Subscriber access provided by Fudan University
Article
Interaction of Droplets Separated by an Elastic Film Tianshu Liu, Xuejuan Xu, Nichole K. Nadermann, Zhenping He, Anand Jagota, and Chung-Yuen Hui Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b03600 • Publication Date (Web): 08 Dec 2016 Downloaded from http://pubs.acs.org on December 15, 2016
Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.
Langmuir is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.
Page 1 of 16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
Interaction of Droplets Separated by an Elastic Film Tianshu Liu1, Xuejuan Xu2, Nichole Nadermann3, Zhenping He3, Anand Jagota3, Chung-Yuen Hui1* 1
Field of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853,USA Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA 3 Department of Chemical & Biomolecular Engineering and Bioengineering Program, 111 Research Drive, Lehigh University, Bethlehem, PA 18015, USA 2
Abstract The Laplace pressure of a droplet placed on one side of an elastic thin film can cause significant deformation in the form of a bulge on its opposite side1–3. Here, we show that this deformation can be detected by other droplets suspended on the opposite side of the film leading to interaction between droplets separated by the solid (but deformable) film. The interaction is repulsive when the drops have a large overlap and attractive when they have a small overlap. Thus, if two identical droplets are placed right on top of each other (one on either side of the thin film) they tend to repel each other, eventually reaching an equilibrium configuration where there is a small overlap. This observation can be explained by analyzing the energy landscape of the droplets interacting via an elastically deformed film. We further demonstrate this idea by designing a pattern comprising a big central drop with satellite droplets. This phenomenon can lead to techniques for directed motion of droplets confined to one side of a thin elastic membrane by manipulations on the other side.
Introduction The ability to manipulate and transport droplets is important in a variety of industrial applications such as micro-fluidic devices4–6 and self-cleaning surfaces7–9. Motion of individual droplets on a surface by the Marangoni effect10 typically requires some form of spatially varying surface tension force10 which can be achieved by gradients of geometry11,12 or interfacial energy13. There are various ways to create a gradient in interfacial energy, from chemical14–19 to electrical20 and thermal21–25. However, these mechanisms require careful preparation of the surface or substrate. Recently, Cira et al. have demonstrated that two-component droplets of well-chosen miscible liquids can sense and move in response to the vapor emitted by neighboring droplets26, with no other gradient on the surface. For the phenomena discussed above, the substrate is basically un-deformable and there is little interaction between droplets unless they make contact. A contrasting situation is one in which rigid particles float on a liquid. In this case, particles distort the liquid-vapor interface, leading to long range interactions, a phenomenon dubbed the “Cheerios effect” because of the common observation that toruses of the breakfast cereal by that name tend to agglomerate27. Related phenomena have recently been reported in which rigid particles placed in very soft but solid gels exhibit long-range interaction28. When a liquid drop is placed on a soft solid, recent experiments and theories have shown that capillary force can significantly deform soft materials, such as elastomers and hydrogels29–33. The large deformation caused by liquid drops themselves can provide a driving force for drop transportation. This mechanism was first proposed by Style et al. 34 where they created a gradient of substrate stiffness by changing its thickness. In their experiments, they found the droplets prefer to move toward a softer substrate where the local apparent contact angle is lower. This idea was further pursued by Karpitschka et al. who reported both attraction and repulsion between droplets on a soft elastic substrate, depending on the thickness of the substrate solid35, a reverse Cheerios effect36. To explain their observation, they computed the deformed
ACS Paragon Plus Environment
Langmuir
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
shape of the substrate using a small strain linear elasticity theory and found that the imbalance of surface tension in horizontal direction provides the driving force. High substrate deformation in the forgoing examples is achieved by using compliant materials; it can also be achieved using a relatively stiffer material but with slender aspect, such as a thin film1–3,29,37–40 or a slender rod41,42. In this work, we show that droplets placed on a uniform thin film can sense and move in response to the deformation of the film caused by the droplets on the other side of the film. This interaction mechanism results in droplet motion without chemical modification of surfaces, nor is it necessary to have two-component droplets. The droplets are separated by an impermeable thin elastic film which prevents them from coalescing. Further to demonstrate this mechanism, we designed and experimentally observed a “satellite” droplet pattern where a larger droplet on one side of the film is surrounded by a series of droplets on the other side. A full, large-deformation, analysis of the total system energy landscape is conducted to establish a quantitative basis for droplet interaction and stability. The new mode of droplet interaction we report here can also be intuitively understood as being driven by droplets seeking a softer regions of the substrate. However, unlike the results of Style et al. 34, in which individual droplets move on an externally created stiffness gradient, the phenomenon we report is active only for interaction between two or more droplets where each droplet moves in the effective potential energy landscape created by film deformation caused by all the droplets. This phenomenon could lead to new techniques for directed motion of droplets confined to one side of a thin elastic membrane by manipulations on the other side. In addition to its relevance for droplet fluidics, we believe that the phenomenon we report could potentially be helpful in understanding and studying other processes, such as the role of substrate compliance on behavior of cells, where it is well known that cell motility and fate is strongly dependent on substrate stiffness43–48.
Methods: The samples used in our experiments comprise a circular thin Vinyl Polysiloxane (VPS, Zhermack Elite Double 8, 1:1 mass ratio between base and catalyst) film suspended on a VPS substrate with a hole 8mm in diameter. A schematic cross-sectional view of the sample is shown in Fig. 1. To prepare this sample, a VPS substrate a few millimeters thick with a hole 8mm in diameter was molded in a Plexiglas mold. The VPS film was fabricated by spin-coating liquid VPS onto a Polystyrene-coated glass slide at 2000rpm for 2 minutes. The VPS substrate was placed onto the thin film immediately, and they were cured together at room temperature for over 30 minutes. The thickness of the film was measured by a white-light interferometer (Zegage; Zygo Corporation) to be 20.5µm.
ACS Paragon Plus Environment
Page 2 of 16
Page 3 of 16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
Fig. 1 Schematic side view of the supported-film sample. Both film and support are made of Vinyl Polysiloxane (VPS). The film has a diameter of 8mm and a thickness of 20.5µm.
Experiments: Interaction of two droplets We study the interaction between two droplets placed on different sides of the thin elastic VPS film. A 5 µL de-ionized water (DI) droplet was placed on the bottom side of the film using an air displacement pipette as shown in Fig.2 (a). In this configuration, the film bulges upwards due to the Laplace pressure p of the liquid drop, as shown by Nadermann et al.1. After this, another DI droplet of the same volume was placed on the top side of the film. The position of this droplet was controlled so that the two droplets, to begin with, were practically on top of each other (see Figs. 2(b, d)). Since the two droplets were nearly identical, neglecting gravity, the film became flat everywhere and went back to its undeformed state. However, this equilibrium configuration was not stable; the two droplets started to separate after a few seconds and eventually stopped at a stable configuration with a small overlap (see SI, movies 1,2).
(a)
(b)
(c)
(d) (e) Fig.2 Process of repulsion between two droplets on either side of a VPS film. (a) Using an air displacement pipette, a DI droplet was placed underneath the film, and the film bulged out due to the Laplace pressure P (see insert). (b) Another DI droplet of same volume was placed on the top side of the film, overlapping the bottom droplet. (c) The configuration in (b) was unstable and the two droplets separated. (d) Snapshot of the initial (unstable) overlapped configuration (top view). (e) Snapshot of the stable configuration with a small overlap (top view). When two droplets overlap exactly (Fig. 2b), the out-of-plane deformation of the film caused by one droplet is cancelled by the other. Therefore, the influence of the second droplet is effectively to increase the stiffness of the film. When two droplets are separated, the film again bulges out due to Laplace pressure (Fig. 2c); both droplets are “sensing” the compliant elastic film. According to Style et al.34, droplets tend to move toward a configuration where the substrate is softer. This durotaxis arises because
ACS Paragon Plus Environment
Langmuir
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 4 of 16
a gradient in stiffness translates into a gradient in contact angle, and a gradient in contact angle drives a drop towards the region of smaller apparent contact angle49. So, qualitatively, this argument suggests that the separation process can be understood as droplets seeking a softer place on which to sit.
Energy landscape and stability of the droplet pair To understand our observation quantitatively, we analyze the energy landscape of two droplets that are infinitely long in the out-of-plane direction (Fig. 3). This plane problem is mathematically much more tractable than the fully 3D problem of interacting circular drops. Usually, the results of a plane strain problem differ from those of an axisymmetric or 3D calculation only within constant factors on the order of unity. These droplets lie on the opposite sides of an elastic film clamped at its edges. The initial undeformed width of the film is 2 l0 . We assume the droplets are sufficiently small so we can ignore gravity. Initially, the droplets are right on top of each other (Fig. 3 (a)). In this position, the droplets have identical width 2 cr and the film is completely flat since the Laplace pressure acting on the liquid/film interface exactly cancels as shown in Fig. 3 (a). In the absence of residual stress (due to fabrication), the film should have no elastic strain in the overlap configuration. From this initial configuration, we allow the droplets to separate and compute the total energy of the system as a function of separation (see Fig. 3(b)). Any separation will cause the film to deform and the drops to change their shape. Specifically, the portion of the film (labeled 2 and 6 in Fig. 3 (b)) that is exposed to air on one side and liquid on the other will bulge into circular arcs due to Laplace pressure1,3, while the region of film between the liquid drops (labeled 4 in Fig. 3 (b)) is flat since the pressure is the same on both its sides. Note that the film exterior to the drops (3 in Fig. 3 (b)) also remains flat since we ignore gravity. However, this part of the film is stretched due to the bulging of the film. The extent of the film jointly covered by the two droplets is denoted by 2a, and a / c r serves as a parameter that characterizes the separation between the two droplets. (Note that a / c r = 1 when the two droplets overlap completely.) To compute the elastic energy and the deformation of the film, we assume the film is a membrane with no bending stiffness. (This approximation is accurate if the length
EI / σ 1mm.) To model large
deformation, the film is taken to be an incompressible neo-Hookean solid with shear modulus µ . The strain energy density W of a neo-Hookean membrane subjected to uniform stretch ratio λ is given by W (λ ) =
µh 2
(λ
2
+ λ −2 − 2 )
(1)
where h is the undeformed thickness of the film. To reduce the number of parameters in the problem, we further assume that the surface stresses of the film that is exposed to air and liquid are isotropic, independent of surface strains and are numerically equal to the surface energies γ SV ,γ SL , respectively. We define an elasto-capillary number β as the ratio of liquid-vapor surface energy γ LV to the effective modulus of the film µh . For our system, β is on the order of 0.1.
ACS Paragon Plus Environment
Page 5 of 16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
(a) (b) Fig. 3 (a) Initial configuration of the two drops in which 2 c r is the equilibrium contact width at zero separation. The film has no deformation as the Laplace pressure and surface tension at the contact line from two droplets cancel each other. (b) Configuration of two separated droplets. Part 1 is identical to part 5 and part 2 is identical to part 6 due to symmetry. In the following, we summarize the key steps in our calculation of the total energy, which consists of contributions from the surface and elastic stretching. Details of our analysis are given in the Appendix. The liquid drops (1 in Fig. 3 (b)) and the film (2 in Fig. 3 (b)) deform into circular arcs with unknown radii
r1 and r2 ,
respectively. In addition, the unknown stretch ratios in each region (2,3,4, and 6) are
constant and denoted by λ2 ,λ3 ,λ4 , and ( λ 6 = λ 2 due to symmetry), respectively. For any given values of a and b, one can fully constrain the deformed geometry by specifying r1 , r2 and the angles θ 1 , θ 2 in Fig. 3 (b). As a result, the energy of the system is completely determined by 7 unknowns ( r1 , r2 , θ 1 ,θ 2 , λ 2 , λ 3 , λ 4 ). Specifically, the surface energies of the droplets and the deformed films (part 2 and 3) can be directly obtained from the deformed geometry, they are given by the first four terms in (A12). The elastic energies in part 2, 3 and 4 of the film can be determined using (1), the unknowns λ2 ,λ3 ,λ4 and the undeformed length of these parts, and these energies are given by the last three terms in (A12). To determine these 7 unknowns as a function of a, b and known surface energy densities γ SV ,γ SL , γ LV , we enforce force balance and volume conservation. From the free body diagram of A and B (shown in Fig. 3 (b)), the equilibrium equations can be written as
uuur uuur uuur r ( A) T2( A) + T3(A) + γ LV =0 uuur uuur uuur r (B) T2(B) + T4(B) + γ LV =0
(2) (3)
r r r where T2 , T3 and T4 denote the tensions in parts 2, 3 and 4 of the film respectively; the superscript () denotes that these tensions are to be evaluated at points A and B. These tensions (unit = force/length) are given by
T2 = µ h(λ2 − λ2−3 ) + γ SV + γ SL T3 = µ h(λ3 − λ3−3 ) + 2γ SV T4 = µ h(λ4 − λ4 −3 ) + 2γ SL
ACS Paragon Plus Environment
(4)
Langmuir
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 6 of 16
Equation (4) indicates that the tensions have two contributions, one is due to elastic stretch of the neoHookean membrane51,52 and the rest is from the surface stress of the interface. Since the droplet volume is a constant during separation, we impose the condition V (droplet) = V0
(5)
where V0 is the volume of the liquid drop before deformation. In addition, since the boundary of the film is clamped at the edges, the total undeformed width of the film should be l3
λ3
+
l2
λ2
+
2l0 . This condition is
l4 = l0 2λ4
(6)
where l2 ,l3 and l4 are the deformed length of parts 2,3 and 4 of the film respectively. The fact that the x coordinate of point B is equal to b yields one more equation, that is, the y coordinate of point B must be given by
r1 cosθ1 − r12 − (b + a − r1 sinθ1 )2 = r2 cosθ2 − r22 − (b − a + r2 sinθ2 )2
(7)
The equilibrium equations (2) and (3) provides 4 scalar equations, which, together with (5-7) give 7 equations which allows us to solve numerically for the 7 unknowns r1 , r2 , θ 1 ,θ 2 , λ 2 , λ 3 , λ 4 .
(a)
(b)
Fig. 4 (a) Total energy of the system vs a / cr for different elasto-capillary number β . The local energy minimum is achieved when a / c r ≈ 1.7 7 for β = 0.1 , and 1.75 for β = 1 . (b) Minimum energy configuration for β = 0.1 . The red line is the thin film and the blue line is the surface of the droplets. Next, for a given separation a, we minimize the energy of the system by varying b. Repeating this calculation for different values of a, we obtain the result in Fig. 4 (a). Fig. 4 (a) shows that, once the system is perturbed from the complete overlap case, the total energy starts to decrease indicating that the overlap scenario is unstable. Once separation starts, the two droplets will continue to repel each other until a local energy minimum is achieved, as shown in Fig. 4 (a). That there is a local energy minimum for partial overlap between the two drops explains the experimental observation that there is always a
ACS Paragon Plus Environment
Page 7 of 16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
small overlap between drops on two sides of the film, as shown in Fig. 2 (e). A computed equilibrium configuration for the case of β = 0.1 is shown in Fig. 4 (b). When the film is more compliant i.e. β becomes larger, the energy well will be deeper, but the local energy minimum position only changes slightly. This suggests that for a softer film, it will be easier to observe the separation process but the final equilibrium configuration will not change much. The presence of an energy minimum confirms that interaction between droplets is repulsive for large overlap and attractive for small overlap. It also suggests that motion of a droplet on one side of the membrane, say the lower one in Figure 4(b), can be controlled by directed motion of the droplet on the other side (the upper one in Figure 4(b)). Indeed, the ability of a droplet on one side to “drag along” the droplet on the other side of the membrane is evident in video 2.
Satellite droplet pattern To demonstrate further the mechanism for inter-droplet interaction just described, we show its use to create a “satellite” droplet pattern where small drops surround a large one. In this experiment, a single 1µ L DI droplet was placed on the top side of the film. The sample together with the droplet was then placed on the top of a glass beaker filled with DI water as shown in Fig. 5 (a). We heated the glass beaker to increase the evaporation rate, and small droplets continued to condense on the bottom of the film as shown in SI (movie 3). As these small droplets coalesced to form bigger droplets under the pre-placed droplet on the top side of the film, they were repelled to the boundary of the top droplet. This process continued until a ring of droplets formed around the top droplet as shown in Fig. 5 (b).
(b)
(a)
Fig. 5 (a) Schematic of the experimental setup. The glass beaker was heated to increase the evaporation rate, and the sample together with a droplet was held by two glass slides on top of the glass beaker. Small droplets condensed on the bottom of the film due to evaporation (not shown in figure) (b) Satellite droplet pattern: four “satellite” droplets underneath the film formed around the top droplet.
Conclusion and Discussion We report a short range interaction between droplets sitting on different sides of a thin elastic film. When droplets overlap completely, the system is in an unstable state, and a small perturbation leads to
ACS Paragon Plus Environment
Langmuir
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
rapid separation between the two droplets. The process stops when there is a small overlap between the two droplets (about 1/3 of droplet size). The phenomenon is explained quantitatively by computing the surface and elastic energy of the system and finding the configuration where the total energy is minimized. This new interaction mechanism can be potentially useful for controlled motion of droplets on one side of a barrier by manipulations on the other side. It also demonstrates non-trivial interactions between liquidvapor surface tension and elasticity of the compliant substrate. It should be noted that our plane strain model assumes that the droplets are infinitely long in the out of plane direction, which is not true in most cases. Indeed, if the two droplets were completely separated, the plane strain model, in the absence of gravity, predicts that the film stretches uniformly and remains flat outside the droplets. This is not the case for 3D droplets. Such non-uniform deformation would result in spatially varying apparent contact angle leading to more complex interaction between droplets. This interaction is short range since we did not observe any interaction between droplets that are placed more than one diameter away from each other. The deformation of the film caused by two 3-D droplets and how their interaction depends on the spacing between them will be studied in a future work.
ASSOCIATED CONTENT Supporting information Top view video for the separation process of two overlapped droplets. Side view video for the transition from repulsion to attraction between two droplets separated by a thin elastic film. Experimental video of the formation of “satellite” droplet pattern. Acknowledgement This work is supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-FG02-07ER46463. We would also like to acknowledge the assistance of Mr. Theodore Lee of Liberty High School in Bethlehem, PA, with the experiments. We are grateful to Prof. Pedro Reis of the Massachusetts Institute of Technology for introducing us to the elastomeric material used to fabricate our samples.
ACS Paragon Plus Environment
Page 8 of 16
Page 9 of 16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
References (1) (2) (3) (4) (5) (6) (7) (8)
(9)
(10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20)
(21) (22) (23) (24)
Nadermann, N.; Hui, C.-Y.; Jagota, A. Solid Surface Tension Measured by a Liquid Drop under a Solid Film. Proc. Natl. Acad. Sci. U. S. A. 2013, 110 (26), 10541–10545. Schulman, R. D.; Dalnoki-Veress, K. Liquid Droplets on a Highly Deformable Membrane. Phys. Rev. Lett. 2015, 115 (20), 206101. Hui, C.-Y.; Jagota, A. Planar Equilibrium Shapes of a Liquid Drop on a Membrane. Soft Matter 2015. Squires, T. M.; Quake, S. R. Microfluidics: Fluid Physics at the Nanoliter Scale. Rev. Mod. Phys. 2005, 77 (3), 977–1026. Srinivasarao, M. Three-Dimensionally Ordered Array of Air Bubbles in a Polymer Film. Science (80-. ). 2001, 292 (5514), 79–83. Pollack, M. G.; Shenderov, A. D.; Fair, R. B. Electrowetting-Based Actuation of Droplets for Integrated Microfluidics. Lab Chip 2002, 2 (2), 96–101. Zhang, X.; Shi, F.; Niu, J.; Jiang, Y.; Wang, Z. Superhydrophobic Surfaces: From Structural Control to Functional Application. J. Mater. Chem. 2008, 18 (6), 621–633. Kim, P.; Wong, T.-S.; Alvarenga, J.; Kreder, M. J.; Adorno-Martinez, W. E.; Aizenberg, J. LiquidInfused Nanostructured Surfaces with Extreme Anti-Ice and Anti-Frost Performance. ACS Nano 2012, 6 (8), 6569–6577. Schellenberger, F.; Xie, J.; Encinas, N.; Hardy, A.; Klapper, M.; Papadopoulos, P.; Butt, H.-J.; Vollmer, D.; Gennes, P. G. de; Bonn, D.; et al. Direct Observation of Drops on Slippery Lubricant-Infused Surfaces. Soft Matter 2015, 11 (38), 7617–7626. Edwards, D. A.; Brenner, H.; Wasan, D. T. Interfacial Transport Processes and Rheology; Butterworth-Heinemann, 1991. Lv, C.; Chen, C.; Chuang, Y.-C.; Tseng, F.-G.; Yin, Y.; Grey, F.; Zheng, Q. Substrate Curvature Gradient Drives Rapid Droplet Motion. Phys. Rev. Lett. 2014, 113 (2), 26101. Prakash, M.; Quéré, D.; Bush, J. W. M. Surface Tension Transport of Prey by Feeding Shorebirds: The Capillary Ratchet. Science 2008, 320 (5878), 931–934. Chaudhury, M. K.; Chakrabarti, A.; Daniel, S. Generation of Motion of Drops with Interfacial Contact. Langmuir 2015, 31 (34), 9266–9281. Chaudhury, M. K.; Whitesides, G. M. How to Make Water Run Uphill. Science 1992, 256 (5063), 1539–1541. Ichimura, K. Light-Driven Motion of Liquids on a Photoresponsive Surface. Science (80-. ). 2000, 288 (5471), 1624–1626. Zheng, Y.; Bai, H.; Huang, Z.; Tian, X.; Nie, F.-Q.; Zhao, Y.; Zhai, J.; Jiang, L. Directional Water Collection on Wetted Spider Silk. Nature 2010, 463 (7281), 640–643. Stone, H. A.; Stroock, A. D.; Ajdari, A. ENGINEERING FLOWS IN SMALL DEVICES. Annu. Rev. Fluid Mech. 2004, 36 (1), 381–411. Sumino, Y.; Magome, N.; Hamada, T.; Yoshikawa, K. Self-Running Droplet: Emergence of Regular Motion from Nonequilibrium Noise. Phys. Rev. Lett. 2005, 94 (6), 68301. Daniel, S.; Sircar, S.; Gliem, J.; Chaudhury, M. K. Ratcheting Motion of Liquid Drops on Gradient Surfaces †. Langmuir 2004, 20 (10), 4085–4092. Gallardo, B. S.; Gupta, V. K.; Eagerton, F. D.; Jong, L. I.; Craig, V. S.; Shah, R. R.; Abbott, N. L. Electrochemical Principles for Active Control of Liquids on Submillimeter Scales. Science (80-. ). 1999, 283 (5398), 57–60. Cazabat, A. M.; Heslot, F.; Troian, S. M.; Carles, P. Fingering Instability of Thin Spreading Films Driven by Temperature Gradients. Nature 1990, 346 (6287), 824–826. Daniel, S.; Chaudhury, M. K.; Chen, J. C. Fast Drop Movements Resulting from the Phase Change on a Gradient Surface. Science 2001, 291 (5504), 633–636. Lazar, P.; Riegler, H. Reversible Self-Propelled Droplet Movement: A New Driving Mechanism. Phys. Rev. Lett. 2005, 95 (13), 136103. Brzoska, J. B.; Brochard-Wyart, F.; Rondelez, F. Motions of Droplets on Hydrophobic Model
ACS Paragon Plus Environment
Langmuir
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
(25) (26) (27) (28)
(29) (30)
(31) (32) (33) (34)
(35)
(36) (37) (38) (39)
(40) (41) (42)
(43)
(44)
(45)
Page 10 of 16
Surfaces Induced by Thermal Gradients. Langmuir 1993, 9 (8), 2220–2224. Lagubeau, G.; Le Merrer, M.; Clanet, C.; Quéré, D. Leidenfrost on a Ratchet. Nat. Phys. 2011, 7 (5), 395–398. Cira, N. J.; Benusiglio, A.; Prakash, M. Vapour-Mediated Sensing and Motility in TwoComponent Droplets. Nature 2015, 519 (7544), 446–450. Vella, D.; Mahadevan, L. The “Cheerios Effect.” Am. J. Phys. 2005, 73 (9), 817. Chakrabarti, A.; Chaudhury, M. K. Surface Folding-Induced Attraction and Motion of Particles in a Soft Elastic Gel: Cooperative Effects of Surface Tension, Elasticity, and Gravity. Langmuir 2013, 29 (50), 15543–15550. Jerison, E. R.; Xu, Y.; Wilen, L. A.; Dufresne, E. R. Deformation of an Elastic Substrate by a Three-Phase Contact Line. Phys. Rev. Lett. 2011, 106 (18), 186103. Style, R. W.; Boltyanskiy, R.; Che, Y.; Wettlaufer, J. S.; Wilen, L. A.; Dufresne, E. R. Universal Deformation of Soft Substrates near a Contact Line and the Direct Measurement of Solid Surface Stresses. Phys. Rev. Lett. 2013, 110 (6), 66103. Hui, C.-Y.; Jagota, A. Deformation near a Liquid Contact Line on an Elastic Substrate. Proc. R. Soc. A Math. Phys. Eng. Sci. 2014, 470 (2167), 20140085–20140085. Limat, L. Straight Contact Lines on a Soft, Incompressible Solid. Eur. Phys. J. E. Soft Matter 2012, 35 (12), 9811. Lubbers, L. A.; Weijs, J. H.; Botto, L.; Das, S.; Andreotti, B.; Snoeijer, J. H. Drops on Soft Solids: Free Energy and Double Transition of Contact Angles. J. Fluid Mech. 2014, 747, R1. Style, R. W.; Che, Y.; Park, S. J.; Weon, B. M.; Je, J. H.; Hyland, C.; German, G. K.; Power, M. P.; Wilen, L. A.; Wettlaufer, J. S.; et al. Patterning Droplets with Durotaxis. Proc. Natl. Acad. Sci. U. S. A. 2013, 110 (31), 12541–12544. Karpitschka, S.; Pandey, A.; Lubbers, L. A.; Weijs, J. H.; Botto, L.; Das, S.; Andreotti, B.; Snoeijer, J. H. Liquid Drops Attract or Repel by the Inverted Cheerios Effect. Proc. Natl. Acad. Sci. U. S. A. 2016, 113 (27), 7403–7407. Jagota, A. Role Reversal: Liquid "Cheerios" on a Solid Sense Each Other. Proc. Natl. Acad. Sci. U. S. A. 2016, 113 (27), 7294–7295. Huang, J.; Juszkiewicz, M.; de Jeu, W. H.; Cerda, E.; Emrick, T.; Menon, N.; Russell, T. P. Capillary Wrinkling of Floating Thin Polymer Films. Science (80-. ). 2007, 317 (5838). Lagubeau, G.; Fontelos, M. A.; Josserand, C.; Maurel, A.; Pagneux, V.; Petitjeans, P. Flower Patterns in Drop Impact on Thin Liquid Films. Phys. Rev. Lett. 2010, 105 (18), 184503. Schroll, R. D.; Adda-Bedia, M.; Cerda, E.; Huang, J.; Menon, N.; Russell, T. P.; Toga, K. B.; Vella, D.; Davidovitch, B. Capillary Deformations of Bendable Films. Phys. Rev. Lett. 2013, 111 (1), 14301. Py, C.; Reverdy, P.; Doppler, L.; Bico, J.; Roman, B.; Baroud, C. N. Capillary Origami: Spontaneous Wrapping of a Droplet with an Elastic Sheet. Phys. Rev. Lett. 2007, 98 (15), 156103. Mora, S.; Maurini, C.; Phou, T.; Fromental, J.-M.; Audoly, B.; Pomeau, Y. Solid Drops: Large Capillary Deformations of Immersed Elastic Rods. Phys. Rev. Lett. 2013, 111 (11), 114301. Roman, B.; Bico, J.; Adamson A W, G. A. P.; de Gennes P G, B.-W. F. Q. D.; J, M. M.; H, M. C. H. and H. C.; L, A. T. and R. M.; Mayer T M, de B. M. P. S. N. D. C. P. J. and M. T. A.; van Spengen W M, P. R. and D. W. I.; Raccurt O, T. F. A. d’Avitaya F. and V. T.; et al. ElastoCapillarity: Deforming an Elastic Structure with a Liquid Droplet. J. Phys. Condens. Matter 2010, 22 (49), 493101. Choi, Y. S.; Vincent, L. G.; Lee, A. R.; Kretchmer, K. C.; Chirasatitsin, S.; Dobke, M. K.; Engler, A. J. The Alignment and Fusion Assembly of Adipose-Derived Stem Cells on Mechanically Patterned Matrices. Biomaterials 2012, 33 (29), 6943–6951. Trichet, L.; Le Digabel, J.; Hawkins, R. J.; Vedula, S. R. K.; Gupta, M.; Ribrault, C.; Hersen, P.; Voituriez, R.; Ladoux, B. Evidence of a Large-Scale Mechanosensing Mechanism for Cellular Adaptation to Substrate Stiffness. Proc. Natl. Acad. Sci. U. S. A. 2012, 109 (18), 6933–6938. Discher, D. E.; Janmey, P.; Wang, Y. Tissue Cells Feel and Respond to the Stiffness of Their
ACS Paragon Plus Environment
Page 11 of 16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
(46) (47) (48) (49) (50) (51)
(52)
Substrate. Science (80-. ). 2005, 310 (5751). Lo, C. M.; Wang, H. B.; Dembo, M.; Wang, Y. L. Cell Movement Is Guided by the Rigidity of the Substrate. Biophys. J. 2000, 79 (1), 144–152. Zhang, R.; Ni, L.; Jin, Z.; Li, J.; Jin, F. Bacteria Slingshot More on Soft Surfaces. Nat. Commun. 2014, 5 (May), 5541. Guo, W.; Frey, M. T.; Burnham, N. A.; Wang, Y. Substrate Rigidity Regulates the Formation and Maintenance of Tissues. Biophys. J. 2006, 90 (6), 2213–2220. de Gennes, P.-G.; Brochard-Wyart, F.; Quéré, D. Capillarity and Wetting Phenomena; Springer New York: New York, NY, 2004. Style, R. W.; Jagota, A.; Hui, C.-Y.; Dufresne, E. R. Elastocapillarity: Surface Tension and the Mechanics of Soft Solids. 2016. Srivastava, A.; Hui, C.-Y. Large Deformation Contact Mechanics of a Pressurized Long Rectangular Membrane. II. Adhesive Contact. Proc. R. Soc. London A Math. Phys. Eng. Sci. 2013, 469 (2160). Srivastava, A.; Hui, C.-Y. Large Deformation Contact Mechanics of Long Rectangular Membranes. I. Adhesionless Contact. Proc. R. Soc. London A Math. Phys. Eng. Sci. 2013, 469 (2160).
ACS Paragon Plus Environment
Langmuir
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 12 of 16
Appendix – Energy landscape calculation To simplify the geometry, we consider a 2-D system where both the droplets and film are assumed to be infinitely long in the out of plane direction. The droplets are assumed to be partially separated as shown in Fig. 3 (b). As gravity is neglected, the pressure inside the drop is a constant. Hence, part 1 and 2 are both circular arcs51,52. Part 4 has to be flat as the pressure from different sides cancels each other. The origin of our coordinate system is shown in Fig. 3 (a). Note, after deformation, the material point at the origin remain fixed and is located at the midpoint of part 4. For any given value of a and b, one can fully constrain the geometry by determining the radius of curvature for part 1 and 2 ( r1 , r2 ) and the angles θ 1 , θ 2 . The amount of elastic stretch λ 2 , λ 3 , λ 4 needs to be determined separately. The total energy of the system can be calculated once these 7 unknowns ( r1 , r2 , θ 1 , θ 2 , λ 2 , λ 3 , λ 4 ) are known. We select c r as the characteristic length scale for this problem and normalize all the lengths by it, i.e., r1 =
l r1 r , r2 = 2 , l0 = 0 . The force per unit out of plane length is normalized by γ LV . The condition of force cr cr cr
balance at point A can be written as:
σLV cosθ1 +T2 cosθ2 =T3
(A1)
σLV sinθ1 =T2 sinθ2
(A2)
γ where σ LV = σ LV = 1 , T2 = 1 (λ2 − λ2 −3 ) + σ SV + σ SL and T3 = 1 (λ 3 − λ 3 − 3 ) + 2σ SV . Here β = LV is the β β µh γ LV elasto-capillary number for this system, σ L V , σ S V and σ SL are the liquid-vapor, solid-vapor and solidliquid interface tension. In our system, interface surface stress is assumed to be isotropic, independent of surface strain and numerically equal to surface energy.
Using the coordinate system defined in Fig. 3(a), the position of O 1 and O 2 can be determined by geometry, they are: O1 : (−a + r1 sinθ1 , r1 cos θ1 )
(A3)
O2 : (a − r2 sinθ2 , r2 cos θ2 )
Point B is the intersection between arcs 1 and 2. The x coordinate of B is b while its y coordinate
yB can
be determined from both arc 1 and arc 2. Using (A3), we find 2
2
y B = r1 cos θ 1 − r1 − (b + a − r1 sin θ 1 )2 = r2 cos θ 2 − r2 − (b − a + r2 sin θ 2 )2
ACS Paragon Plus Environment
(A4)
Page 13 of 16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
Fig. A1 Parameter θ 1* and θ 2* can be used to determine the parameter equation of arc 1 and 2. The equation of the circular arc 1 connecting A and B can be determined by the angle θ 1* (see Fig. A1):
x1 (θ1* ) = −a + r1 sinθ1 + r1 sinθ1* , y1 (θ1* ) = r1 cosθ1 − r1 cosθ1* , θ1* ∈ (−θ1 ,sin−1 (
b + a − r1 sinθ1 )) r1
(A5)
Similarly, the equation of arc 2 can described by the angle θ 2* (see Fig. A1)
b − a + r2 sinθ2 ),θ2 ) (A6) r2 Using these parametric equations, we integrate to obtain the volume of the droplet. This volume is conserved through the separation process: x2 (θ2* ) = a − r2 sinθ2 + r2 sinθ2* , y2 (θ2* ) = r2 cosθ2 − r2 cosθ2* , θ2* ∈(sin−1 (
−∫ y1dx1 − ∫ y2dx2 = V0
(A7)
where V0 = V02 is the normalized volume of a droplet. cr
Next we enforce force balance at B. The forces acting there are shown in Fig. 3b: uuur uuuuur uuur uuur γ (B) uuur T (B) uuur T4(B) (B) ( B ) = T4 e4 , LV = eσ , 2 = T2 e2(B)
γ LV
γ LV
γ LV
(A8)
uuur uuur uuur (B) (B) (B) e e e where 4 , σ and 2 are unit vectors pointing in the direction of these forces respectively. They are given by
uuur e4(B) =
r 1 y − r cosθ uuur 1 r cosθ − y −b uuu 1 2 2 B , eσ(B) = B 1 , e2(B) = 2 2 −y r r θ θ r sin − a − b b − a + r sin 1 2 B 1 2 2 1 yB + b 1
2
(
where yB = r2 cosθ2 − r2 − b − a + r2 sinθ2
)
(A9a)
2
is the y coordinate of point B. The magnitude of the forces,
T2,T4 are: T2 =
1
β
(λ2 − λ2 −3 ) + σ SV + σ SL , T4 =
1
β
(λ 4 − λ 4 − 3 ) + 2σ SL ,
Using (A8-A9) and (2-3), the force balance at point B is
ACS Paragon Plus Environment
(A9b)
Langmuir
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 14 of 16
−b 1 yB − r1 cosθ1 T2 r2 cosθ2 − yB 0 + = + 2 b − a + r sinθ 0 2 −y r1 r sinθ − a − b r 2 B 1 2 2 1 yB + b T4
(A10a,b)
Finally, the total undeformed length of the system should remain as a constant 2 l0 , and this constraint is (6) in the main text and can be expanded as:
−1 b − a + r2 sinθ2 θ2 − sin r2 2 r2 b 2 + yB l0 − a + + = l0
λ3
λ2
λ4
(A11)
Equations (A1), (A2), (A4), (A7), (A10a,b), (A11) allows us to determine r1 , r2 ,θ1 ,θ2 , λ2 , λ3 , λ4 for any given a , b . Because of symmetry, we consider the energy stored in half of the film. This energy is: b − a + r2 sinθ2 2 2 Γ = 2 l0 − a σ SV + r2 θ2 − sin−1 σ SL + σ SV + 2 yb + b σ SL r2 b + a − r1 sinθ1 l −a 1 2 1 + sin−1 + θ1 r1 + λ3 + 2 − 2 0 2β λ3 r1 λ3
(
)
(
)
(A12)
−1 b − a + r2 sinθ 2 θ 2 − sin r2 2 r2 b 2 + yB 1 2 1 1 2 1 + + λ4 + 2 − 2 λ2 + 2 − 2 2β 2β λ4 λ2 λ4 λ2
First, we compute the energy for a fixed a and varying b . For example, if we set a = 1.5 , β = 0.1 , the energy landscape is shown in Fig. A2:
ACS Paragon Plus Environment
Page 15 of 16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
Fig. A2 Energy landscape for a = 1.5 , β = 0.1 The lowest energy is defined as the equilibrium energy for a = 1.5 . We repeat this process for different ato obtain Fig. 4(a) in the main paper.
ACS Paragon Plus Environment
Langmuir
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
TOC Graphic
ACS Paragon Plus Environment
Page 16 of 16
