Driving Droplet by Scale Effect on Microstructured Hydrophobic Surfaces

Nov 11, 2012 - contact angle decreases with increasing the scale of the micropillars along the moving direction under disturbance. These new findings ...
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Driving Droplet by Scale Effect on Microstructured Hydrophobic Surfaces Cunjing Lv†,‡ and Pengfei Hao*,† †

Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China Center for Nano and Micro Mechanics, Tsinghua University, Beijing 100084, China

‡

S Supporting Information *

ABSTRACT: A new type of water droplet transportation mechanism on a microstructured hydrophobic surface is proposed and investigated experimentally and theoretically: a water droplet could be driven by scale effect under disturbance and vibration, which is different from the traditional contact angle-gradient-based method. A scale-gradient microstructured hydrophobic surface is fabricated in which the area fraction is kept constant, but the scales of the micropillars are monotonically changed. When additional water or horizontal vibration is applied, the original water droplet could move unidirectionally in the direction from the small scale to the large scale. A new model with line tension energy developed very recently could be used to explain these phenomena. When compared with the traditional contact angle-gradient smooth surface, it is also found that dynamic contact angle decreases with increasing the scale of the micropillars along the moving direction under disturbance. These new findings will deepen our understanding of the relationship between topology and dynamic wetting properties, and could be very helpful in designing liquid droplet transportation devices in microfluidic systems.

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movement.23 For example, the direction and velocity of the droplet motion could be manipulated reversibly by varying the direction and steepness of the gradient in light intensity; the reason for the driving force is an imbalance in contact angles generated on both edges of a droplet.9 When a surface tension gradient is designed into the substrate surface by a chemical method, the random movements of the droplets were biased toward the more wettable side of the surface.22 Driven by nonequilibrium noise, periodic motion of droplet was produced along the glass substrates over several tens of seconds.3 Relationships between movement of liquid droplets and the shape, frequency, and amplitude of the vibration were also systematically studied.11−13 In Daniel’s experiments,18 powered by the energies of coalescence and by the forces of the chemical gradient on a radial surface tension gradient, small drops display speeds that are hundreds to thousands of times faster than those of typical Marangoni flows.24 Except for liquid droplet transportation on contact angle-gradient-based smooth surfaces, researchers have been made significant progress to realize

INTRODUCTION The spontaneous motion of liquid on a solid surface has attracted great interest in recent years. For example, microfluidics is becoming a hot field and has obtained wide applications, and researchers have developed various methods to realize liquid self-transportatoin.1−10 So understanding how the topology of substrates influences the dynamic behaviors of droplets is essential to clarify the underlying mechanism and practical application. The main obstacle to droplet motion on a solid surface arises from the hysteresis of contact angles that pins the droplet edge. In order to surmount hysteresis and drive droplet motion, additional energy should be supplied to the droplet in order to produce net force. However, the surface energy of the droplet cannot be converted to mechanical energy spontaneously, so a special method should be used, for example, wetting gradient surfaces were prepared by mechanical vibration,11−15 electrical,16,17 chemical,3,6 thermal,7,8,18−20 electrochemical,2,10 and photochemical methods,1,9 and so on. The possibility of droplet movement due to a surface tension gradient was first predicted by Greenspan21 and experimentally demonstrated by Chaudhury and Whitesides.22 Since then, researchers have developed various methods to realize droplet © 2012 American Chemical Society

Received: October 14, 2012 Revised: November 7, 2012 Published: November 11, 2012 16958

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spontaneous droplet transportation on rough surface. On asymmetric textures, a droplet could be transported unidirectionally under electric field and vibration.14−17 Surprisingly, Lagubeau et al. extended the famous Leidenfrost effect to solids very recently.8 Moreover, liquid droplets could even move uphill on elaborately designed shape-gradient composite surfaces, and by using a chemically patterned surface in combination with nanotexture, the current record for spontaneous droplet motion under ambient condition is about 0.5 m/s.25,26 Today, the combination of roughness gradient-designed substrate and surface tension effect has been applied in micropumps for various applications.4,5,27,28 To some extent, in most of the above studies, droplet transportation was realized by means of contact angle-gradient. Here, different from the previous method,3,7−9,11,14,23,25 we propose a new way to drive water droplets by scale effect on hydrophobic micropillar-like substrates. We first designed a group of micropillar-like substrates in which the area fractions were kept constant, but the scales of the micropillars decreased from one side to the other side. When additional water or horizontal vibration was exerted on the original droplet, the droplet could move from the region with small-scale micropillars to the region with large-scale micropillars. To the authors' best knowledge, realizing water droplet transportation depending on scale effect has never been systematically studied. Furthermore, this interesting phenomenon could not be understood just by the traditional Cassie−Baxter model,29 because it is now well-known that the traditional Cassie−Baxter model is valid only for uniformly rough surfaces.30,31 Our research demonstrates that scale is very important and should not be ignored, especially in small scale. The model including the very recently developed line tension models32−34 can not only be applied to explain these phenomena, but also gives new insights to the wetting theory.

Figure 1. Pillar-like microstructural surfaces: (a) top schematic view of the square pillars; (b) side schematic view of the square pillars.

were designed with the width and length W = 8 mm and l = 2 mm, respectively. All of the micropillars had square shape with height H ≈ 30 μm, and the area frictions were designed at f = a2/(a + b)2 = 0.16, where a is the side length of the square pillar, b is the spacing between the neighboring pillars. Widths ai (i = 1,...,7) of the micropillars were 2, 4, 6, 10, 20, 30, and 40 μm, respectively. Then the seven substrates were arranged in order of their sizes (Figure 2) and fabricated in a silicon wafer. By this method, we produced a scale-gradient substrate. Here, we use S = A/L as a shape-dependent roughness scale,33 given by the boundary length L and area A of the pillar cross sections. For the above square-shaped micropillars, Si = ai/4. Experiment 1: Droplet Motion Realized by Water Coalescence. Previously, Daniel reported that coalescence could induce droplet motion from the condensation of steam on a contact angle-gradient smooth surface.12,18 However, to the best of our knowledge, spontaneous motion induced by coalescence on a size-gradient superhydrophobic surface under atmospheric conditions for a large droplet has never been reported. In the first experiment, we will show droplet motion driven by adding water into the original droplet. As shown in Figure 3, in the beginning, a 5 μL water droplet was first injected from the pinhead, and the substrate was adjusted horizontally in order to ensure the droplet gravity center stand at the middlemost bottom of the pinhead after it was produced. Then, we fixed the substrate, and added 5 μL of additional water into the original droplet one time at the rate of 0.5 μL/s. When the 5 μL of additional water was discharged from the pinhead, there was a short break in order to ensure that the droplet arrived at an equilibrium state. Then, 5 μL of additional water was added at the same rate, and the cycles continued until the droplet became big enough. All the processes were recorded by a high-speed camera. Surprisingly, we can see clearly from Figure 3 that the water droplet could always move in the large-scale direction. When the volume of the total droplet is small, this moving behavior did not seem obvious, but when the volume of the droplet was larger than 20 μL, the unidirectional motion of the droplet seemed remarkable. In the above experiment, when the droplet deviated from the middlemost bottom of the pinhead, outflow of additional water might produce hydraulic force, and the original droplet might be pushed to move under its horizontal component. In order to make sure that the motion was caused by the scale effect of the micropillar-like substrate, and not by the horizontal component of the hydraulic force, we conducted further experiments below. When the gravity center of the water droplet moved on the right of the pinhead (Figure 4d), we moved the substrate horizontally gently enough to make sure that the gravity center of the water droplet stands on the left of the pinhead (see Figure 4e). Then, additional water was added on the droplet. In this case, if hydraulic force existed, the direction of its horizontal component would be in the opposite direction to

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DRIVING DROPLET BY SCALE EFFECT  EXPERIMENTAL OBSERVATION Sample Preparation. The micropillar-like substrates were fabricated by standard photolithography and inductively coupled plasma (ICP) etching techniques, and then a selfassembled monolayer (SAM) of octadecyltrichlorosilane (OTS) of formula C18H37Cl3Si (Acros Organics) was adopted to realize superhydrophobicity by a standard procedure.35 After the chemical modification, the apparent contact angle on flat surfaces was 105 ± 1°. The apparent static and dynamic contact angles were measured with a commercial contact angle meter (OCAH200, Dataphysics, Germany). The images of the water droplets on the microstructured substrates were observed and recorded by a high-speed charge-coupled device (CCD; 400 fps), and sequential photographs of the wetting behaviors of the water droplets were taken every 2.5 ms. We use a homemade oscillator in which the frequency ranges from 0 to 200 Hz, and the amplitudes were measured directly from the experiments. Different from Shastry’s14 and Reyssat’s15 work in which the solid−liquid contact area fraction was introduced as a control variable to manipulate the droplet, the feature in our experiment is that a group of micropillar-like substrates were fabricated in which the area fractions were kept constant, but the sizes of the pillar width were decreased from one side to the other side. Figure 2 shows the detailed information of the gradient substrate. First, seven independent small substrates 16959

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Figure 2. Designed parameters of the gradient micropillar-like substrate: (a) SEM images of the gradient substrates, the scale bar is 10 μm; (b) schematic of the gradient substrates with increasing scales in the x direction. The total width and the length of the substrate is W = 8 mm, L = 14 mm (not to scale), respectively. The gradient substrate is composed of seven substrates; the width of each single substrate is l = 2 mm.

the motion of the droplet, and should play a role in resistance. However, in the process of dropping, the droplet always moved in the large-scale direction (see Movie 1, Supporting Information). So, we can exclude that the “driving force” pushing the droplet motion originated from hydraulic force. Another interesting phenomenon was that the left and the right apparent contact angles, θL and θR, on the rear and front of the contact boundary was different from each other not only in the process of the droplet motion, but also in the break of dropping. In order to give detailed information, we plot the relationship between time t and the values of θL and θR (Figure 5). From Figure 5 we can see θL and θR exhibiting distinguished features in the process of droplet motion: (i) when the original droplet was disturbed by adding water, fluctuation of θL and θR were around 20°; (ii) for certain time, θL was always larger than θR, which means that the right contact boundary of the droplet (in the forward direction) was always more wettable than the left contact boundary; (iii) during the 15th second to the 23rd second, both θL and θR increased with increasing of the droplet volume, which means gravity probably influenced the apparent contact angle; (iv) during the 23rd second to the 43rd second, both θL and θR decreased with time, which means that, along the moving direction, the scale of the micropillar was larger, and the value of the apparent contact angle is lower. Experiment 2: Droplet Motion Realized by Vibration. In the second experiment, we will show droplet motion driven by mechanical vibration. The substrate was first adhered on a solid platform, and a droplet was released (see Figure 6). We then produced a constant sine horizontal vibration on the platform, and kept the vibration direction parallel to the scale-

Figure 3. Droplet profiles at different positions with increased volume. The positions of the pinhead and the substrate were always fixed.

Figure 4. Droplet profiles at different positions with sustained increase in volume: (a−d) the gravity centers of the water droplet were always on the right of the pinhead; (e) the substrate was moved horizontally to let the gravity center of the water droplet stand on the left of the pinhead. The position of the pinhead was always fixed (see Movie 1, Supporting Information).

Figure 5. Apparent dynamic contact angles θL and θR in the process of droplet motion and in the break of dropping. The red squares represent θL; the black circles represent θR. 16960

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Figure 6. Droplet vibration experiment: (a) sketch of the experimental setup; (b) droplet profiles at different positions with constant volume (20 μL) and constant vibration (the frequency and the amplitude of the oscillator were 80 Hz and 0.75 mm, respectively). The droplet moved in the small-scale to large-scale direction. The time bars are shown in the process of the vibration (see Movie 2, Supporting Information).

Figure 7. Apparent dynamic contact angles θL and θR in the process of the droplet motion. The red squares represent θL; the black circles represent θR.

gradient direction. Interestingly, when the volume of the droplet was around 20 μL, and the frequency and the amplitude of the oscillator were around 80 Hz and 0.75 mm, the droplet started to move on the same scale-gradient substrate from the small-scale to the large-scale. We tried again many times, and this phenomenon could always happen. In order to give detailed information, we also measured the dynamic contact angles θL and θR varying with time (see Figure 7). Different form Figure 5, θL and θR in Figure 7 exhibit distinguished features: (i) when the droplet was disturbed by vibration, the amplitude of the apparent contact angles was larger than in Figure 5, and the fluctuation of θL and θR was around 35°; (ii) the maximum/minimum values of θL at different times is the same as or a little larger than the maximum/minimum values of θR; (iii) in the process of the droplet motion, the trend of the maximum/minimum values of θL and θR at different times decreased with the increasing value of the scale of the micropillars, but was not so obvious.

cos θC = − 1 + f (cos θY + 1)

(2)

According to eq 2, we can see that the apparent contact angle θC on a rough substrate in the fakir state is only relative with θY and the area fraction f. In other words, once the material systems (γSV, γSL, and γLV) are given, the apparent contact angle θC is only determined by f. On the contrary, from the above two experiments, the wetting characteristic of the designed scale-gradient substrates was not kept constant when the droplets were subjected to small disturbance or vibration even though we fixed f, which is completely different from the mechanism14,15 in which the driven droplet depends on the variable area fractions directly. Even though asymmetrically structured geometries (sawtooth periodic structures) were used to realize droplet transportation successfully driven by electrical field,16,17 vibration,16,17 Leidenfrost effect,8 and so on, none of them include the same mechanism as revealed in our experiments. Actually, the Cassie−Baxter equation was deduced from an equivalent energy form on rough surfaces, such as γ ̅ − γSL̅ cos θC = SV γLV (3)

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THEORETICAL ANALYSIS AND DISCUSSION As we know, the contact angle θY reflects wetting characteristic of a smooth surface and is given by the well-know Young equation:36 γ − γSL cos θY = SV γLV (1)

where γS̅ V = fγSV and γS̅ L = (1 − f)γLV + fγSL are the equivalent solid−vapor surface tension and solid−liquid surface tension on rough surfaces, respectively. Putting γS̅ V and γS̅ L into eq 3, we can easily get eq 2. According to Cassie−Baxter’s idea, for the scale-gradient substrate in our experiment, the equivalent surface tension γS̅ V and γS̅ L of each small substrate (Figure 2) is the same as each other. While this is not the case, there must be other mechanisms that control the wetting behaviors of a water droplet on a rough substrate. Where does the driven force induced by the scale-effect come from? Very recently, Zheng32 and Wong33 recognized the importance of the three-phase contact line tension on the

where γSV, γSL, and γLV are the surface tension coefficients on solid−vapor, solid−liquid, and liquid−vapor interfaces, respectively.37 When the droplet stands on a hydrophobic rough substrate and is in the fakir state,38 the contact areas are composed of water−solid interface and water−vapor interface. The wetting characteristic of such a surface was first addressed by Cassie and Baxter,29 and the apparent contact angle θC was predicted by 16961

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parameters. Shiu45 observed experimentally that the contact angle of a water droplet increased on various size-reduced polystyrene surfaces. We should emphasize that in Yang’s work,46 the contact angle increased with decreasing silica particle sizes on modified silica-coated paper, but, interestingly, the value of contact angle decreased when the particle size decreased further (Figure 9 in ref 46). Recently, Yang47 studied the wetting behaviors of picoliter water droplets on surfaces with grooves of different widths; he not only obtained −4 × 10−7J/m and 3 × 10−5J/m contact line tension on hydrophilic and hydrophobic surfaces, but also observed that that the contact angle increased with increasing droplet volume and decreasing groove width (see Figure 4 in ref 47). Chen et al. revealed that the line tension at the triple line becomes important and is not negligible in the energy analysis, and a line tension value on the order of 105 J/m was calculated in their experiments on microscale pillar-structured hydrophobic surfaces during droplet evaporation.48 Here, we want to emphasize that later Raspal’s work34 further enhanced the idea that nanoporous surfaces may allow the effect on line tension to be visible for micro- to macrodroplets, and that the line-tension model has a physical foundation to solve the contact-angle problem. All the above studies imply the important role of the scale effect for wetting, and this effect is exactly represented in our experiments. On the basis of the above understanding, our experiments and eq 4 can be used to explain why wetting characteristics could be varied with the topology of the micropillar substrates when the area fractions are kept constant.41,43−46 The influence of the line tension is very important for the static and dynamic wetting behaviors of a water droplet, and should be taken into consideration for studying the wetting property of a rough substrate, especially with small-scale structures. Next, we will estimate the value of the driving force on scalegradient micropillar structures. On the basis of eq 3 and 4, the scale-effect will induce

liquid−vapor−solid phase boundary, and they constructed a new model to predict the apparent contact angle of water droplet on a small-scale rough substrate, independently. On the basis of their idea, the equivalent solid−liquid surface tension γ̃SL including the line tension term can be expressed as γ̃SL = (1 − f)γLV + fγSL + fτ/S, where τ is the line tension, and S = a/4 for a square-shaped pillar, which reflects the influence of scale effect on the wetting properties of the rough substrate. Putting γS̅ V and γ̃SL into eq 3, we can get Zheng and Wong’s model:32,33 cos θ* = − 1 + f (cos θY + 1) −

fτ SγLV

(4)

On the basis of eq 4, it is not difficult to understand why a water droplet could move from the small-scale to the large-scale under disturbance and vibration. If we keep area fraction f constant, S is smaller, the length of the contact line in units of the contact area is longer, so the contribution of the line tension energy is larger, which means the small-scale region is physically more hydrophobic. Even though eq 4 is applied quantitatively to predict the apparent contact angle in ref 32, it is still qualitative to compare dynamic apparent contact angles for substrates with different S; this conclusion could be validated in Figures 5 and 7. In Figure 5, in the process of droplet motion under disturbance of adding water, θL is almost always larger than θR. After the 23rd second, both θL and θR decrease obviously, which means they decrease with increasing the scale of the micropillars. Additionally, in this time the contact area is large enough to occupy substrates with several different S. Before the 23rd second, this phenomenon is not distinguished because, on one hand, the contact area is not large enough to occupy different S; on the other hand, the volume of the droplet is too small to withstand the disturbance, so the influence of gravity may be larger than the scale effect. In Figure 7, the contact area of the 20 μL water droplet is large enough to occupy two different S or more during vibration. For the minimum value of apparent contact angle at a certain time, θL is almost always larger than θR. As we know, a lot of previous studies believed that receding contact angle and receding line control the dynamic behaviors of wetting.35,39,40 In our experiment, the minimum value of θL and θR can be treated as receding contact angles of the rear and the front contact lines, so the scale-effect is validated further. What’s more, the maximum values of θL and θR at a certain time can be treated as advancing contact angles of the rear and the front contact lines. According to some previous studies,41 advancing contact angle could be very large and have no relationship with the geometrical parameters of the substrate. In Figure 7, most θL and θR are larger than 165°, which are too close to each other to distinguish their difference. Our result for the advancing contact angles are consistent with Dorrer’s41 result. Moreover, a lot of researchers recognized the importance of the topology of the microstructure and the three-phase line tension on the wetting properties of the droplet. Chen42 believed that the topology of the roughness is important for the wetting characteristics. In Ö ner’s experiment,43 the advancing contact angle increased with decreasing width of the micropillars when the area fractions were kept constant (Table 1 in ref 43). In Dorrer’s work,41 and Zhang’s44 work, the receding contact angles were increased with decreasing scale of the micropillars when area fractions were kept constant (Figure 8a in ref 41 and Figure 3a in ref 44), but the advancing contact angle seemed have no relationship with the geometrical

fτ ∂S ∂ (cos θ*) = 2 ∂x S γLV ∂x

(5)

If we ignore the influence of gravity, the liquid−vapor interface of the water droplet will be a spherical surface. The total surface energy could be written as E = γLV(ALV − ASL cos θ*) =

3VγLV (6)

R

where ALV, ASL, V, and R are the areas of the liquid−vapor interface on the spherical surface and the area inside the boundary of the contact line, and the volume and radius of the spherical surface, respectively. After some calculations, it's not difficult to obtain the driving force caused by the scale effect along the x-direction (Figure 2) Fx = −

∂E ∂x

= (3V )2/3 π 1/3

fτ 2

(1 + cos θ*) 1/3

S (1 − cos θ*)

2/3

(2 + cos θ*)

∂S ∂x (7)

Actually, in our experiment, S is not a continuous function of x; we define ∂Sij/∂x = (Sj − Si)/l and Sij = (Si + Sj)/2 (i, j = 1,...,7), and here we can estimate the scale-effect caused by S and ∂S/ ∂x, for example, the upper and lower limits of Fx. Herein, S12 = 0.75 μm, ∂S12/∂x = 0.25 × 10−3, θ12 * ≈ 160° and S67 = 8.75 μm, 16962

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∂S67/∂x = 1.25 × 10−3, θ*67 ≈ 150° (see Figure 2 and ref 32), and we can give the upper limit driving force F12 = 1.17 × 10−6N and the lower limit driving force F67 = 9.21 × 10−8N. If we let Fxunit = Fx/(2R sin θ*) (here, 2R sin θ* is the width of the solid−liquid contact area), we can get the driving force per unit length: F12unit = 0.001 N/m and F67unit = 5.45 × 10−5N/m, which is too small compared with the liquid−vapor surface tension γLV = 0.073 N/m; that is the reason why we should depend on disturbance or vibration to help the water droplet overcome the drag force. When the droplet is moving at a steady state, slippage of the liquid near the contact line will generate viscous drag force Fd12,49,50 Fd = 3ηπRvx

∫y

ymax

min

dy = 3ηπRvxld ξ(y)

Figure 8. Relationship between the position and the time; PL (red squares) and PR (black circles) are the real-time locations of the left and right contact boundaries of the droplet, and PO (black line) is a reference point on the vibrated substrate. The data is partially enlarged in the lower right inset.

(8)

where η is the viscosity of the liquid, vx is the droplet velocity along the x-direction, ξ(y) is the thickness of the droplet in its height direction, ymax and ymin are two cutoff lengths, and ld is a logarithmic factor including the cutoffs in the integration, which can often be treated as a constant.51 Another drag force comes from the contact angle hysteresis, such as θL = θ* ± ΔθL/2 and θR = θ* ± ΔθR/2 (see Figure 7); ΔθL and ΔθR represent the contact angle hysteresis in the two contact boundary lines of the vibrated droplet. Daniel11 studied rectified motion of a liquid droplet on contact angle-gradient smooth surface induced by vibration. The net force acting on the droplet during the deceleration phase of the droplet is zero, but the expression of the net force induced by vibration during the acceleration phase is Fh = 2πR2γLV(d cos θ*/dx), actually, Fx = Fh in our experiment in Figure 6. So, when we combine eqs 7 and 8, we can get a steady velocity driven by scale-effect vx =

V1/3 (1 + cos θ*)(1 − cos θ*)1/3 1 · · ηld 3 π (2 + cos θ*)1/3 ⎛ fτ ∂S ⎞ ⎜ 2· ⎟ ⎝ S ∂x ⎠

droplet would move spontaneously with higher velocity and longer distance, which seems amazing. Interestingly, from Figure 8, we can see that the oscillation amplitudes of PL and PR are much smaller than PO when compared with a contact angle-gradient smooth surface,15,54,55 and also, the phase of PL and PR is not consistent with PO. These observed phenomena mean that large slippage happens between the contact area of the droplet and the superhydrophobic substrate; this conclusion could be verified by the fact that we can observe a layer of air cushion between the droplet and the substrate (see Figure 6b, and Movie 2, Supporting Information), in this case, scale-effect could drive the droplet to move easily. Moreover, different from Reyssat’s experiments15 in which the displacement mainly takes place in the dewetting state (the rear of the drop moves significantly, but the front remains nearly fixed), in our experiment (Figures 6−8), both the rear and the front contact lines move significantly. Dynamic wetting behaviors of the contact line on superhydrophobic substrates should be the focus of further study. Even though droplet transportation driven by the scale-effect could be explained using the recently developed line tension models,32−34,48 in reality, the droplet may also obtain energy from the disturbance and lose it due to dissipation, so, a more comprehensive model needs to be developed in future.

2/3 1/3

(9)

We will give a rough estimation about vx theoretically. As shown in Figure 2, during vibration (Figure 5), the droplet occupied only five pieces of substrates on its path of movement (S2 to S6); let the average value of S and ∂S/∂x be equal to S̅ = ∑6i=2Si/5 = 3.5 μm and ∂S̅/∂x = (∑5,6 i=2,j = 3∂Sij/∂x)/4 = 0.8125 × 10−3, respectively. At room temperature, η = 0.001 N·s/m−2 for water. Let V = 20 μL and θ̅* ≈ (θ12 * + θ67 * )/2 = 155°. The value of line tension in eq 4 was determined to be τ = 1.57 × 10−8 J/ m in ref 32. Hoffman showed that the logarithmic factor ld is on the order of 1552 for a liquid spreading on a dry surface, but it is reduced to a value of about 5 if the solid is prewetted by a micrometric film.53 We can further get vx = 1.1 mm/s and vx = 3.3 mm/s for ld = 15 and ld = 5, respectively. However, for a droplet on a superhydrophobic surface, the contact boundary lines are not continuous, so the viscous drag would be even smaller; maybe we can expect a small value of ld, for example, when ld = 2.1, we get vx = 8.0 mm/s, which is consistent with our experimental results dealing with the least-squares method in Figure 8. Equation 9 revealed that the scale-effect not only comes from S, but also comes from ∂S/∂x. If a similar scalegradient surface with much smaller micropillars is fabricated (for example, nanoscale) and larger ∂S/∂x is designed, the moving phenomenon should be more significant, and the

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CONCLUDING REMARKS In this paper, we revealed that the “scale-effect” could be considered as a “driving force” to realize water droplet transportation on a hydrophobic substrate with scale-gradient microstructures. When vibration or additional water was applied on the original droplet as a disturbance, the droplet always moved from the small-scale to the large-scale region in order to decrease its total surface energy. However, the traditional Cassie−Baxter model does not take into consideration the influences of topology and scale-effect on the wetting properties of rough surface, so it could not be applied directly to predict apparent contact angle and droplet transportation behaviors in our experiments. Different from the previous understanding, we revealed that the line tension could be considered as the mechanism and should not be ignored, especially in small scale. With the recently developed model32−34 including line tension on the solid−liquid contact 16963

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boundary, we can estimate the droplet transportation velocity very well. What’s more, our discovery may give new qualitative explanations as to why so many plants and insects that possess excellent hydrophobic/superhydrophobic properties have small-scale micronano structures:56,57 small-scale could help them to keep away from wetting when external perturbation happens. Our work could be a guide to design optimal microstructures to realize excellent superhydrophobic properties and special functions in the microfluidics field for practical applications.

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ASSOCIATED CONTENT

S Supporting Information *

Movie 1: When additional water is adding into the original droplet, it always moves from the small scale to the large scale on the substrate no matter the relative position between the pinhead and the gravity center of the water droplet. The area fraction of this scale-gradient substrate is kept constant at f = 0.16. Movie 2: The droplet moves on the scale-gradient substrate in which the area fraction is kept constant at f = 0.16, but the scales of the micropillars increases from the left side to the right side. When steady vibration is produced, the droplet moves from the small scale to the large scale. The volume of the water droplet, the recording speed of the high-speed CCD, and the frequency and the amplitude of the oscillator are 20 μL, 400fps, 80 Hz, and 0.75 mm, respectively. This information is available free of charge via the Internet at http://pubs.acs.org/.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from the NSFC under Grant Nos.10672089, 10872114, 10832005, and 11072126 is gratefully acknowledged.

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REFERENCES

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