Spontaneous Droplet Motion on a Periodically Compliant Substrate

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Spontaneous Droplet Motion on a Periodic Compliant Substrate Tianshu Liu, Nichole K. Nadermann, Zhenping He, Steven H. Strogatz, Chung-Yuen Hui, and Anand Jagota Langmuir, Just Accepted Manuscript • Publication Date (Web): 27 Apr 2017 Downloaded from http://pubs.acs.org on April 28, 2017

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Langmuir

Spontaneous Droplet Motion on a Periodic Compliant Substrate Tianshu Liu1, Nichole Nadermann2, Zhenping He2, Steven H. Strogatz3, Chung-Yuen Hui1, Anand Jagota2*. 1

Field of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY, 14850, USA.

2

Department of Chemical & Biomolecular Engineering and Bioengineering Program, 111 Research Drive, Lehigh University, Bethlehem, PA, 18015, USA.

3

Department of Mathematics, Cornell University, Ithaca, NY, 14853-4201, USA

KEYWORDS: Surface tension, droplet motion, soft material, chaos, instability, evaporation, periodic structure.

ABSTRACT: Droplet motion arises in many natural phenomena, ranging from the familiar gravity-driven slip and arrest of raindrops on windows, to the directed transport of droplets for water harvesting by plants and animals under dry conditions. Deliberate transportation and manipulation of droplets is also important in many technological applications, including droplet-based microfluidic chemical reactors and for thermal management. Droplet motion usually requires gradients of surface energy or temperature, or external vibration to overcome contact angle hysteresis. Here, we report a new phenomenon in which a drying droplet placed on a periodically compliant surface undergoes spontaneous, erratic motion in the absence of surface energy gradients and external stimuli such as vibration. By modeling the droplet as a massspring system on a substrate with periodically varying compliance, we show that the stability of equilibrium depends on the size of the droplet. Specifically, if the center of mass of the drop lies at a stable equilibrium point of the system, it will stay there until evaporation reduces its size and this fixed point becomes unstable; with any small perturbation, the droplet then moves to one of its neighboring fixed points.

Introduction Spontaneous droplet motion is of great importance in a variety of industrial applications such as the design and fabrication of microfluidic devices1–3 and superhydrophobic coatings4. Typically, droplet motion is induced using a surface with gradients in curvature5, in droplet apparent contact angle, or by external stimuli. By

external stimuli we mean a gradient provided by the substrate or an external source of vibration6–8 . Spatial gradient in apparent contact angle can be induced by variation in thermal9–12, chemical13–17, electrical18 and elastic19–21 proper-

ties of the surface. Continuous transportation of a droplet requires a continuous input of free energy, such as by vibration; it can also be provided by evaporation of the droplet itself. For example, by controlling the direction of vapor flow, a droplet placed on a hot ratchet can propel itself in a specified direction22. Evaporation can also cause pining and un-pining of contact line on a structured surface and thus generate droplet motion23. More recently, Cira et al.24 have demonstrated that two-component droplets of well-chosen miscible liquids on a clean glass slide can move and interact with each other by emitting vapor.

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Here, we report a new phenomenon in which a water droplet moves spontaneously on a surface with periodic variation in compliance. The motion is spontaneous in the sense that it requires no surface energy gradient or external stimuli. Specifically, we fabricated an elastomeric substrate with periodic variation of surface compliance. We found that a small droplet placed on such a substrate

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tion less likely, not more. However, we observe the reverse. Experimental Procedure Figs. 1 (a-d) show a series of still images from a top-view video of the phenomenon: a droplet of deionized water moves spontaneously on a compliant surface whose stiff-

Fig. 1 Droplet on a film-terminated fibrillar surface. (a-d) Beginning at t = t0, micrographs of a droplet of de-ionized water placed atop a film-terminated fibrillar surface show (i) significant distortion of interferometric contours, indicating out-of-plane deformation driven by liquid surface tension, and (ii) spontaneous movement of the droplet. In this series of optical micrographs, the droplet moves over a distance much larger than its own size and the interfibrillar distance. (e) Scanning electron micrograph image of the film-terminated fibrillar surface. The fibril height is 40 , the interfibrillar spacing is 115 , and the diameter of the fibrils is 10 . (R1.5) (f) A schematic showing a droplet on the structured surface, and the deformation caused by droplet surface tension. The experiments are repeated on samples with height h varied from 10 – 40 and interfibrillar spacing a varied from 20 – 110 . (g) Contours and 3D representation of the vertical displacement of the substrate caused by surface tension of the drop. undergoes random motion over distances significantly larger than the drop size and the spatial period over which compliance varies. This phenomenon is surprising because it is unclear what drives it; there is no long range gradient in surface energy or in surface compliance. In particular this phenomenon is quite different from that reported by Style et al.19, where droplet motion is driven by a long-range gradient in substrate stiffness. If the motion observed in our system were driven by a stiffness gradient, the droplet would be trapped at a position of minimum substrate stiffness and would not be able to move a distance of more than one period of compliance variation. This indicates an agent other than substrate stiffness alone drives droplet motion. What makes the phenomenon even more counterintuitive is that increasing surface compliance and surface undulations both usually lead to an increase of contact angle hysteresis compared to a flat unstructured control surface25. Therefore, compared to a flat, unstructured control, on which drops show no long-range motion, a structured surface with a periodic variation in compliance should make drop mo-

ness varies periodically in both directions. Before analyzing the details and the cause of the motion, let us clarify the experimental setup. A typical micrograph of the type of structure used in our studies, and a schematic of the cross-section with a droplet on top, are shown in Figs. 1(e) and 1(f), respectively. The structure comprises a thin poly(dimethylsiloxane) (PDMS) film (thickness t ≈ 10 µ m ) bonded to a periodic square array of fibrils with center to center spacing a. The fibrils are circular pillars 10µm in diameter. The architecture of this surface was inspired by the structure of contact surfaces in many insects and small animals that have arrays of fibrils to make contact, for enhanced and modulated adhesion and friction 26,27. Naturally, the specimen surface is stiffer if probed directly above a fibril and relatively compliant if probed between fibrils. Therefore, the surface compliance of this structure varies periodically with a period equal to inter-fibril spacing. The sample preparation method is similar to the process described in Glassmaker et al.28. Briefly, using standard photolithography and deep reactive ion etch techniques,


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Langmuir arrays of circular holes of diameter 10 µ m are formed on silicon wafers. Next, the etched master silicon wafers are coated by a hydrophobic self-assembled monolayer so that the molded PDMS structure can be easily peeled off. After this step, liquid PDMS (Sylgard 184; Dow Corning; 10:1 mass ratio of elastomer base to curing agent) is cast into the master silicon wafer. A clean glass slide is used to confine the material and is separated from the master by 0.635 mm using a clean spacer to form the backing layer.The liquid PDMS is cured at 80oC for 1 hour and manually removed from the master. The resulting PDMS array with backing layer is then placed on a thin liquid PDMS film spin-coated on a glass slide. The entire structure is cured at 80oC for an hour and then peeled off from the glass slide. In our experiment, the height h of the fibrils is set to be 10, 20, 30 or 40 µ m and the center to center spacing a between the fibrils is 20, 35, 50, 65, 80, 95 or 110 µ m . Figure 1 (f) shows a schematic of this structure.

Initially, the droplet has the shape of a spherical cap and the contact line between the droplet and the film is roughly circular (Fig. 1(g)), with small undulations near the fibrils. Typical snapshots of the deformed and moving droplet are shown in Figs. 1(a)-(d). The droplet gradually becomes smaller as it evaporates and the contact line becomes irregular. As evaporation continues, the droplet starts to move spontaneously, as shown in the SI movie, SM1. The motion of the droplet is relatively slow at first, then picks up speed as it becomes smaller with further evaporation, and eventually slows down and stops, usually when the size of the droplet becomes smaller than the spacing between fibrils. The surface tension and Laplace pressure of the droplet can induce significant deformation of the substrate (see Fig. 1(g)). As expected, deformation is smaller at or near the fibrils, where the compliance is lowest. This variation of spatial deformation leads to local variation of apparent contact angle along the contact line.

A single deionized water droplet of volume 0.1 µL was placed on the film. The volume of the droplet was controlled by an air displacement pipette. The sample together with the droplet was then placed under a white light interferometer (Zegage; Zygo Corporation) for observation.

As the video in the Supplementary Information (SM1) indicates, droplet motion has the appearance of a random walk. To test if the motion is truly random, or if there is some unknown gradient on the surface driving the motion, we repeated the experiment several times using a new

19

Results and discussion

(a)

(b)

Fig. 2 (a) Trajectories and (b) velocities of the droplet centroid for different drops placed on the same substrate. Different colors correspond to different experiments. The fibril spacing a is 20 µ m and the fibril height is 10 µ m . Measured trajectories for samples with different fibril spacing and fibril heights are given in the SI. The circular markers are centered at the droplet’s centroid. The marker size is proportional to the contact area of the drop (100 times smaller than real contact area), thus showing how the drop shrinks by evaporation. The time between two consecutive markers is 3 sec. For a specific region of the sample, the experiments could only be repeated about 3-7 times, after which new droplets lost their ability to move. This is likely due to the deposition of some impurities during droplet drying, which increases the contact angle hysteresis. droplet deposited on the same sample, and recorded the motion of the droplet. (A Matlab® program was used to detect the droplet contact line and to compute the centroid of its contact area for every frame of the video.) Tra-

jectories of the centroid for different drops placed on the same substrate are shown in Fig. 2 (a). In Figs. 2 (a) and (b), different colors correspond to different trials. The size of the markers (circles centered at droplet’s centroid) are


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proportional to the contact area between the drop and the substrate. The time between two consecutive markers is 3 seconds. From both the SI movie SM1 and Fig. 2(a), we see that the motion of the drop is quite substantial and that the distance it travels is typically much larger than the interfibrillar spacing a. From the recorded trajectories, the drop motion is apparently random and is very sensitive to initial conditions. For example, the black and the red paths in Fig. 2(a) start at almost the same location, but the trajectories are completely different. In several instances, for example the case illustrated by the purple and cyan paths, the droplets follow similar paths in the beginning but deviate from each other later on. These observations rule out a hidden gradient driving the process and suggest that the dynamics of droplet motion may be sensitive to initial conditions and small perturbations along the way. Large droplets move slowly or not at all; their sluggishness is probably due to inertia or contact angle hysteresis. As a drop evaporates and becomes smaller, the speed of its motion increases. When the size of a droplet becomes smaller than the inter-fibrillar spacing a, drop motion slows down and it eventually stops and dries. The SI movie SM1 and the velocity data in Fig. 2(b) show that the motion of droplets can be jerky. Fig. 2 (b) indicates that most of the time the drop moves slowly and continuously but suddenly speeds up and darts across the surface at certain locations, as indicated by the spikes in Fig. 2(b). Similar trends are observed for samples with different fibril spacing and height, as shown in SI. To support the notion that evaporation is needed for droplet motion, we repeated our experiment (a) with water in a vapor-saturated environment where droplet shrinkage is prevented, and (b) using a liquid (DMSO, dimethyl sulfoxide) that does not dry over the time-scale of the experiment. (Details are provided in SI.). No drying was observed, nor was there any drop motion. Model for droplet motion The erratic motion observed in our experiments rules out the possibility that an overall gradient drives the droplet motion. Furthermore, compared with a flat surface, the structured surface should make contact angle hysteresis larger 25. So, what provides the force to overcome the hysteresis and induce the motion? A clue comes from Fig. 1(g), in which the film deformation caused by surface tension can be as large as 4 µ m (consistent with a simple calculation given in SI). Such large deformation can reduce the apparent contact angle θ from its equilibrium value on a rigid substrate, θ c (a material constant). This is illustrated in Fig. 3 for a particular position of the droplet. When the contact line sits right on top of a fibril (, ), the local substrate stiffness is much higher and the apparent contact angle is approximately equal to . On the other hand, at B, the contact line lies on the

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compliant film and the apparent contact angle is smaller than . As reported by Chaudhury et al. 29, the difference between contact angles on two sides of a drop, say θ < θ c on one side and θ = θ c on the other as in the example drawn in Fig. 3, will give an unbalanced force, which drives the motion of the drop. In our system, this force is periodic and depends on the shape of the contact line as well as the position of the drop on the periodic structure.

Fig. 3 Schematic of a cross-section of a droplet on a periodic surface, which consists of a thin film on top of identical fibrils, with inter-fibrillar spacing a. The contact line on the left causes little deformation since it lies directly on top of a fibril, where the film is least compliant. In contrast, the contact line on the right lies between two ridges, where the film is most compliant. Note that since the contact line can move, the contact angle at A can be fixed at θc while θ decreases at B without violating volume conservation. Surface tension pulls the film upward, thereby deforming the structured surface. This uplift reduces the apparent contact angle, and the difference in contact angle at the two contact lines provides an unbalanced driving force to move the drop. In our experiments, the contact line of the droplet is irregular; parts of it lie on fibrils and the rest are on the compliant film. The elastic deformation of the film is coupled to the dynamics of motion. Furthermore, the amount of hysteresis depends on the position of the contact line and it can be very sensitive to surface impurities. Due to all these complications, it is extremely difficult to develop a comprehensive quantitative model to compare directly against the experiment. Instead, in the following, we present a simple one-dimensional model that captures some of the key features of this system, namely, the periodic driving force and the shrinking volume of the droplet. We use it to show that the combination of shrinking size due to drying and periodic variation in surface compliance sets up spontaneous motion of the drop with dynamics of droplet motion that can be extremely sensitive to its shape, size, and position. Consider a 2D droplet (a droplet that is infinitely long in the out-of-plane direction) on a one-dimensional periodic structure consisting of a thin film on top of identical parallel ridges with center-to-center spacing a. The geometry of the cross-section of the sample is the same as drawn


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Langmuir schematically in Fig. 3. The droplet is constrained to move along the x-axis, which is perpendicular to the ridges. Since there is only a weak dependence of PDMS’s moduli on temperature, we neglect the variation of stiffness due to temperature change caused by evaporation. We choose the origin x = 0 to be midway between two ridges, where the film is most compliant. At time t , the coordinates of the contact lines are x2 ( t ) , x1 ( t ) , respectively. The size of the droplet is l(t ) ≡ x2 ( t ) − x1 ( t ) , which decreases linearly

µ

ε (1) Where

where l0 is the initial size and β is an evaporation rate. As argued above, the net driving force for droplet motion is given by the difference in the horizontal component of liquid vapor surface tension γ LV at the two ends of the droplet. We take the configuration θ = θc as a reference and define a local force F on each of the two contact lines such that the difference between the forces, thus defined, is the net driving force. Periodicity of the structure implies that the variation of F has the same period, a . Specifically, when the contact line is on top of a ridge, that is, when x = ( 2n + 1) a / 2 , where n is an integer, θ = θ c and F is at its minimum. When the contact line lies in between two ridges, ≈ , θ is at its minimum θmin and F reaches its maximum. Since is, by definition, the apparent contact angle at a specific location, it is a constant for a given structure. However, it is important to note that θmin depends on the elasticity of the substrate and the thickness of the film. As an estimate of the value of we measured the receding contact angle on an unstructured PDMS substrate to be about 94°. Based on experimental measurements of film deformation (Figure 1), the value of is estimated to be about 5o-10 o lower than . As the contact line moves, the contact angle varies between these bounds. In our 2D model, we assume F varies sinusoidally between these two limits, that is, 1 F ( x ) = γ LV ( cos θmin − cosθ c ) (1 + cos ( 2π x / a ) ) + γ LV cosθ c 2 (2)

(3)

The left hand side in (3) models damping, including contributions from fluid viscosity and substrate viscoelasticity30. We also assume that the droplet is relatively stiff so it is effectively shrinking rigidly in size. With these approximations, (1-3) can be combined into one equation (We also assume that drop motion is slow enough that one may ignore inertia, see SI for details):

according to (see SI) l (t ) = l0 (1 − β t )

dxcm = F (x2 ) − F ( x1 ) = ∆F dt

dX cm = sin ( L / 2 ) sin X cm dL

!" # $ %

(4)

is a dimensionless parameter,

and X cm = π ( x2 + x1 ) / a and L = 2π l / a are the normalized center-of-mass coordinate and length of the droplet, respectively. The dimensionless parameter represents a ratio of characteristic relaxation ( ) and %

!" # $

drying (1/β) times. For small values of this parameter drying is slower than relaxation and the droplet center of mass is predicted to move jerkily. For large values of this parameter, drying is faster, motion is less jerky and eventually the drop dries too fast for it to move much And, since L is monotonically related to t via Eq. (1), it can play the role of a new time variable, as it does in Eq. (4), though it is important to keep in mind that L shrinks as t grows; hence, L plays the role of a backward time variable. dX cm = 0 , and these are therefore Note that for X cm = nπ , dt fixed or equilibrium points. Also, notice that as L decreases monotonically in time, the time-dependent coefficient sin(L/2) in Eq. (4) oscillates periodically from positive to negative values. This has the effect of alternately stabilizing or destabilizing any given fixed point. For example, the fixed point at Xcm= 0 (mod 2π) is stable when the coefficient sin(L/2) > 0, and unstable when sin(L/2) < 0. Define

Ω = (0,2π ] ∪ (4π ,6π ] ∪ (8π ,10π ] ∪L ,

Ω = (2π ,4π ] ∪ (6π ,8π ] ∪ (10π ,12π ] ∪L . Then, the stability of a general fixed point X F = nπ , n ∈ Z can be summarized as: C

where γ LV is the surface energy of the liquid/air interface. Since the net horizontal force acting on the droplet is F(x2 ) − F(x1) , the center of mass xcm = ( x1 + x2 ) / 2 moves according to

 stable if L ∈Ω X F = 2kπ  C unstable if L ∈Ω

(5a)

 stable if L ∈Ω X F = 2kπ + π  unstable if L ∈Ω

(5b)

C


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(a)

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(b)

Fig. 4 (a) The blue lines and red lines represent locations of the droplet center of mass that are stable and unstable, respectively. The black arrows represent the velocity of the center of mass ( dX cm / dt ) if it deviates from the stationary point. (b) Potential energy landscape (red denotes high potential and blue low potential). where k is any integer. If the center of mass of the drop lies at a stable fixed point, it will stay there until evaporation reduces its size by a so this fixed point becomes unstable; any perturbation will cause the droplet to move to one of its neighboring fixed points. At which one of the neighboring fixed points the droplet eventually ends up is extremely sensitive to the slightest perturbation of the environment. Fig. 4 (a) plots the slope field of (4) for ε = 0.1 which indicates the direction of movement once the droplet is perturbed from the fixed points. Typical droplet motion predicted using this model with ε = 0.1 is recorded as a video in Supplementary Information (SM2). Another way to visualize the dynamics is to note that the right hand side of (4) can be interpreted as the force due to the potential −sin( L / 2) cos Xcm which depends on both the normalized droplet size and its position. Fig. 4(b) shows the periodic energy landscape. Regions of high or low potential energies are labeled red or blue, respectively. As a drop shrinks, it will move towards regions of lower potential. In our one-dimensional system, this means that it can move to the right or left. Note that this simple one dimensional model would predict a “stick and go” motion, whereas in the experiment the droplet motion is usually smoother even though some sudden fast motion can be observed (see Fig. 2(b)). This can be partly attributed to the three dimensional nature of the droplet where it can be stuck in one direction but ready to go in another direction. Summary and Conclusion We report spontaneous random motion of water droplets on a periodic film-terminated fibrillar structure by a novel mechanism that operates with no external stimulus or long-range gradient. A droplet placed on such a substrate gradually becomes smaller due to evaporation. When the size of the droplet becomes small enough, it overcomes inertia and moves spontaneously and erratically. A drop-

let can move a considerable distance before it finally halts and vanishes by evaporation. This motion is counterintuitive for a number of reasons. First, it occurs in the absence of any overall gradients or external excitation, which often drive droplet motion. Second, surface undulation and compliance usually increase contact angle hysteresis and hence might be expected to work against drop motion. However, these are the very features that are necessary here for drop motion. In our model, the position of the contact line depends on the size of droplet as well as the position of its center of mass. Droplet motion is caused by an imbalance of forces acting on the contact lines on either side of the droplet. These forces are determined by the liquid surface tension, surface properties of the substrate (such as surface energies and stresses) and local elastic deformation. The imbalance arises due to the periodic variation in substrate compliance, which manifests as a periodic variation of apparent contact angle. The direct role of evaporation is to reduce the droplet size, driving the system towards an instability that results in droplet motion. The role of evaporation is to provide a means to change the size of the droplet. Many of the key features in our experiments, such as spontaneous motion and its sensitivity to initial condition can be explained using a simple one-dimensional lattice model in which a 2D evaporating droplet is subjected to an unbalanced (sinusoidal) force with period equal to the spacing between fibrils. The source of this force is the difference in apparent contact angles on the two sides of the drop due to differences in deformability. The motion is damped by a viscous force that is much greater than the inertial force. This simple model explains the observed spontaneous and erratic motion. There are obvious limitations in our 2D model. For example, the contact line can only occupy two points and


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Langmuir hence the drop can only move left or right, whereas in reality, the contact line is irregular, and both it and the drop itself can move in different directions. To capture the irregularity of the contact line, a 3-D model has to be established which will be pursued in the future. Our model also does not account for effect of fluid dynamics, which can alter contact angles and instantaneous driving forces. We believe that these two forgoing effects can account for the fact that in experiments the drop tends not to completely reverse direction. Also, our analysis neglects contact line hysteresis, which is an important damping mechanism. It also imposes the constraint that maximum apparent contact angle change due to substrate deformation must exceed the contact angle hysteresis for motion to occur. However, we believe that our model answers the basic question of why motion occurs in the first place in terms of the stability of fixed points but, because of these simplifications, it cannot be used to quantify observations.

ASSOCIATED CONTENT Supporting Information. Detailed model set-up and simplification process for our 1-D droplet motion model. Experimental video for a droplet spontaneously and erratically moving on our periodic structure. Video that shows the droplet behaviors predicted by our 1-D model. This material is available free of charge via the Internet at http://pubs.acs.org.

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Corresponding Author *[email protected]

ACKNOWLEDGMENT The contributions of TL, NN, ZH, CYH, and AJ were supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under award DEFG02-07ER46463. The research of SHS was supported in part by NSF grants DMS-1513179 and CCF1522054.

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Insert Table of Contents artwork here Visualization of Substrate Uplift Due to the Liquid-Vapor Surface Tension of a Drop

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Spontaneous Droplet Motion on a Periodically Compliant Substrate

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