Droplets sliding down a vertical surface under increasing horizontal

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Droplets sliding down a vertical surface under increasing horizontal forces Sirui Tang, Yagnavalkya Bhimavarapu, Semih Gulec, RATUL DAS, Jie Liu, Hartmann E. N'guessan, Taylor Whitehead, Chun-Wei Yao, and Rafael Tadmor Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b04157 • Publication Date (Web): 15 Apr 2019 Downloaded from http://pubs.acs.org on April 21, 2019

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Droplets sliding down a vertical surface under increasing horizontal forces Sirui Tang,† Yagnavalkya Bhimavarapu,† Semih Gulec,† Ratul Das,† Jie Liu,† Hartmann N’guessan,† Taylor Whitehead, ‡ Chun-Wei Yao,§ Rafael Tadmor*†#

†

Dan F. Smith Department of Chemical Engineering, Lamar University, P.O. Box 10053,

Beaumont TX 77710 ‡ Department

of Physics, Lamar University, Beaumont, TX 77710

§ Department

of Mechanical Engineering, Lamar University, Beaumont, TX 77710

# Department

of Mechanical Engineering, Ben Gurion University, Beer Sheva, Israel

Keywords: Centrifugal Adhesion Balance, Adhesion, Tribology, Retention force, Interfacial modulus, Wetting properties, Water drops on surfaces Corresponding Author *E-mail: [email protected]

Abstract

We have investigated the retention forces of liquid drops on rotating, vertical surfaces. We considered two scenarios: In one, a horizontal, centrifugal force pushes the drop towards the surface (“pushed drop” case), and in the other, a horizontal, centrifugal force pulls the drop away from the surface (“pulled drop” case). Both drops slide down as the centrifugal force increases, although one expects that the pushed drop should remain stuck to the surface. Even more surprising, when the centrifugal force is low, the pushed drop moves faster than the pulled drop, but when the

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centrifugal force is high, the pushed drop moves much slower than the pulled drop. We explain these results in terms of interfacial modulus between the drop and the surface.

INTRODUCTION The thermodynamic equilibrium of a liquid drop that rests on a horizontal surface is well understood in terms of Young’s equation,1, 2 which results from the minimization of the interfacial energies.3–5 However, the relation between the force that is required to move a liquid drop and properties of the solid surface, such as its ability to reorient its molecules upon interaction with the liquid, is still not well understood. The Furmidge equation does not address this, and the Tadmor equation, which does, requires verification. Understanding such phenomena is important in many applications, such as printing,6 microfluidics,7 steam generator,8 etc. Methodologies that have been used in previous investigations to study the forces associated with wetting phenomena include magnetic force,9 air flow,10 vertical deflectable capillary,11, 12 and tilt stage13 etc., each of which has its unique strengths. In this paper, we study the motion of drops sliding on a vertical surface using the Centrifugal Adhesion Balance (CAB). Through a combination of centrifugal and gravitational forces, the CAB is able to decouple normal and lateral body forces. A previous study14, 15 demonstrated that the lateral force required to slide pendent drops on a surface is greater than the lateral force required to slide sessile drops, in spite of the larger contact area of sessile drops, and seemingly contrary to intuition. In these studies, the normal force (gravitational force) on the drops was assumed constant during the experiment.14, 15 The purpose of the present paper is to extend those studies to cases where the load on the drops can be varied during the experiment.


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In this study, we refer to a “pushed” (or “pulled”) drop as a drop on a vertical surface faces towards (or away from) the axis of the rotation and the drop is pushed towards (or pulled away from) the surface (see Figure 1). To compare the behavior of drops on horizontal surfaces with that on vertical surfaces, we need to realize that the pushed drop is the analog of the sessile drop, whereas the pulled drop is the analog of the pendant drop. These analogies correspond to the direction of the force perpendicular to the surface. For the sessile drop, the gravity is pushing the drop into the surface, while for the pendent drop, the gravity is pulling the drop away from the surface. From the results of Refs. 14 and 15, one would expect that the pulled drop may experience stronger pinning than the pushed drop. While this is true for the onset of motion, our results show that the behavior is more complex and depends on the stage of the motion as the strength of the normal force (centrifugal force) increases with time. We found that, as the centrifugal force starts pulling or pushing the drops, both pulled and pushed drops slide along the surface. Surprisingly, at early stages, when the centrifugal (normal) force is small, the pushed drop slides down faster than the pulled drop, but at later times stages, when the centrifugal force is stronger, the pushed drop slides down slower than the pulled drop.

EXPERIMENTAL SECTION Materials. The substrate used were silicon wafers (P-doped with Boron; orientation: ; thickness: 406 - 480 μm; resistivity: 0 - 100 Ohm-cm) purchased from University Wafers. The silicon wafers were cut into 12 mm×12 mm pieces.

Surface preparation. We utilized and modified the similar technique developed by Einati et al16 for the silanization of the surface. Before the procedure of silanization, the silicon substrates were


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rinsed with the following sequence of solvents: deionized water, acetone, deionized water. The silicon substrates were dried in a StableTemp vacuum oven (model: 5053 - 10 from Cole-Parmer) at 100 ℃ for 30 min and then cleaned by Ultraviolet-Ozone Surface Treatment in a UV/ozone (model: Procleaner 110) for 45 min to remove the contaminants from surface and decompose organic molecules on surface immediately prior to use. Silanization was applied by coating from 1 wt% C18 Silane (Trimethoxyl(octadecyl) silane)/tolune solution. The silicon surface was immersed in this solution for 3 hr at 70 ℃. The silanized surface was rinsed by deionized water and dried in the vacuum oven for 45 min at 80 ℃.

The Polydimethylsiloxane(PDMS) and curing agent(Sylgard 184, Dow Corning) were mixed in the ratio of 10:1(weight : weight). The PDMS was put in the vacuum desiccators for degassing (20 min) and then was made to coat the silicon surface by spin coating at 500 rpm for 15 s. The surface was placed in an oven for 20 min at 110 ℃.

Experimental procedure. Several satellite droplets were put around the surface on the sample holder to reduce evaporation and thus maintain a constant drop volume of the main (central) droplet.17 Also to keep the drops deposition process consistent,18 we used the same type of the syringe and placed the drop at the same height from the surface. Thus, the drops have an identical history from placement to tilt for both pulled and pushed cases. Following the placement of the central droplet, the chamber was sealed with a glass dome to protect against wind and suppress evaporation. The evaporation suppression system increased the relative humidity around the drop to about 94% within 2 min.17 Then the surface was tilted to a vertical position, and the centrifugal arm started rotation with constant acceleration 0.6 rpm/s. 4 ACS Paragon Plus Environment

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Centrifugal Adhesion Balance (CAB). The CAB (Wet Scientific) has a shaft that rotates with the help of a motor (Figure S1). A centrifugal arm is attached to the shaft. When the shaft rotates, the centrifugal arm rotates in the horizontal plane. At the end of the centrifugal arm, there is a plate that can be tilted at any desired angle with respect to the horizontal direction with the help of another motor. The plate is inside a chamber in which a camera is mounted to monitor the motion of the drop (up to 60 frames per second). A sample holder is mounted on the plate. The entire assembly of motors and camera is controlled by a computer. More information about CAB is available in the Supporting Information (SI).

Lateral and normal forces acting on pulled and pushed drops. Figure 1A shows a schematic representation of the drop resting on the sample plate of the CAB,14, 19 and the lateral (𝑓 ∥ ) and normal (𝑓 ⊥ ) forces can be calculated as follows, 𝑓 ∥ = 𝑚(𝜔2𝐿 𝑐𝑜𝑠 𝛼 ― 𝑔𝑠𝑖𝑛𝛼)

(1)

𝑓 ⊥ = 𝑚(𝜔2𝐿 𝑠𝑖𝑛 𝛼 + 𝑔𝑐𝑜𝑠𝛼)

(2)

where 𝑚 is the mass of the drop, 𝑔 is the acceleration of gravity, 𝜔 is the angular velocity, 𝐿 is the distance from the center of the drop to the axis of the rotation, and 𝛼 is the tilt angle, which can be adjusted from 0º to ±360º.


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Figure 1. A) Schematic representation of a drop resting on the rotating, tilted plate of the CAB. Diagrams of B) the pulled and C) the pushed drops. The normal force is the centrifugal force (𝑚𝜔2 𝐿), and the lateral force is the weight (𝑚𝑔). The insets in B) and C) show how the clockwise rotation of the CAB produces a normal force on the drop.

To have the centrifugal force pulling the drop away from or pushing the drop into the surface, the initial horizontal surface is tilted to the vertical position as shown in Figure 1B and 1C. When the drop is tilted to the vertical position, the tilting angle, 𝛼, in Eq. (1) and (2) becomes 90° (or 270°). Considering gravitational force of the drop as the lateral force and the centrifugal force as the normal force acting in our pulled case and pushed case and considering only the magnitude of the 6 ACS Paragon Plus Environment

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forces (the omitted minus and plus signs correspond to the pulled and the pushed drops, respectively), Eqs. (1) and (2) become, 𝑓 ∥ = 𝑚𝑔

(3)

𝑓 ⊥ = 𝑚𝜔2𝐿

(4)

Figure 1B and 1C displays the forces of Eqs. (3) and (4) acting on a pulled and a pushed drop. The lateral force (𝑓 ∥ = 𝑚𝑔) is constant, as the mass lost for a 3 μL water drop due to evaporation in the course of an experimental run (~5 min) is negligible under these conditions: room temperature 23 ℃ and 1 atm with 94% relative humidity.17 The normal force (𝑓 ⊥ = 𝑚𝜔2𝐿) increases at the same angular acceleration.

We define the front edge maximal contact angle as the advancing contact angle (𝜃𝐴), the rear edge minimal contact angle as receding contact angle (𝜃𝑅), and the as-placed contact angle18 as the resting contact angle (𝜃𝑟𝑒𝑠𝑡), see Figure 2.20


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Figure 2. A) Image of a water drop on a horizontal surface. The dashed line represents the interface between the drop and the surface. B) Schematic representation of a drop resting on an inclined surface, and top view of its contact line. 𝑙 denotes the drop’s length and 𝑤 denotes the drop’s width.

RESULTS AND DISCUSSION Drop motion. We placed a water drop on a C18 silanized silicon surface as shown in Figure 2A and tilted the surface to a vertical position as shown in Figure 1B and 1C. For a 3 μL water drop resting on a C18 silanized silicon surface, the Bond number is Bo = 0.18 (Bo =

∆𝜌𝑔𝑅2 21 ) , 𝛾

where 𝑅

is the radius of the curvature of the drop at its apex.22 The apparent contact angle is 98° ± 1° for the drops as they are placed on a horizontal surface (soon to be pulled or pushed). After the surface is tilted to the vertical position, the advancing and receding contact angles are 101.5 ± 1° and 93 ± 1° respectively. When the rotation speed is zero, these contact angles are still the same for the pulled and the pushed cases. They begin to experience different normal forces only after the rotation of CAB starts. The angular velocity of the rotation increases linearly at the rate of 0.6 rpm/s (Figure S2). As the angular velocity increases, both pushed and pulled drops slide down, though at different speeds. Figure 3 displays six images taken at different instants (𝑡 = 0, 50 s, and 300 s) for both pulled (left-hand side of Figure 3) and pushed (right-hand side of Figure 3) drops (videos 1 and 2). We choose 𝑡 = 0 as the moment when the centrifugal force starts to be applied, and 𝑥 = 0 as the initial position of the receding edge of the drops (see Figure 3a).


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Figure 3. Selected images of 3 μL water drops on a vertical surface while being pulled away from (left-hand side) and pushed towards (right-hand side) a C18 silanized silicon surface at times a) 𝑡 = 0, b) 𝑡 = 50 s, and c) 𝑡 = 300 s. The horizontal arrows represent the normal (centrifugal) forces, 𝑓 ⊥ , whose magnitude is shown on each arrow. The thin, vertical arrows represent the lateral (gravitational) force, 𝑓 ∥ = 29 µN for all cases (the drop’s weight).

At 𝑡 = 50 s, the centrifugal force is about 1/6 the weight of the drop (𝑓 ⊥ = 5 µN, 𝑓 ∥ = 29 µN). At this instant, the pulled drop has barely moved (about 0.36 mm), but surprisingly the pushed drop has moved significantly (about 1.07 mm) (see Figure 3b). Thus, contrary to what one would expect, the pushed drop moves faster than the pulled drop at the initial stages of the low centrifugal forces. At 𝑡 = 300 s, the centrifugal force is 170 µN, which is about six times the weight of the 9 ACS Paragon Plus Environment

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drop. By this time, the pushed drop has moved about 2.85 mm, but the pulled drop has slid down the surface a significant 5.34 mm (see Figure 3c). Overall, we observed that the pulled drop started to move significantly faster than the pushed drop when the normal force (centrifugal force) reached a value of about 170 μN. Thus, when the normal force is weak, the pushed drop moves faster than the pulled drop, but when the normal force is strong, the pushed drop moves slower than the pulled drop.

In Figure 4 we can distinguish between three stages in the motion of the drops: At the first stage (𝑡 = 0 - 50 s, and small centrifugal force 𝑓 ⊥ = 0 - 5 µN), the velocity of the pushed drop is significantly larger than that of the pulled drop; at the second stage (𝑡 = 50 - 275 s, and intermediate normal force 𝑓 ⊥ = 5 - 150 µN), the velocities of the pulled and the pushed drop are comparable; and at the third stage (𝑡 > 275 s, and large normal force 𝑓 ⊥ > 150 µN), the speed of the pulled drop increases profoundly, whereas the speed of the pushed drop experiences only a minor increase (see Figure 4b). The velocity during the first five seconds is plotted in the inset of Figure 4b. When the rotation speed is zero, the velocity of both pulled and pushed drops (which are still the same) is 7.2 µm/s on average. Once rotation starts, the velocities increase to different values as shown in the inset to Figure 4b. These results seem counterintuitive, especially the fact that the pushed drop moves faster than the pulled drop during the first stage of the motion, and the fact that the pushed drop continues to move down the surface during the third stage even though the normal force acting on it increases.


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Figure 4. Plots of a) position of advancing and receding edges, b) velocity and centrifugal force, c) drop length, and d) contact angles as the function of time for the pulled and the pushed drops (c.f. Figure 3) for 3 μL water drops on vertical C18 silanized silicon surfaces. The inset in b) shows the velocities in the first 5 s. Red - pushed drop, blue - pulled drop, black - centrifugal force. Time 𝑡 = 0 is the instant when the rotation starts.

As can be seen in the Figure 4d, the contact angles remain relatively constant during the first stage of the motion of the drops, albeit some fluctuations. At the second stage the contact angles of pulled drop vary more than that of pushed drop, with the advancing angle increasing slightly and 11 ACS Paragon Plus Environment

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the receding angle decreasing slightly. However, at the third stage of the motion, the advancing and receding contact angles of the pulled drop significantly increase and decrease respectively.

As the normal force increases, the shape of the drop changes, especially at the third stage, see Figure 3c. As the centrifugal force increases, the elongation in the lateral direction of the pushed drop increases, while that of the pulled drop decreases; meanwhile, the height of the pushed (pulled) drop decreases (increases), which moves the center of mass of the drop towards (away from) the surface.

In Figure 4c, we can see how the length of the drops remains constant during the first stage of the motion, while it starts increasing (for pushed drops) and decreasing (for pulled drops) at the second stage. However, this change is not correlated with the change in the speed. For example, at the first stage of the motion, the length of the pushed drop remains relatively constant, whereas its speed is (relatively) high. At the second stage of the motion, the speed of pushed drop decreases with increasing length, while the speed of pulled drop has minor change with decreasing length. Finally, at the third stage, both the speed and the elongation in normal direction of pulled drop increase.

The features of Figure 4 can be seen in other systems. As an example of a polymeric surface, we measured 2 μL and 3 μL pulled and pushed drops sliding on a Polydimethylsiloxane(PDMS) surface (Figure 5). These results agree with the features of water-silanized silicon surface system as shown in Figure 4 and are clearer than the silanized system because it has a long period of zero motion at the first stage. There, following a long and steady part in which there is no motion of the water drops on the PDMS surfaces, it is clear to see that the pushed drops start moving at a smaller


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normal force compared to the pulled drop. The slowdown of the pushed drop during the 3rd state is also clear in this system, as well as the steady increase in the velocity of pulled drop with rising normal force.

Figure 5. Plots of a) position of advancing and receding edges and b) velocity as the function of time for pulled and pushed water drops sliding on the PDMS surface. Blue dots or triangles – pulled; red dots or triangles – pushed. Down triangles – advancing; up triangles – receding. Black – normal force. A) 2 μL, B) 3 μL.

Experiments like that shown in Figures 4 were repeated for a number of drops and a number of volumes. Figure 6A shows the average initial velocities which refer to the average velocities of the initial linear portion of the position curve. The average initial velocities of pushed drops are higher than those of the pulled drops, which seems more pronounced for the cases of 3 µL and 4 µL drops. In addition, for both pulled and pushed cases, the average initial velocity is proportional 13 ACS Paragon Plus Environment

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to the size of the drop. Yet, during the last stage the velocities of the pulled drops are much higher than that of the pushed drops (Figure 6B).

Figure 6. The initial (A) and final (B) average velocities of pulled (blue) and pushed (red) water drops sliding on vertical C18 silanized silicon surfaces.

As a side note, we mention that for some small drops (i.e. 2 μL) or dirty surface, the pulled drop occasionally flew away before it completed the sliding length available in our camera, as shown in Figure S3 and video 3. The phenomenon of flying drops is considered in another study.23

Interpretation of the results. The Young equation1, 24 is a balance over the lateral components of the surface tension in a symmetric drop. However, to balance the normal component, γsinθ, there is a need for a deformation (protrusion) at the triple line.25–27 Although this protrusion is topographically negligible, the stresses associated with it enhance the reorientation of the molecules of the solid at the triple line. In addition, capillary forces can break chemical bond while having a small topographic effect.28 Gupta et al.29 and Bormashenko et al. 30 also show that solid molecules reorient due to exposure to a liquid7 and reorient again due to the renewed exposure to 14 ACS Paragon Plus Environment

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air6. In turn, this reorientation facilitates the interactions between these molecules and the molecules of liquid. Since the protrusion and associated reorientation take time, they vary both with the time that the drop is in contact with the surface,25, 26 and with the velocity of the sliding drop . In an earlier publication, the proportionality to sin𝜃 of this protrusion was used to explain why the lateral retention force on sessile drops is smaller than that on pendant drops whose contact angles are always closer to 90°, i.e. higher γsinθ.14

Figure 7 shows schematic representations of pushed and pulled drops, and a possible protrusion that the normal component of their surface tensions produce on the solid. It is accompanied by molecular reorientation, a possible visualization of which is shown in Figure 7A and 7B. In these schematics, the end groups of solid molecules are depicted as having a higher affinity to the liquid than the rest of the chain. Namely, the solid end group is more hydrophilic (more polar) and the rest of the chains are more hydrophobic. When the solid is exposed to air, the more hydrophobic part of the chain is extended to the interface,30 and the more polar group is buried deeper in. When a liquid drop is introduced, the Shanahan – de Gennes type deformation26 at the triple line helps these functional groups move, and the chain reorients so that the polar end groups reach the interface for both pushed and pulled cases.29 However, since the contact angles of the pulled drop is closer to 90°, the higher value of 𝛾𝑠𝑖𝑛𝜃 causes a stronger force, and therefore allows for a quicker, more pronounced, molecular reorientation of the solid. This reorientation facilitates the molecular interaction between the solid and the liquid.


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Figure 7. Schematic illustrations of the solid-drop contact as a result of forces acting on pulled (A & C) and pushed (B & D) drops. A possible Shanahan - de Gennes type protrusion26 and resultant molecular reorientation at the triple line are shown for A) the pulled drop and B) the pushed drop at the first and second stages. The light blue spheres represent a relatively more polar functional groups and the black chains represent a relatively less polar functional groups. At the third stage, the high centrifugal forces cause a shape change shown in C) and D) for the pulled and the pushed drops. The red dots inside the drops in C) and D) represent the center of mass, to show qualitatively, its relative distance from the solid surface. The centrifugal force is directed in the same (respectively, opposite) direction as γsinθ for pulled (respectively, pushed) drops. Thus, the contact angles of pulled drops are closer to 90°, which intensify the protrusion, while that of pushed drop are further from 90°, which subside such


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protrusion. Hence, during the first stage of the motion (when the centrifugal force is not too strong), the retention force on the pulled drop is larger than that on the pushed drop.

When the drop is not moving, the retention force is determined by the triple line rather than the area. However, when the drop moves, both the triple line and the contact area affect the retention force. An increase of the contact area, especially during the second and third stages of the motion, enhances the retention force and vice versa.

For drops on a vertical surface, the position of the center of mass of the drops has a strong influence on the drop’s shape, especially at the third stage of the motion. See Figure 7C and 7D for an exaggerated representation of the shift of the center of mass (represented as a red dot) in the two cases. At this stage, the torque produced by the drop’s weight increases for a pulled drop, due to an increase in the lever arm that is caused by the shift of the center of mass of the drop away from the surface. The higher torque at this stage is part of the reason for the higher speeds of the pulled drops in this stage.

Thus, the retention force on the pulled and pushed drop is affected by three factors: 1. The normal force, which strengthens the retention force of the pulled drop by bringing their contact angles closer to 90º (compared to the pushed drop); 2. The torque, which weakens the retention force of the pulled drop; 3. The reduction of the pulled drop’s contact area, which weakens the force required for moving/sliding these drops.


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The Furmidge equation31–34 for the drop’s lateral retention force, Eq. (5), cannot explain our results, but we bring it here, due to its prevalent use in the literature, to relate to this fact, 𝐹 ∥ = 𝑘 ∙ 𝑤 ∙ 𝛾(𝑐𝑜𝑠𝜃𝑅 ― 𝑐𝑜𝑠𝜃𝐴)

(5)

where k is a shape factor, 𝑤 is the width of the drop (see Figure 2), 𝛾 is the surface tension, and 𝜃𝐴 and 𝜃𝑅 are the advancing and receding contact angles. In Figure 8A, we plot the 𝑘 factors as a function of time according to Eq. (5) assuming 𝑤 = 𝑙 (c.f. Figure 2) and 𝐹 ∥ = 𝑚𝑔 for a 3 μL water drop on a vertical C18 silanized silicon surface. Since the maximal change of speed at the last stage of the motion is 0.33 – 0.24 = 0.09 mm/s over a period of about 2 s, it corresponds to an acceleration of the order of 0.09/2 = 0.045 mm/s2, i.e. 4.5×10-5 m/s2 which is negligible compared to 9.8 m/s2. Thus, we assume the weight of the drop is the retention force in this case. In this case, 𝑘 values are greater than 1 for most of the first and the second stage for both pulled and pushed drop which is contrary to the theory.33,35–37 If we use the actual width of the drop (which should be less than the length), 𝑘 would be even higher (which makes no sense). But the greater problem is that the values of k go down as the forces increase both for the pulled and the pushed cases. Since k is a geometric correction and since pulling and pushing change the geometries in opposite directions, k cannot account for both. Therefore, the Furmidge equation could not explain our results for two reasons: (1) theoretically predicted k-factor requires k < 1, 33,35–37 but in our results k > 1; and more importantly, (2) the value of k varies in the course of our experimental runs in a way that cannot be justified. We present the Furmidge equation here to emphasize these problems, and turn to a different theory to explain our results. Note that in this paper, lowercase 𝑓 is the actual force generated by CAB and capital 𝐹 is the drop retention force as predicted theoretically (either by the Furmidge or the Tadmor equations).


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Figure 8. Calculated values of a 3 μL water drops sliding on a vertical C18 silanized silicon surface. A) 𝑘 factor that is required to fit Eq. (5), B) Interfacial modulus (𝐺𝑠) that is required to fit Eq. (6). Red for pushed, blue for pulled, and black for 𝑘 = 1.

We mentioned in our discussion of Figure 7 that the solid molecules have less time to reorient themselves at higher speeds. Below, we turn to a theory that considers this interaction. The property that defines the ability of the solid molecules to reorient in response to a contacting liquid is called interfacial modulus and is noted by 𝐺𝑠. Eq. (5) can be modified to incorporate the protrusion26 created by the molecular interaction between the solid and the liquid. Two such modifications are written as38 𝐹∥ =

4𝛾2𝑠𝑖𝑛 𝜃(𝑐𝑜𝑠𝜃𝑅 ― 𝑐𝑜𝑠𝜃𝐴)

(6)

𝐺𝑠

and (Xu et al39) 𝐹∥ =

𝛾𝑤(𝑐𝑜𝑠𝜃𝑅 ― 𝑐𝑜𝑠𝜃𝐴)Δ𝑃

(7)

𝐺𝑠 19 ACS Paragon Plus Environment

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where 𝜃 is the contact angle of the drop resting on the surface with given time, 𝐺𝑠 is the interfacial modulus, which describes the re-orientation of the outer layer of molecules of the solid surface and Δ𝑃 is the Laplace pressure difference between the inside and the outside of the drop near the triple line.

Compared to Eq. (5), Eqs. (6) and (7) can take the effect of the protrusion into account through the interfacial modulus, 𝐺𝑠. Essentially, a solid surface that is either stiffer on a molecular level, or that lacks different functional groups (or has similar functional groups) has a higher interfacial modulus and hence produces a smaller retention force (if all functional groups are identical, there is no ability to pin the drop to the surface). We consider Eq. (7) to be more accurate than Eq. (6), since Eq. (6) assumes that the radius in the Laplace equation is proportional to the drop’s width, while Eq. (7) doesn’t assume that. However, both equations use the interfacial modulus as the main parameter to describe retention forces. Thus, for the sake of simplicity, we will use Eq. (6) rather than Eq. (7) to explain our results. A recent paper 40 developed a method to independently evaluate 𝐺𝑠 using a Surface Force Apparatus (SFA).

In order to obtain 𝐺𝑠 in Eq. (6), we have assumed that the retention force in Eq. (6) is equal to the weight of the drop. This is a reasonable approximation, as shown earlier, since the maximal acceleration is negligible (4.5×10-5 m/s2 𝐺𝑠, 𝑝𝑢𝑠ℎ at the later stages of the run. This further increase the speed of the pulled drop.

The difference between 𝐺𝑠,𝑝𝑢𝑙𝑙/𝐺𝑠,𝑝𝑢𝑠ℎ and 𝐹Furmidge,𝑝𝑢𝑙𝑙/𝐹Furmidge,𝑝𝑢𝑠ℎ arises from the 𝑠𝑖𝑛𝜃 factor that exists in the 𝐺𝑠 prediction but not in the Furmidge prediction. The resultant change in the values of 𝐺𝑠 makes sense and can form a basis to establish an independent prediction to calculate 𝐺𝑠 while the deviation of the Furmidge prediction from the actual force cannot be explained. 23 ACS Paragon Plus Environment

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ASSOCIATED CONTENT Supporting Information More details about CAB, and the plot of angular velocity of the rotation, normal force and lateral force against time. (PDF) A 3 μL pulled water drop sliding on a C18 silanized silicon surface as shown in Figure 4. (MPEG-4) A 3 μL pushed water drop sliding on a C18 silanized silicon surface as shown in Figure 4. (MPEG4) A flying drop as shown in Figure S3. (MPEG-4)

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]

ACKNOWLEDGMENTS This study was supported by NSF grants CMMI-1405109 and CBET-1428398 and CBET0960229 and supported in part at the Technion by a fellowship from the Lady Davis Foundation.

REFERENCES (1)

Young, T. An Essay on the Cohesion of Fluids. Philos. Trans. R. Soc. London 1805, 95, 65–87.

(2)

Israelachvili, J. N. Intermolecular and Surface Forces, 3rd ed.; Academic Press: London, 2011.


Page 24 of 29

Page 25 of 29 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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(3)

Bormashenko, E. Y. Wetting of Real Surfaces; De Gruyter: Berlin, 2013.

(4)

Tadmor, R. Line Energy and the Relation between Advancing, Receding, and Young Contact Angles. Langmuir 2004, 20 (18), 7659–7664.

(5)

Butt, H.-J.; Kappl, M. Surface and Interfacial Forces; Wiley-VCH Verlag GmbH & Co. KGaA: Weinheim, Germany, 2010.

(6)

Calvert, P. Inkjet Printing for Materials and Devices. Chem. Mater. 2001, 13 (10), 3299– 3305.

(7)

Teh, S.-Y.; Lin, R.; Hung, L.-H.; Lee, A. P. Droplet Microfluidics. Lab Chip 2008, 8 (2), 198–220.

(8)

Zhao, F.; Zhao, C.; Bo, H. Droplet Phase Change Model and Its Application in WaveType Vanes of Steam Generator. Ann. Nucl. Energy 2018, 111, 176–187.

(9)

Huang, C.-Y.; Lai, M.-F.; Liu, W.-L.; Wei, Z.-H. Anisotropic Wettability of Biomimetic Micro/Nano Dual-Scale Inclined Cones Fabricated by Ferrofluid-Molding Method. Adv. Funct. Mater. 2015, 25 (18), 2670–2676.

(10)

Sommers, A. D.; Ying, J.; Eid, K. F. Predicting the Onset of Condensate Droplet Departure from a Vertical Surface Due to Air Flow-Applications to TopographicallyModified, Micro-Grooved Surfaces. Exp. Therm. Fluid Sci. 2012, 40, 38–49.

(11)

Pilat, D. W.; Papadopoulos, P.; Schäffel, D.; Vollmer, D.; Berger, R.; Butt, H. J. Dynamic Measurement of the Force Required to Move a Liquid Drop on a Solid Surface. Langmuir 2012, 28 (49), 16812–16820.

(12)

Gao, N.; Geyer, F.; Pilat, D. W.; Wooh, S.; Vollmer, D.; Butt, H. J.; Berger, R. How Drops Start Sliding over Solid Surfaces. Nat. Phys. 2018, 14 (2), 191–196.

(13)

Lander, L. M.; Siewierski, L. M.; Brittain, W. J.; Vogler, E. A. A Systematic Comparison


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

of Contact Angle Methods. Langmuir 1993, 9 (8), 2237–2239. (14)

Tadmor, R.; Bahadur, P.; Leh, A.; N’Guessan, H. E.; Jaini, R.; Dang, L. Measurement of Lateral Adhesion Forces at the Interface between a Liquid Drop and a Substrate. Phys. Rev. Lett. 2009, 103 (26), 266101.

(15)

de la Madrid, R.; Whitehead, T.; Irwin, G. M. Comparison of the Lateral Retention Forces on Sessile and Pendant Water Drops on a Solid Surface. Am. J. Phys. 2015, 83 (6), 531– 538.

(16)

Einati, H.; Mottel, A.; Inberg, A.; Shacham-Diamand, Y. Electrochemical Studies of SelfAssembled Monolayers Using Impedance Spectroscopy. Electrochim. Acta 2009, 54 (25), 6063–6069.

(17)

Wasnik, P. S.; N’guessan, H. E.; Tadmor, R. Controlling Arbitrary Humidity without Convection. J. Colloid Interface Sci. 2015, 455, 212–219.

(18)

Tadmor, R.; Yadav, P. S. As-Placed Contact Angles for Sessile Drops. J. Colloid Interface Sci. 2008, 317, 241–246.

(19)

N’Guessan, H. E.; Leh, A.; Cox, P.; Bahadur, P.; Tadmor, R.; Patra, P.; Vajtai, R.; Ajayan, P. M.; Wasnik, P. Water Tribology on Graphene. Nat. Commun. 2012, 3, 1242.

(20)

Tadmor, R.; Chaurasia, K.; Yadav, P. S.; Leh, A.; Bahadur, P.; Dang, L.; Hoffer, W. R. Drop Retention Force as a Function of Resting Time. Langmuir 2008, 24 (17), 9370– 9374.

(21)

Rapp, B. E. Fluids. In Microfluidics: Modelling, Mechanics and Mathematics; Elsevier, 2017; pp 243–263.

(22)

Chatterjee, J. Critical Eotvos Numbers for Buoyancy-Induced Oil Drop Detachment Based on Shape Analysis. Adv. Colloid Interface Sci. 2002, 98 (3), 265–283.


Page 26 of 29

Page 27 of 29 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

(23)

Tadmor, R.; Das, R.; Gulec, S.; Liu, J.; E. N’guessan, H.; Shah, M.; S Wasnik, P.; Yadav, S. B. Solid-Liquid Work of Adhesion. Langmuir 2017, 33 (15), 3594–3600.

(24)

Chow, T. S. Wetting of Rough Surfaces. J. Phys. Condens. Matter 1998, 10 (27), L445– L451.

(25)

Carré, A.; Gastel, J.-C.; Shanahan, M. E. R. Viscoelastic Effects in the Spreading of Liquids. Nature 1996, 379 (6564), 432–434.

(26)

Shanahan, M. E. R.; de Gennes, P. G. Equilibrium of the Triple Line Solid/Liquid/Fluid of a Sessile Drop. In Adhesion 11; Springer Netherlands: Dordrecht, 1987; Vol. 315, pp 71– 81.

(27)

Yasuda, T.; Miyama, M.; Yasuda, H. Dynamics of the Surface Configuration Change of Polymers in Response to Changes in Environmental Conditions. 2. Comparison of Changes in Air and in Liquid Water. Langmuir 1992, 8 (5), 1425–1430.

(28)

Sheiko, S. S.; Sun, F. C.; Randall, A.; Shirvanyants, D.; Rubinstein, M.; Lee, H. Il; Matyjaszewski, K. Adsorption-Induced Scission of Carbon-Carbon Bonds. Nature 2006, 440, 191–194.

(29)

Gupta, P.; Loos, K.; Korniakov, A.; Spagnoli, C.; Cowman, M.; Ulman, A. Facile Route to Ultraflat SAM-Protected Gold Surfaces by “Amphiphile Splitting.” Angew. Chemie Int. Ed. 2004, 43 (4), 520–523.

(30)

Bormashenko, E.; Chaniel, G.; Gendelman, O. Hydrophilization and Hydrophobic Recovery in Polymers Obtained by Casting of Polymer Solutions on Water Surface. J. Colloid Interface Sci. 2014, 435, 192–197.

(31)

Furmidge, C. G. L. Studies at Phase Interfaces. I. The Sliding of Liquid Drops on Solid Surfaces and a Theory for Spray Retention. J. Colloid Sci. 1962, 17 (4), 309–324.


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(32)

Kawasaki, K. Study of Wettability of Polymers by Sliding of Water Drop. J. Colloid Sci. 1960, 15 (5), 402–407.

(33)

Dussan V., E. B.; Chow, R. T. P. On the Ability of Drops or Bubbles to Stick to Non Horizontal Surfaces of Solids. J. Fluid Mech. 1983, 137, 1–29.

(34)

Dussan V., E. B. On the Ability of Drops or Bubbles to Stick to Non-Horizontal Surfaces of Solids. Part 2. Small Drops or Bubbles Having Contact Angles of Arbitrary Size. J. Fluid Mech. 1985, 151, 1–20.

(35)

Extrand, C. W.; Gent, A. N. Retention of Liquid Drops by Solid Surfaces. J. Colloid Interface Sci. 1990, 138 (2), 431–442.

(36)

Brown, R. A.; Orr, F. M.; Scriven, L. E. Static Drop on an Inclined Plate: Analysis by the Finite Element Method. J. Colloid Interface Sci. 1980, 73 (1), 76–87.

(37)

ElSherbini, A. I.; Jacobi, A. M. Retention Forces and Contact Angles for Critical Liquid Drops on Non-Horizontal Surfaces. J. Colloid Interface Sci. 2006, 299 (2), 841–849.

(38)

Tadmor, R. Approaches in Wetting Phenomena. Soft Matter 2011, 7 (5), 1577–1580.

(39)

Xu, W.; Xu, J.; Li, X.; Tian, Y.; Choi, C.-H.; Yang, E.-H. Lateral Actuation of an Organic Droplet on Conjugated Polymer Electrodes via Imbalanced Interfacial Tensions. Soft Matter 2016, 12 (33), 6902–6909.

(40)

Yadav, P. S.; Gulec, S.; Tadmor, R.; Jena, A.; Tang, S.; Yadav, S.; Katoshevski, D. Interfacial Modulus and Surfactant Coated Surfaces. Surf. Topogr. Metrol. Prop. 2018, 6 (4), 045007.

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Droplets sliding down a vertical surface under increasing horizontal

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