Friction of Droplets Sliding on Microstructured Superhydrophobic

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Friction of droplets sliding on microstructured superhydrophobic surfaces Shasha Qiao, Shen Li, Qunyang Li, Bo Li, Kesong Liu, and Xi-Qiao Feng Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b03087 • Publication Date (Web): 02 Nov 2017 Downloaded from http://pubs.acs.org on November 3, 2017

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Friction of droplets sliding on microstructured superhydrophobic surfaces 1

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1,2

1

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1

Shasha Qiao , Shen Li , Qunyang Li *, Bo Li , Kesong Liu , Xi-Qiao Feng * 1

AML, CNMM and Department of Engineering Mechanics, Tsinghua University, Beijing

100084, China 2

State Key Laboratory of Tribology, Tsinghua University, Beijing 100084, China

3

Key Laboratory of Bio-Inspired Smart Interfacial Science and Technology of Ministry of

Education, School of Chemistry and Environment, Beihang University, Beijing 100191, China

* To whom correspondence should be addressed. Email: [email protected] or [email protected]

Keywords: superhydrophobic surfaces, liquid-solid interface, friction, hysteresis, retention force

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ABSTRACT:

Liquid transport is a fundamental process relevant to a wide range of applications, e.g. heat transfer, anti-icing, self-cleaning, drag reduction, and microfluidic systems. For these applications, a deeper understanding of the sliding behavior of water droplets on solid surfaces is of particular importance. In this study, the frictional behavior of water droplets sliding on superhydrophobic surfaces decorated with micro-pillar arrays was studied using a nano-tribometer. Our experiments show that surfaces with a higher solid area fraction generally exhibited larger friction, although friction might drop when the solid area fraction was close to unity. More interestingly, we found that the sliding friction of droplets was enhanced when the dimension of the microstructures increased, showing a distinct size effect. The nonmonotonic dependence of friction force on solid area fraction and the apparent size effect can be qualitatively explained by the evolution of two governing factors, i.e. the true length of the contact line and the coordination degree of the depinning events. The mechanisms are expected to be generally applicable for other liquid transport processes involving dynamic motion of three-phase contact line, which may provide a new means of regularing liquid transfer behavior through surface microstructures.

INTRODUCTION

As a common phenomenon in our daily life, liquid transport is an essential process associated with many biological and industrial applications, such as water collection and drag reduction1-3. The ability to control liquid transport on superhydrophobic surfaces has attracted much attention in the society and many novel materials and structures that


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possess unique surface functions, e.g. self-cleaning, anti-icing, heat transfer, oil-water separation and microfluidic systems, have been proposed1-2,

4-11

. It has long been

recognized that interfacial properties of solids, e.g. wetting and adhesion, sensitively depend on the micro- and nano-structures on their surfaces. Recently, emerging evidences show that the morphology of solid surfaces can also significantly affect their liquid transport behavior

5, 12

. For example, although both of lotus leaves and rose petals

have high static contact angles, water droplets slide easily on one surface but adhere firmly to the other because of the drastically different surface microstructures 5. To better understand the liquid transfer behavior and design surfaces with desired liquid transfer properties, a systematical study on how surface microstructures affect the sliding behavior of water droplets, especially the lateral resistance force, is of fundamental importance. The lateral force required to overcome the interfacial shear resistance between a liquid droplet and a solid surface is commonly referred to as friction force or retention force13. The sliding behavior of droplets on solid surfaces and the contact line friction force has been previously explored via various experimental setups, e.g. the inclined plane method14-19, centrifugation method20-22 and direct dragging method13, 23-27. Considering the competition between the work of wetting and the work done by gravity, Furmidge26 proposed a model to predict the retention force for a droplet sitting on an inclined plane. In this model, the liquid-solid contact area of the moving drop is assumed to be nearly rectangular and the retention force F can be calculated from surface tension, advancing and receding contact angles, and width of the contact region. Later, more continuum models have been proposed to describe the retention force by adopting different


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assumptions of contact line geometry and contact angle distribution along the contact line 17-19, 22, 29-34

. Despite the refined description of the retention force, these continuum

theories typically do not explicitly consider the effect of surface microstructures. To include the influence of surface morphology, Lv et al. 15 developed an analytical model to predict the sliding angle of droplets on microstructured surfaces in terms of solid surface fraction, droplet volume, and static Young’s contact angle. In their model, the effect of microstructure was incorporated by considering the real differential contact area when the droplet is depinned, although the shape and motion of the contact line were still assumed continuous and smooth. Unlike the traditional inclined plane or centrifugation methods, Pilat et al13 used a drop adhesion force instrument to measure the real-time lateral resistance force when droplets slid over solid surfaces with micro-pillar arrays. They experimentally showed that the real-time lateral force actually fluctuated locally, in a fashion of stick-slip, with a spatial variation of the micro-pillar array. This behavior was attributed to the individual dipinning events where multiple pillars simultaneously detachted from the contact line along the receding edge. The experimental results suggest that the sliding friction of droplets on solid surfaces indeed relies critically on the dynamic interaction between the contact line and the surface microstructures. In addition to sliding droplets, the motion of three-phase contact line also occurs for droplets evaporating in air or pulled off from solid surfaces. The interplay between the contact line configuration and the surface microstructures seems to be critical for understanding the dynamics of the systems. For an evaporating droplet on micro-pillared surfaces, Xu et al.35 found that the normalized maximal length of the three-phase contact line was the direct factor determining the depinning force. However, the study neglected


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the potential effect of microstructures (e.g, solid area fraction and solid pillar size) on the apparent contact area. More intertestingly, recent experimental observations also suggest that the pinning force and the contact line dynamics also depend on the size of the microstructures

36-37

. Li et al. 37 measured the tensile force when retracting droplets from

solid surfaces with micro-pillars. They found that the maximum pull-off force was enhanced when the size of the micro-pillar increased, which was attributed to the normal force from the local Laplace pressure on the top of the micro-pillars. Since laterally sliding a droplet on solid surfaces also involves movement of local contact line, the resultant friction force may be similarly affected by the configuration of the contact line and size of the microstructures. To systematically investigate the dynamic behavior of the moving three-phase contact line when droplets slide on textured surfaces, we first fabricated a series of superhydrophobic surfaces decorated with micro-pillars of various sizes and area density. The real-time variation of the lateral resistance force experienced by a water droplet sliding on these microstructured surfaces was measured. We found that, although the sliding friction generally increased with increasing solid surface fraction, it might drop unexpectly when solid area fraction was close to 1, espcially for surfaces with relatively large pillars. More importantly, the friction force also exhibited an apparent size scale effect, i.e., it increased monotonically with the size of micro-pillars when the solid area fraction was fixed. Assisted by optical microscopy, we attributed the non-monotonic dependence of friction force on solid area fraction to the different variation trends of contact anlge hysteresis and apparent contact size. Whereas the size scale effect of friction can be understood by considering the different coordination state of the depinning


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events around the receding tail. Although the sliding friction of liquid droplets on microstructured surfaces has been investigated previously, the non-monotonic variation trend and the apparent size effect have not been clearly demonstrated and systematically studied. Our findings provide new insights into the liquid-solid sliding behavior, which may lead to new design of superhydrophobic surfaces with tunable transport properties.

RESULTS AND DISCUSSION

The liquid-solid sliding friction was measured with a nano-tribometer (Anton Paar NTR2) by sliding a water droplet in linear reciprocating mode on microstructured solid surfaces, as illustrated in Figure 1a and Figure S1a. The water droplet, with a constant volume of 2 µL, was suspended at the end of a capillary tube and remained adhered throughout the sliding process. During friction measurements, the lateral displacement of the capillary tube and the lateral resistance force arising from the liquid-solid interface were recorded simultaneously. The solid surfaces comprising squared arrays of micro-pillars (with a lateral dimension a, a neighboring space b) were fabricated from silicon wafers (N-type ) and subsequently treated by fluoroalkylsilane, as shown by the scanning electron microscopy (SEM, Quanta FEG 450, FEI) image in Figure 1b and a schematic in Figure 1c. In the experiments, 15 types of samples with different micro-pillar sizes and solid area fractions (defined as ratio between the pillar top area and the apparent area

φ=

a2 ) were prepared and tested (more information can be found in Tables S1). (a + b) 2

Due to the limitation of the nano-tribometer setup, the advancing and receding contact angles (denoted as θ A and θ R , respectively) and the contact radius were measured offsite by an optical contact angle goniometer (Dataphysics OCA20) while sliding similar6 ACS Paragon Plus Environment

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sized water droplets on corresponding microstructured surfaces. The apparent advancing and receding contact angles were calculated using the tangent leaning method from a macroscopic view, as shown in Figure S2. The static contact angles were measured using the Laplace-Young fitting method 3 times on each surface and the results are shown in Table S1.

Figure 1. (a) Schematic of the experimental setup measuring a droplet sliding on a microstructured surface. (b) An SEM image of a typcial microstructured solid surfaces. (c) A schematic showing the dimensions of microstructures.

Figure 2a presents a typical curve showing the variation of lateral force with slide distance, when a droplet was slid forward and backward in a cycle on a microstructured surface. As suggested by the lateral force traces in Figure 2a, the water droplet was initially pinned when the slide distance was relatively small. During this stage (depicted as Stage-i), the droplet was continuously stretched and the lateral force exerted by the


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capillary tube kept increasing. After certain distance, when the accumulated lateral force was large enough to overcome the retention force from the solid surface, the water droplet would start sliding. During the sliding stage (depicted as Stage-ii), the lateral force entered a steady state and fluctuated around a mean value. Once the droplet slid to the maximum distance and turned backward, the water droplet got pinned again at the turning point (depicted as Stage-iii) before it finally slipped backward (depicted as Stageiv). As shown in Figure 2a, the lateral force traces form a closed loop (known as friction loop) and the averaged friction/retention force, Fr , can be calculated as half of the averaged difference between the forward and backward traces during steady-state sliding.

Figure 2. (a) A typical friction force loop when a droplet slid on a microstructured surface ( a = 5 µm, b = 10 µm, φ = 0.11 ). (b) A series of optical images showing the corresponding sliding process.

To explore the influence of solid area fraction on sliding friction, we first fixed the size of the micro-pillars and only varied the solid area fraction. Figure 3a shows the droplet sliding friction, Fr , as a function of solid area fraction, φ , for surfaces with relatively small microstructures ( a = 5 µm ). We also measured sliding friction for droplets sliding 8 ACS Paragon Plus Environment

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on flat silicon surfaces (similar surface treatment by fluoroalkylsilane) and presented them as data for φ = 1 . As can be seen from Figure 3a, the sliding friction of droplets increases monotonically when the solid area fraction of the microstructured surfaces increases from 0.11 to 1. According to the continuum mechanics models30, for a liquid droplet sliding quasi-statically on a solid surface, the retention force Fr can be estimated as Fr = k γ w(cos θ R − cos θ A ) ,

(1)

where γ is the surface tension of water, w is the half-width of the liquid-solid contact area perpendicular to the sliding direction, θ A and θ R denote the advancing and receding contact angles, respectively, and k is a constant prefactor. Although the specific values of k are scattered in literature, ranging from 4 / π to 2 depending on assumptions of the contact configuration

33-34

, these continuum models imply that the retention force is

largely determined by two factors, i.e. the apparent contact radius and the apparent contact angle hysteresis. To better understand the variation trend of friction, we plotted ( cos θR − cos θA ) , representing the apparent contact angle hysteresis effect, and the contact radius R, representing the apparent size of the liquid-solid contact area, as functions of solid fraction φ in Figure 3b. It is noted that, for water droplets sliding on superhydrophobic surfaces in our experiments, the liquid-solid contact region was approximately circular. Because the sliding was rather stable and smooth, the contact angle data reported in

Figure 3 were obtained by averaging three measurement values at random locations during the steady-state sliding stage. The individual variation of advancing and receding


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contact angles with solid fraction ( a = 5 µm ) are shown in Figure S3a. It is noted that the cosine of receding contact angle cos θ R is dominated over cos θ A . The uncertainty of the apparent high advancing contact angle resulted from the measuring method has a minor influence on the trend of contact angle hysteresis. As shown in Figure 3b, the contact radius increases monotonically with φ on microstructured surfaces ( a = 5 µm ) and it reaches a maximum value on smooth surface ( φ = 1 ). This trend can be understood by the classic Cassie-Baxter model38, where the apparent static contact angle θ of a microstructured solid surface is related to the intrinsic Young’s contact angle θ 0 of the solid and the solid area fraction φ by the following expression cos θ = −1 + (1 + cos θ 0 )φ .

(2)

If one further assumes that the droplet on superhydrophobic surfaces has a truncated spherical shape, then the contact radius R can be calculated as 1/3

  3V R=  sin θ , 3  π (2 − cosθ + cos θ ) 

(3)

where V is the droplet volume and θ is the apparent static contact angle. According to Eqs. (2)-(3), the contact radius increases with the solid area fraction, which is qualitatively consistent with our experiment observations. The static contact angle θ and contact radius R measured in the experiments for surfaces with a = 5 µm are shown in

Figure S4 (a) and (b) together with their theoretical prediction. It is worth noting that the static contact angle lies in a certain range and its value depends on how the droplet is deposited on the solid surface and the sizes of the apparent contact area in static state and sliding state can be different. Despite these, the experimentally measured variation trends


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of contact angle and contact radius were qualitatively consistent with the idealized theoretical analysis. This analysis is to infer that the increasing friction with increasing solid area fraction for microstructured surfaces with smaller pillars can be qualitatively explained by enhanced surface wettability. Unlike the monotonic increase of contact radius, the variation of contact angle hysteresis effect, represented by ( cos θ R − cos θ A ) , was more complicated: it increased initially with

φ for microstructured surfaces and then decreased eventually for smooth surface. Early studies35 have showed that the contact angle hysteresis is closely related to the pinning capability of microstructures at the corrugated receding contact line. The initial enhancement in contact angle hysteresis could be due to stronger pinning effect from the micro-pillars when their density (i.e. solid surface fraction) was increased. However, as the density of micro-pillar reached a critically high value, the corrugated contact line might become smoother again leading to a net reduction in its true total length, although the apparent contact radius might still be increasing. Because the pinning capability is largely determined by the true length of contact line as found by Xu and Choi35, this net reduction might lead to a slight decrease in contact angle hysteresis when the solid surface fraction changed from φ = 0.44 to φ = 1 as shown in Figure 3b.

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Figure 3. (a) Variation of sliding friction with solid area fraction and the corresponding ratio of Fr / [γ R (cos θ R − cos θ A )] for water droplets sliding on microstructured surfaces with a = 5 µm ; the red dashed line represents the averaged value of Fr / [γ R (cos θ R − cos θ A )] for four samples. (b) The corresponding contact angle hysteresis and contact radius measured via an optical contact angle goniometer; the uncertainties were calculated as the standard deviation.

To examine the applicability of the continuum models for our experiments, we calculated the value of Fr / [γ R (cos θ R − cos θ A )] and plotted them against solid surface fraction in

Figure 3a. As can be seen in Figure 3a, Fr / [γ R (cos θ R − cos θ A )] stays approximately constant at ~1.5 for all samples, including microstructured surfaces with a = 5 µm and smooth surface. This implies that the friction force measured in our experiments can be well described by the continuum model, i.e. Equation (1), if the prefactor k is chosen to 1.5. Previously, Brown et al.29 and Carre et al.

39

showed that the prefactor should be

around π / 2 if the contact area is circular shaped and the contact angle varies as a smooth function of position along the contact line. We also compared the retention force measured from the experiments with the ones calculated by the continuum model (

a = 5 µm ) and a good agreement was obtained as shown in Figure S5. The good consistency between the experiments and the theoretical model suggests that the continuum approach works well for the microstructured surfaces with pillar size

a = 5 µm . We also performed similar friction experiments for microstructured surfaces with pillar sizes a = 10 µm , 20 µm and the results are shown in Figures 4a–d. It is seen that the variation trends of friction force, contact angle hysteresis and contact radius for samples with a = 10 µm , 20 µm are qualitatively similar to those observed for samples with a = 5 µm . Although the ratio Fr / [γ R (cos θ R − cos θ A )] is no longer a constant, its


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value fluctuates moderately between 4 / π and 2, which still lies in the range of continuum estimations as indicated by the shaded regions30, 33.

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0.0 0.0

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Figure 4. (a) and (b) Variations of friction with solid area fraction and the corresponding ratio of Fr / [γ R(cos θ R − cos θ A )] for surfaces with a = 10 µm and 20 µm , respectively; the shaded regions in (a) and (b) represent range of k predicted by the continuum estimations. (c) and (d) The corresponding contact angle hysteresis and contact radius measured in experiments, respectively.

In contrast to the smooth sliding on surfaces with small micro-pillars, the droplets sliding on surfaces with relatively large pillars ( a ≥ 40 µm ) exhibited distinctively different dyanamic behavior, as illustrated by the typical snapshots in Figures 5a–c. When a droplet slid on surfaces with large pillars, the contact line would get stuck initially at the rear edge (state 1, Figure 5a); when the lateral force exceeded a threshold value (state 2, 13 ACS Paragon Plus Environment

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Figure 5b), the rear edge of the water droplet suddenly depinned and it jumped forward unstably to the next pillar (state 3, Figure 5c). The corresponding lateral force traces are shown in Figure 5d, where a regular saw-tooth pattern reflecting the periodic stick–slip can be clearly seen. Because the stick-slip is induced by the unstable depinning of the solid-liquid contact line from the micro-pillar, its period is consistent with the spacing between neighboring micro-pillars. For example, the stick-slip period for the lateral force curve shown in Figure 5d is ~160µm, which matches well with the spacing of the micropillars. We also confirmed that the period of the stick-slip behavior remained the same upon changes in the sliding velocity, as shown in Figure S6, suggesting that the instabilities are indeed spatially controlled but not time-controlled40.

Figure 5. (a)–(c) Typical snapshots during water droplets sliding on a surface with large pillars ( a = 40 µm，b = 80 µm ). (d) The corresponding friction force traces of a water droplet sliding on a surface with large pillars ( a = b = 80 µm ); the three red circles represent the typical states shown in (a-c).

Because of the particular stick-slip behavior of droplets on surfaces with relatively large micro-pillars, significant fluctuations were observed in both contact radius R and contact angle hysteresis. The large fluctuation in friction and contact angle hysteresis and the discrete nature of contact radius make it a challenging task to quantify them


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unambiguously. To this end, we simply took their averaged values during the steady-state sliding stage in the following analyses. The contact angle and contact radius data in

Figure 6 were obtained by averaging 4-10 group measurements and each group includes values both “before” and “after” a jump.

Figures 6a and 6b show the variations of

averaged friction force with solid surface fraction φ for samples with larger pillars ( a = 40 µm and 80 µm , respectively). The values measured on smooth surfaces are again included as a reference. For samples with a micro-pillar size of a = 40 µm (Figures 6a), although the friction force still increases monotonically with φ , the increasing trend is not as rapid as those for a = 5, 10 and 20 µm . For example, Fr on smooth surface ( φ = 1 ) is only slightly larger than that on microstructured surfaces with φ = 0.44 . By comparing

Figure 6a with Figures 3a, 4a, 4c, one can see that a direct cause for this slower increasing trend comes from the overall enhancement of friction for samples with larger pillar sizes. This effect becomes more pronounced for samples with micro-pillar size of

a = 80 µm . As shown in Figure 6b, the friction force at φ = 0.44 is already comparable to that on smooth surfaces and the friction at φ = 0.64 is noticeably larger than the smooth case, leading to a non-monotonic friction variation with φ .


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a = 80µm

0.0 0.0

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Figure 6. (a) and (b) Variations of friction with solid area fraction and the corresponding ratio of Fr / [γ R(cos θ R − cos θ A )] for surfaces with a = 40 µm and 80 µm ; the shaded regions in (a) and (b) represent the range of k predicted by the continuum estimations. (c) and (d) The corresponding contact angle hysteresis and contact radius measured in experiments, respectively. Each data point represents averaged value within four stick-slip periods.

To understand the unusual frictional behavior, we again examined the variations of contact radius and contact angle hysteresis for these two sets of samples. As shown in

Figures 6c and 6d, the variation trends for samples with larger micro-pillars are qualitatively similar to the cases with smaller micro-pillars: the contact radius R increases monotonically with φ , while contact angle hysteresis effect ( cos θ R − cos θ A ) increases initially with φ and then decreases as φ approaches one. Despite the similarity, one can readily

see

that

both

the

contact

angle

hysteresis

effect

and

the

ratio

Fr / [γ R(cos θ R − cos θ A )] for surfaces with large micro-pillars are enhanced noticeably. 16 ACS Paragon Plus Environment

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Despite this, the reasonable agreement between experiments and the theoretical estimations suggests that the continuum model can still capture the essential physical process even for samples with larger microstructures. In other words, the dependence of friction force on solid area fraction can be understood by consideirng the combined effect of different variation trends of contact anlge hysteresis and apparent contact size.

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10

0.6

0.4 5 0.2 0 0

10

20

30

40

a(µm)

50

60

70

80

90

0

10

20

30

40

50

60

70

80

90

a(µm)

Figure 7. (a) Variations of the friction force and (b) the ratio λ with pillar size a ( φ = 0.25 ).

According to the above discussions, the sliding friction of droplets depended not only on solid area fraction of the microstructured surfaces, but also on size of the micro-pillars. To explore the size effect, friction as a function of the micro-pillar size is given in Figure

7a when the solid area fraction is fixed at φ = 0.25 . As the micro-pillar size changes from

5 µm to 80 µm , the friction force increases from 8.75µN to 14.58µN showing a clear size dependence. We also performed experiments for sample surfaces with solid fraction

φ = 0.11 and φ = 0.44 and similar friction size effect was obtained as shown in the supporting information as Figure S8. Previously, Zheng and Lv41 reported that scaling down the microstructures of a surface would enhance its superhydrophobicity, hypothetically due to the line tension that is typically operating under sub-micron scales42. 17 ACS Paragon Plus Environment

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In our experiments, the dimension of all microstructures were 5 µm or above, therefore the line tension effect was expected to be negligible. We plotted the variations of receding contact angles with pillar size a for water droplets sliding on microstructured surfaces with

φ = 0.11 , 0.25 and 0.44 and the results are shown in Figure S7a-c. In

general, the averaged receding contact angles decreased moderately when the pillar size a increased from 5µm to 80µm . It is noticed that the size effect exhibited by the receding angle is not as pronounced as that of the friction force. We hypothesize that the decrease of the apparent receding angle at the tail of the contact line (i.e. the contact angle hysteresis) only partially accounts for the size effect of friction and other mechanisms may play a role. To unveil the origin of this unique size dependence, we focused on the configurational changes and the local pinning forces along the actual contact line. Because frictional resistance is closely related to the pinning state of the trailing edge but not the leading edge43-44, the instantaneous friction force can be calculated as n

Fr (t ) = ∑ Fi (t ) ,

(4)

i =1

where Fi (t ) is the local pinning force of the ith micro-pillar at time t and n is the total number of micro-pillars at the receding contact line. The maximum pinning force of each individual pillar can be estimated by45- 46

Fi max = απγ a sin θR0 cos

θR 2

,

(5)

Where α is a correction factor and α = 0.46 as reported in the literature [45], θ R0 represents the intrinsic receding contact angle on smooth silicon surfaces and θ R is the


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apparent receding contact angle. If all the pinning forces from different micro-pillars acting in concert, then the maximum total resistance force along the receding contact line can be calculated as Fr max = Fi max ⋅ n .

(6)

As a rough approximation, the total number of micro-pillars at the receding contact line can be estimated as n =

πR

47

a +b

, where R is the apparent contact radius measured in the

experiments. Since the pinning forces at the micro-pillars do not reach their maximum values at the same time, the instantaneous friction force Fr (t ) and the averaged friction during steady-state sliding Fr are smaller than Frmax . Therefore, the actual magnitudes of the pinning forces at the micro-pillars should lie in the range of 0 ~ Fi max .To quantify the degree of coordination among the pinning forces, we define the coordination parameter

λ=

Fr Fr , which is typically less than one and gets larger when pinning effects = max Fr nFi max

among different micro-pillars are becoming more in concert. As clearly shown in Figure

7b, the coordination parameter λ is enlarged monotonically when the micro-pillar size a increases from 5 µm to 80 µm even though the solid area fraction stays at φ = 0.25 . Because λ is not a direct variable measured from the experiments, the uncertainty of λ was calculated using the error propagation equation as shown in the Supporting Information. The relatively small values of λ observed in the experiments suggest that the pinning forces at different micro-pillars were indeed not acting in synchronization, instead, they reached their own maximum values at different moments. However, as the size of the micro-pillars increased, the pinning effect from different micro-pillars behaved more in phase. 19 ACS Paragon Plus Environment

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Figure 8. Snapshots of contact line configurations when water droplets sliding on surfaces with small and large micro-pillars: (a) a = 10 µm, b = 10 µm , and (b) a = 40 µm, b = 40 µm . The arrows indicate the locations of the pillars, where the three-phase contact line is at the critical state and is about to slip in the processes from (i) to (ii) and (ii) to (iii).

Although a quantitative comparison of the coordination parameter by directly measuring the depinning forces on all individual pillars cannot be made with the available experimental techniques, the coordination effect can be illustrated by the visualization of the dynamic contact line. To assess the real-time coordination state among the micropillars, we performed another set of comparative experiments, as schematically depicted in Figure S9, where a water droplet suspended under a transparent cover glass was pressed and slid against superhydrophobic microstructured surfaces. Using an optical microscope, the evolution of the three-phase contact line of the sliding droplet could be clearly monitored. For droplets sliding on surfaces with smaller microstructures (Movie


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S1, the snapshots shown in Figure 8a for a = b = 10 µm ), the receding contact line remained nearly circular and the contact line on the micro-pillars slip alternately. As a consequence, the receding contact line on these micro-pillars were not loaded in a same pace during droplet sliding except the tail end of the receding contact line (e.g., in Figure

8a, from state (ii) to (iii) ), suggesting that the variation of the individual pinning forces were less coordinated, as schematically depicted in Figure 9a. In Figure 9, the relative magnitude of the pinning force at individual pillar was qualitatively assessed from the local deformed configuration of the contact line. The distribution of the local pinning forces at individual pillars (the color scheme) in Figure 9 was estimated by inspecting the relative position between the dynamic contact line and the center of individual pillars at the receding end shown in Figure 8 and Movie S1 and S2. We want to emphasize that the coordination status shown in Figure 9 is intended to qualitatively describe, rather than strictly quantifying the synchronization level of the depinning forces. In contrast, for droplets sliding on surfaces with larger micro-pillars (Movie S2, snapshots shown in

Figure 8b for a = b = 40 µm ), the contact line was severely distorted and deviated from the circular shape due to stronger interference from neighboring micro-pillars. Meanwhile, the corrugation of contact line on larger micro-pillars led to a larger true contact line length (both the apparent contact radius and the corrugation of the contact line increase with the size of the pillars, leading to an increasing of true contact line length). Because of the strong interaction, the slip motion of the contact line on different micro-pillars was more concurrent (e.g., in Figure 8b, from state (ii) to (iii), as the arrows signed, the contact line on four adjacent pillars slipped synchronously to the next row of pillars along the droplet sliding direction, suggesting that the individual pinning


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Page 22 of 33

forces are more synchronized.) and each slip would lead to a noticeable drop in total pinning force, as schematically illustrated in Figure 9b. As the size of the micro-pillars increased, the pinning effect from different micro-pillars behaved more in phase, which is qualitatively consistent with the trend shown in Figure 7b. Therefore, we propose that the change in coordination state of the local pinning forces due to contact line interaction of neighboring micro-pillars is the primary mechanism for the size-scale dependence in our experiments. The degree of synchronization depends on the deformation of the three phase contact line with respect to the pillars and the maximum pinning force the pillars can provide. The maximum pinning force that larger pillars can exert is higher and a sudden contact line depinning on these larger pillars will result in a stronger disturbing effect on their neighboring pillars leading to collaborative depinning events. Therefore, a larger micro-pillar size will generally result in stronger interference, more synchronized pinning effect and more significant overall pinning effect. This may be a general trend as qualitatively similar effect has also been observed for evaporation experiments for water droplets on microstructured surfaces, as shown in Figure S10. It is worth mentioning that similar fundamental mechanism can be found in many phenomena (e.g. adhesion) in biological and bio-inspired systems

48-50

. The interplay and the synchronization effect

among asperities are also very critical in understanding stick-slip friction behavior of rough solid interfaces51.


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(a) (a)

(b)

Fi

0

Fi

Fi max

0

Fi max

Figure 9. Schematics showing the configuration of the receding contact line and the corresponding pinning forces at individual micro-pillars when water droplets sliding on surfaces with (a) small and (b) relatively large micro-pillars. The blue arrows indicate the sliding direction. The dashed circular lines represent the apparent droplet contact line, while the solid curves represent the real configuration of the receding contact line. The color of the individual micropillars along the rear half of the contact line signifies the magnitude of the pinning forces.

CONCLUSION

The sliding friction of water droplets on superhydrophobic solid surfaces decorated with micro-pillars was studied using a nanotribometer. By systematically varying the geomertry of microstructures, we found that the droplet sliding friction largely depended on the solid area fraction of the surfaces. Surfaces with higher solid area fraction generally exhibited larger friction, although the retention force might decrease again when the solid area fraction was close to one (i.e. the smooth case). In addition, our experiments clearly demonstrated that the sliding friction of droplets was criticially affected by the dimension of the microstructures with larger micro-patterns resulting in higher friction force. Assisted by optical microscopy, we attributed the non-monotonic dependence of friction force on solid area fraction and the apparent size effect to the two governing factors, i.e. the true length of the contact line and the coordination state of the depinning events during sliding. Our proposed mechanisms are expected to be generally 23 ACS Paragon Plus Environment

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applicable for other liquid transport processes involving dynamic motion of three-phase contact line; and the findings may offer a new strategy of tuning liquid transfer beavhior using microstructured surfaces.

AUTHOR CONTRIBUTIONS

S.Q. performed experiments and obtained the experimental data. S.Q., S.L. and K.L. analyzed the data with input from all other authors. Q.L. and X.F. supervised the experimental and analytical work. The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

ACKNOWLEDGMENT

We gratefully acknowledge the supports from the National Basic Research Program of China (2013CB933003 and 2015CB351903) and the National Natural Science Foundation of China (11432008 and 11422218).

MATERIAL FABRICATION

Micro-pillars of different sizes were prepared on silicon wafers via photo etching. The micro

structured

surfaces

were

modified

with

1H,1H,2H,2H-

perfluorodecyltrimethoxysilane using chemical vapor deposition after treatment with air plasma. We designed 15 samples with different pillar widths and spacings between pillars. The samples can be divided into five groups, with each group having the same pillar width and different solid surface fractions (Table S1). Due to the micro-fabrication process, the height of the pillars is not uniform ranging from 15µm to 30µm and the 24 ACS Paragon Plus Environment

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larger pillars are typically taller. The heights are measured by the 3D Laser Scanning Microscope (KEYENCE-VK-X-100) and an example is shown in Figure S1b for micropillar structured surface with a = b = 40 µm . Because the pillars are made of stiff silicon, the maximum deflection of pillar under the maximum friction force of droplet is −8 estimated to be on the scale of 10 nm . Therefore, the pillars can be essentially regarded

as rigid when water droplets sliding on the pillars and the elastic deformation should not affect the sliding process. The droplet volumes are kept constant as 2 µL and the diameter of the droplet is about 1.56 mm. The averaged numbers of pillars under the diameter of the contact region are calculated according to the contact radius measured in the experiment and the results are shown in Figure S11. As the pillar size increases, the number of pillars in contact reduces. This is part of reasons why sliding on surfaces with relatively large pillars, show more severe fluctuation compared to others.

Table S1. Pillar sizes (a) and spacings (b) of the samples and the corresponding contact angles. a

b

φ (solid

Contact

a

b

φ (solid

Contact

a

b

φ (solid

Contact

(µm)

(µm)

fraction)

angle(°)

(µm)

(µm)

fraction)

angle(°)

(µm)

(µm)

fraction)

angle(°)

5

10

0.11

164.1±3.1

10

5

0.44

154.4±1.3

40

40

0.25

155.0±1.0

5

5

0.25

158.0±3.0

20

40

0.11

157.1±2.3

40

20

0.44

152.3±2.1

5

2.5

0.44

157.3±0.5

20

20

0.25

157.0±0.7

80

80

0.25

153.6±1.3

10

20

0.11

159.9±2.8

20

10

0.44

154.1±1.0

80

40

0.44

152.4±0.9

10

10

0.25

157.6±1.1

40

80

0.11

158.4±0.5

80

20

0.64

151.0±0.8


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METHODS

The measurements of water-droplet friction on solid surfaces were performed using a nano-tribometer (Anton Paar NTR2) equipped with a capillary glass tube (as shown in

Figure S1a). The nano-tribometer has two independent high-resolution dual-beam cantilever sensors to measure the normal and friction force up to 10 mN (the resolution of the force sensor is about 3 nN with a noise floor about ~50nN). As shown in Figure S1a, a fluoroalkylsilane-treated capillary glass tube with an inner diameter of 0.5 mm was mounted on the end of the cantilever, and a 2-µL droplet of pure water was dispensed onto the end of the glass tube using a micropipette (the tube was pre-sealed with glue so that water did not get into the tube). The sample was mounted on a stage and brought in contact with the suspended water droplet. The normal compressive load was set as 40 µN. During friction measurement, the sample stage was moved with reciprocating motion at 0.05 Hz with a 3-mm lateral stroke. The sampling rate of data acquisition was 100 Hz. The lateral force versus sliding position were recorded at each cycle. To minimize the system noise, the force curves were smoothed using the adjacent-averaging method (5point). The experiments were conducted in an environment at ambient conditions with a relative humidity between 30%~60%. In this work, friction Fr was taken as the average value of 3-5 reciprocating cycles and uncertainty was the sum of within-group and between-group standard deviations.


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Supporting Information

Photo of the experimental setup of the nano-tribometer; 3D morphology of micro-pillar structured surface; optical images of sliding water droplets; variations of advancing contact angle, receding contact angle, the static contact angle and contact radius with solid area fraction, variation of sliding friction and the theoretical retention force with solid area fraction; friction force under different sliding speeds; variation of receding contact angle and friction force with pillar size; experimental setup for observing contact line configuration; snapshots of water droplets during evaporation; number of pillars within the contact region; the calculation method of the uncertainty for coordination parameter λ ; movies of droplets sliding on surfaces with smaller and larger microstructures.

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Table of Contents Graphic


Friction of Droplets Sliding on Microstructured Superhydrophobic

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