fig. 3. The red numbers on contours are values. Black circles correspond to values of the bridge in Fig. 3 during compressing and stretching (some are labeled with letters based on Fig. 3 snapshots). The blue-dotted line is the line of = . Dashed-lines are to guide the eyes.

3.2.

Unidirectional motions

3.2.1. Moving an O bridge towards the cusp For the first time, we will demonstrate the possibility of moving an ‘O bridge’ unidirectionally towards the cusp in a loading cycle. To achieve this goal, the value of should be kept in the Region I during the entire compressing and stretching cycle. Figure 6 shows a bridge between 16 ACS Paragon Plus Environment

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two Teflon AF surfaces with = 2° undergoing compressing and stretching starting from ∗ = ∗ = 0.86. The bridge was compressed and stretched with ∆ ∗ = 0.16. The small amount of ∆ ∗ at a large ∗ meant the value of for the entire cycle fell in the Region I, and so remained larger than during the cycle (see Fig. 6 (a)). As a result, the bridge moved towards the cusp in both compressing and stretching phases (see Fig. 6 (b)). The same behavior was also observed for a liquid bridge between OTS surfaces undergoing a small amount of compressing and stretching at a large ∗ (see details in SI Section S2). Video 1 as an electronic companion to this paper provides an example of such motion.

Fig. 6 (a) Contour plot of versus and for the system ( = 2°, Teflon AF surfaces) undergoing the compressing and stretching depicted in (b). Black circles are labeled according to Fig. 6 (b). In (b), the orange lines highlight the surfaces contact area with the liquid. The yellow dashed-line shows the origin where the bridge movement is measured.


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3.2.2. Moving an O bridge away from the cusp In the Region II, the pinning and depinning occurs in the opposite sides of a bridge compared to the Region I, causing the bridge to move away from the cusp during a loading cycle. To move an ‘O bridge’ unidirectionally away from the cusp, one can start compressing and stretching the bridge from an initial ∗ = ∗ where value falls within the Region II. By compressing and stretching, the bridge moves away from the cusp as long as remains within the Region II. At ∗ = 0.46, a bridge between Teflon AF with = 2°, satisfies the above conditions; = 1.09 and falls inside the Region II (see Fig. 7 (a) point 1). Starting from ∗ = ∗ = 0.46, the bridge was compressed and then stretched with ∆ ∗ = 0.2. For this amount of ∆ ∗ , is kept inside the Region II during the entire loading cycle (see Fig. 7 (a) processes ‘1’ to ‘3’). This caused the bridge to move away from the cusp in both compressing and stretching stages (see Fig. 7 (b)). Under a similar condition, the same mechanism of motion was also observed for liquid bridges between OTS surfaces which details can be found in SI Section S3. Video 2 as an electronic companion to this paper provides an example of such motion.


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Fig. 7 (a) Contour plot of versus and for the system (Teflon surfaces, = 2° ) undergoing the compressing and stretching depicted in (b). Black circles are labeled according to Fig. 7 (b). In (b), the red dashed-lines highlights the surfaces contact area with the liquid. The yellow dashed-line shows the origin where the bridge movement is measured.

3.2.3. Sliding motion of an O bridge away from the cusp Considering the range of

, the maximum value of (denoted as ) occurs when

= and = :

∗ ∗ cos 2 , , = = ∗

, cos − 2

(7)

∗ ∗ At , ∗ and ∗ have their maximum values (i.e. ∗ = , and ∗ = , ). If the bridge is

compressed such that = . For any extra compression, ∗ and ∗ cannot be larger, 19 ACS Paragon Plus Environment

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therefore, the only way for the bridge to obey Eqn. (7) is to recede on the narrow side (where = ) and advance on the wide side (where = ) simultaneously; as such, a sliding motion towards the less confined region will result. Simultaneous motion of the wide side and ∗ ∗ the narrow side of the bridge are such that , and , (and as a result) remain

unchanged. Since = , and remain unchanged, from trigonometry, it can be shown that ∗ the amount of horizontal movement of the bridge during sliding (∆5BC>D>?E ) is equal to:

∗ ∆5BC>D>?E

∗ ∆ BC>D>?E ≈ tan

(8)

∗ is the amount of compressing from the moment = . where ∆ BC>D>?E

In Fig. 8, a bridge between Teflon surfaces with = 2° is shown with ∗ = 0.16. At this point, ≈ = 1.41 (see Fig. 8 (a)). Then, the compression was continued for another ∗ ∆ BC>D>?E = 0.078. The bridge started sliding away from the cusp by advancing on the wider

side, and simultaneously receding on the narrow side of the bridge (see Fig. 8 ‘b’ and ‘c’). Since ∗ ∗ ∗ = , and ∗ = , were unchanged during the process, the shape of the bridge also ∗ remains unchanged (compare the shapes of the bridge from ‘a’ to ‘c’ in Fig. 8). The ∆5BC>D>?E

calculated from Eqn. (8) and the experiments where in good agreement as well (see Fig. 8). The error (~5%) can be due to the measurement and/or evaporation of the liquid volume as it is used for the length scaling. Video 3 as an electronic companion to this paper provides an example of such motion.


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Fig. 8 (a) A liquid bridge (Teflon surfaces, = 2°) was formed at ∗ = 0.16 where = . (b) The bridge slides away from the cusp to maintain . (c) As compressing continues, the bridge keeps sliding away from the cusp while maintaining . The red dashed-lines highlights the surfaces contact area with the liquid. The time (I) of the snapshots are given on the right side of the figure.

The for an ‘O bridge’ between OTS surfaces is much larger than that of the system with Teflon AF ( = 2.87 versus 1.41). Therefore, the bridge must be compressed much more to achieve . With the current experimental setup, the surfaces narrow ends came in contact with each other before reaching , preventing us to observe the sliding phenomenon for the systems with OTS surfaces. Note that this sliding motion is different from the spontaneous motion of a bridge between hydrophilic surfaces reported in Refs. 1-8. The spontaneous motion is an unstable uncontrollable


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motion (i.e. not in the mechanical equilibrium), while the sliding motion reported here is quasistatic and controllable with ∆ ∗ .

3.3. Effect of J and CAs on the direction of motion during a loading cycle In the previous section, we demonstrated that the bridge can be unidirectionally moved in three distinct ways. However, in certain scenarios, only one type of the motions may be desired, while others should be avoided e.g. only motion towards the cusp may be desired. We discussed how ∗ can be adjusted to induce one of the motions (i.e. by making the system more and less confined with a change in ∗ ). Another question may arise here: How , and can be adjusted to favour one of the motions? To answer this question, imagine a scenario where a stable bridge is formed with an initial . Depending on the value of , it can be in the Region I, Region II, or can be equal to . 3.3.1. Scenario 1: Assume is in the Region I (similar to the case in Fig. 6) and the goal is to move the bridge only towards the cusp. In compression stage, the narrower side of the bridge advances with = . The advancing of the narrower side continues until passes ‘ = ’ line into the Region II. Since, = , the intersects the line ‘ = ’ at = = . At this point, the value of = K (see K in Fig. 5; subscript ‘c’ stands for compression) will be:

cos 2 K =

cos − 2

(9)

so K is only governed by at a fixed . At = K , for any larger compression, the direction of motion changes (bridge will move away from the cusp) as falls in the Region II. To avoid 22 ACS Paragon Plus Environment

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the motion of the bridge away from the cusp (in favour of motion of the bridge towards the cusp) K should increase. In the next subsection, we will discuss how this can be done. 3.3.2. Scenario 2: Assume is in the Region II and < (like the case in Fig. 7). By compressing the bridge, remains in the Region II and the bridge keeps moving away from the cusp with = ; the direction of motion will not change with an increase of . On the other hand, in stretching, the narrower side will recede with = and decreases. If the stretching continues, may intersect the line ‘ = ’ at = = ; will fall in the Region I where the direction of the motion changes towards the cusp. At this point, = B (subscript ‘s’ stands for stretching)

cos 2 B =

cos − 2

(10)

In favor of bridge’s motion away from the cusp in the stretching stage, one can reduce B (i.e. to keep B ) to avoid the motion of the bridge towards the cusp. The derivatives of K with respect to and are given in Eqns. (11) and (12), respectively. The denominators of both equations are the same and positive. In addition, sin 0 and in the

range of , sin2 < 0. Therefore, MK ⁄M 0 and MK ⁄M < 0. MK −2 sin = M 1 cos 2 −


(11)

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MK − sin2 = M

1 cos 2 −

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(12)

As discussed, for a favorable motion towards the cusp, K should increase. Therefore, one can increase and/or decrease . Since the formulations of Eqns. (9) and (10) are similar, B has a comparable response to the change in and i.e. MB ⁄M 0 and MB ⁄M < 0. Thus, to reduce B with the goal of favouring the motion away from the cusp, one should reduce and/or . 3.3.3. Scenario 3: Assume = thus, the bridge will slide away from the cusp for any further compression (e.g. see Fig. 8). The sliding motion can be achieved until the surfaces’ edge intersect (contact) as ∗ is reduced. Recall that increases with the reduction of ∗ . Therefore, a smaller would occur at a larger ∗ . For a smaller , the sliding motion can be induced in a larger range of ∗ before surfaces contact each other, hence allowing to move the bridge further from the cusp. As such, in inducing the sliding motion, a smaller would be more desirable. In Fig. 9 (a) (b), and (c), the effects of , and on are shown in isolation, respectively. The value of is calculated with Eqn. (7). In examining the effect of each of the parameter, other parameters were fixed and their constant values are shown above each plot. Increasing and , increases , while increasing reduces . Among wettability parameters, the influence of on is significantly larger compared to . Though it can be technically more challenging to fabricate surfaces with small , reducing of the surfaces is more effective to achieve a smaller than increasing .


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Fig. 9 versus (a) (at = 2° and = 126.7°) (b) (at = 126.7° and = 10°) and (c) (at = 2° and = 10°).

The recommended change in wettability and geometrical parameters for each of the three scenarios are summarized in Table 1. Table 1 A summery of the recommended changes in wettability and geometrical parameters for each scenario discussed in Sections (3.3.1-3.3.3) parameter

Scenario 1: Motion towards the cusp

Scenario 2: Motion away from the cusp

Scenario 3: Sliding motion

Geometrical

Increase

Decrease

Decrease

Wettability

Decrease

Increase /decrease

Increase /decease

3.4.

Motion of a liquid bridge between ideal surfaces

An ideal surface is defined as one that does not show a contact angle hysteresis. An example can be liquid infused textured surfaces that show ≈ 0; such surfaces can also have contact angles as high as 150° [16, 17]. The motion of an ‘O bridge’ between such surfaces can only be

of sliding type for a constant . This is because = is a constant, which means that the distance of a bridge from the cusp should remain constant for any compression or 25 ACS Paragon Plus Environment

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stretching. Therefore, the bridge should slide away or slide towards the cusp in compression or stretching, respectively. Since ∗ and are constant, the amount of horizontal movement of the bridge can be calculated as ∗ ∆5BC>D>?E =

∆ ∗ tan

(13)

where a positive ∆ ∗ (stretching) results in a horizontal movement towards the cusp, and vice versa. To examine Eqn. (13), we used Surface Evolver which is a numerical tool to predict equilibrium shapes of liquid interfaces subjected to various forces and constraints [15]. Surface Evolver allows finding the equilibrium geometry of a liquid by minimizing the surface energy using a gradient descent method [15]. To perform the simulations, first, the equilibrium shape of a bridge between two tilted surfaces was found for three cases with = 115°, 130°, and 150°. The was

taken to be 2°, 5°, 10° and 20°, which satisfy constraint. Then, the top surface was

moved up with 0.1 increments until ∆ ∗ = 0.5. In each step, the new equilibrium shape of the ∗ bridge was found and the value of ∆5BC>D>?E was calculated with respect to the initial position.

Such a motion resembles the quasi-static motion of the liquid bridge. The simulation was then repeated for compressing with 0.1 increments until ∆ ∗ = −0.5. The results are shown in Fig. ∗ 10 (a). It can be seen that ∆5BC>D>?E is linearly related to ∆ ∗ as predicted by Eqn. (13). Note that ∗ in Fig. 10, simulation results for different values of coincide as ∆5BC>D>?E is only a function of

∆ ∗ and , not . The value of only affects the value of i.e. For larger values of , the bridge finds its equilibrium further from the cusp, and vice versa. See Fig. 11 (b) for examples. Another way of moving an ‘O bridge’ between ideal surfaces is to keep the surfaces fixed, instead changing ; for example, using electrowetting i.e. larger will move the bridge away 26 ACS Paragon Plus Environment

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from the cusp (by decreasing ), and smaller will move the bridge towards the cusp (by increasing ). See Ref. 10 for more details. A 3( water bridge between two oil infused surfaces was used to validate the findings from the numerical simulation and theoretical reasoning explained above ( = 4°). The surfaces were prepared using 5 Sct silicone oil spin coated on a substrate decorated with micro pillars. Such oil infused surfaces have a small contact angle hysteresis; thus, they can be considered as ideal surfaces [16, 17]. As shown in the supplementary ‘Video 4’ accompanied with this paper, the bridge slides (on both narrower side and the wider side) towards the cusp of the solid surfaces after 2&& stretching was applied. We demonstrated that in the absence of , the motion of a bridge can only be of the sliding type. The two methods of moving a bridge by compressing and stretching discussed in Sections 3.2.1 and 3.2.2 require one side of the bridge to remain pinned (while the other side is moving) which is enabled by . Although facilitates these type of motion, high values also means that a large amount of compression or stretching (i.e. ∆ ∗ ) will not cause any motion for a bridge at the beginning of compressing and stretching stages. This is so as a certain amount of ∆ ∗ is needed to overcome the pinning of the contact line either for advancing or receding to be possible.


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∗ Fig. 10 (a) ∆5BC>D>?E versus ∆ ∗ for three systems with ideal surfaces with = 115°, 130° and ∗ 150° each tested for = 2°, 5°, 10°, and 20°. The negative values for ∆5BC>D>?E means sliding away from the cusp. Simulation results for different coincide for each , therefore they are all represented with identical markers corresponding only to the value of . Results are from Surface Evolver simulation. (b) The equilibrium shapes a liquid bridge between ideal surfaces with (1) = 115° (2) = 130°, and (3) = 150°. All surfaces have = 20°. The liquid bridges have the same volume and the images are to the scale.

4. Applications and Prospects Manipulating a small amount of liquid has several industrial applications especially in microfluidics and lab-on-a-chip devices. Electrowetting has shown to be a promising tool to actuate drops and bubbles in these applications [18]. Future works in this area can consider coupling nonparallel surfaces with electrowetting to promote and/or further control liquid motion. Wang and McCarthy [19] synthesized catenoid-shaped particles by polymerizing capillary bridges. The shape and symmetry of the particles can be further controlled with the results 28 ACS Paragon Plus Environment

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presented here. For example, one can synthesize truncated sphere-shaped particles by forming a bridge between two nonparallel hydrophobic surfaces with small CAH, and control its shape using the value of the contact angle (see Fig. 10 (b)). Exploiting the hydrophobicity of surfaces to transport small number of particles has been previously reported [20]. The same goal can also be achieved using capillary bridges between nonparallel surfaces. Particles can be engulfed in the bridge and be moved along the surfaces. In another study, liquid bridges between nonparallel surfaces has been used to separate mixed drop of two immiscible liquids with different wetting properties (e.g. oil and water) [21]. One way to achieve this is by squeezing the mixture between hydrophobic surfaces; the liquids separate as the water moves away from the cusp (due to the surfaces' hydrophobicity) while the oil moves towards the cusp. As shown in this work, the motion of water away from the cusp is only achievable if the mixture is squeezed until the confinement parameter for water reaches Region II.

The present study and two of our previous works [1,2] attempted to explore the behavior of the bridge between identical surfaces. With different top and bottom surfaces, the asymmetric shape of the bridge prevents us to describe the confinement parameter using a closed-form equation such as Eqn. (5). However, the insights from these studies can facilitate understanding the behavior of liquid bridge between non-identical surfaces in the future.


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5. Conclusions We showed for the first time that a liquid bridge between two hydrophobic surfaces can move either towards or away from the cusp, or even have a sliding motion. Defining a new geometrical factor for the system, i.e. confinement of the system (), we showed that the value of is the main determining factor for the direction of the motion. The bridge tends to move towards the cusp, when ∗ is large (a small ), and away from the cusp when ∗ is small (large ). Each system has a maximum value; if a liquid bridge is compressed beyond the , the bridge shows a sliding motion (i.e. simultaneous advancing on the wider side, and receding on the narrower side) away from the cusp. We also concluded that the sliding motion is the only possible type of motion for a liquid bridge between two ideal hydrophobic surfaces (i.e. = 0). For such systems, the bridge slides towards the cusp upon stretching and when compressed it slides away from the cusp. The sliding distance is only related to geometrical parameters such as the extent of compression (or stretching) and the dihedral angle between the surfaces.

■ Acknowledgements The authors acknowledge financial support from the Natural Science and Engineering Research Council (NSERC).

■ Author Information

Corresponding Author 30 ACS Paragon Plus Environment

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*E-mail: [email protected]. Tel: +1-416-736-5901 Notes The authors declare no competing financial interest.

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10. D. Baratian, A. Cavalli, D. van den Ende, F. Mugele, On the shape of a droplet in a wedge: new insight from electrowetting, Soft Matter 11 (2015) 7717–7721. 11. P. Concus, R. Finn, On the behaviour of capillary shapes in the wedge, Proc. Natl. Acad. Sci. 63 (1969) 292–299. 12. P. Concus, R. Finn, Discontinuous behavior of liquids between parallel and tilted plates, Phys. Fluids 10 (1998) 39–43. 13. T. Chen, J.A. Tsamopoulos, J.G. Robert, Capillary bridges between parallel and non-parallel surfaces and their stability, J. Colloid Interface Sci. 151 (1992) 49– 69. 14. D. J. Broesch and J. Frechette, From Concave to Convex: Capillary Bridges in Slit Pore Geometry, Langmuir 28 (2012), 15548-15554. 15. K. Brakke, The Surface Evolver. Exp. Math. 1992, 1, 141-165. 16. E. Jenner and B. D'Urso, Wetting States on Structured Immiscible Liquid Coated Surfaces, Appl. Phys. Lett. 103 (2013) 251606. 17. F. Schellenberger, J. Xie, N. Encinas, A. Hardy, M. Klapper, P. Papadopoulos, H. Butt, D. Vollmer, Direct Observation of Drops on Slippery Lubricant-infused Surfaces, Soft Matter, 2015,11, 7617-7626. 18. B. Pinchasik, J. Steinkühler, P. Wuytens, A. G. Skirtach, P. Fratzl, H. Möhwald, From Beetles in Nature to the Laboratory: Actuating Underwater Locomotion on Hydrophobic Surfaces, Langmuir, 31 (2015), 13734-13742. 19. L. Wang and T. J. McCarthy, Capillary-bridge-derived Particles with Negative Gaussian Curvature, PNAS 112 (2014) 2664–2669.


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20. B. Pinchasik, H. Moehwald, A. G. Skirtach, Mimicking Bubble Use in Nature: Propulsion of Janus Particles due to Hydrophobic-Hydrophilic Interactions, Small 10 (2014), 10.1002/smll.201303571. 21. C. Luo and X. Heng, Separation of Oil from a Water/Oil Mixed Drop Using Two Nonparallel Plates, Langmuir 30 (2014), 10002-10010.


Behavior of a Liquid Bridge between Nonparallel Hydrophobic

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