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A New Drop Fluidics Enabled by Magnetic Field Mediated Elasto-Capillary Transduction

This research introduces a new drop fluidics, which uses a deformable and stretchable elastomeric film as the platform, instead of the commonly used rigid supports. Such a soft film impregnated with magnetic particles can be modulated with an external electromagnetic field that produces a vast array of topographical landscapes with varying surface curvature, which, in conjunction with capillarity, can direct and control motion of water droplets efficiently and accurately. When a thin layer of oil is present on this film that is deformed locally, a centrosymmetric wedge is formed. A water droplet placed on this oil laden film becomes asymmetrically deformed thus producing a gradient of Laplace pressure within the droplet setting it to motion. A simple theory is presented that accounts for the droplet speed in terms of such geometric variables as the volume of the droplet and the thickness of the oil film covering the soft elastomeric film, as well as such material variables as the viscosity of the oil and interfacial tension of the oil-water interfaces. Following the verification of the theoretical result using well-controlled model systems, we demonstrate how the electromagnetically controlled elasto-capillary force can be used to manipulate the motion of single and/or multiple droplets on the surface of the elastomeric film and how such elementary operations as drop fusion and thermally addressed chemical transformation can be carried out in aqueous droplets. It is expected that the resulting drop fluidics would be suitable for digital control of drop motion by simply switching on and off the electromagnetic fields applied at different positions underneath the elastomeric film. We anticipate that this method of directing and manipulating water droplets is poised for its applications in various biochemical reaction engineering, an example of which is Polymerase Chain Reaction (PCR).


INTRODUCTION
Control of the motion 1-4 of small amounts of liquids at meso and macro scales is fundamental to the development of various types of fluidic devices that are of importance in both thermal and water management technologies. In conventional microfluidics [5][6][7][8][9] , flow is controlled in interconnected channels for which an external pressure is applied. By contrast, in various types of discrete flow control 1-4, 10-15 , a curvature gradient is imposed in a droplet that in turn induces a Laplace pressure gradient in it, thus causing it to move. The velocity of the drop is dictated from the interplay of an unbalanced Laplace force and the viscous drag, so that it is proportional to the capillary velocity (ratio of surface tension to viscosity) of the drop. When the viscous force is insignificant, the excess surface free energy of the drop can be converted to its kinetic energy, which has given rise to a plethora of reports related to the jumping [16][17][18][19][20] of droplets. Drop propulsion mediated either by electro wetting 21,22 or dielectrophoretic 23 forces have also been effective in several drop transport processes. A unique situation arises when drops growing on a surface due to the condensation of vapor coalesce with each other, in which the coalesced drop exhibits a diffusive 24 motion. It was found recently 25 that when an absorbing boundary is present to remove these diffusive droplets, a long range directed motion of drops can be generated without the use of a continuous gradient of surface energy. One significant impediment of the droplet motion, however, is defects that pin the contact line of a drop. However, either an external vibration 26,27 or coalescence 3 of several drops have been found to overcome the pinning forces. On a surface with finite wetting hysteresis ratcheting motion [27][28][29][30][31][32][33][34] of drops can also be created either by using an asymmetry of surface in conjunction with an external symmetric vibration or with a homogeneous surface assisted by an asymmetric vibration 32,33 . In spite of these developments, drop fluidic technologies require extensive instrumentations that increase its cost and/or it suffers from the drawback that the surface becomes eventually contaminated that prevents its repeated use. What would be ideal is a drop fluidic device that would be inexpensive to manufacture, the speed of operation would be relatively fast, and its surface would be free from contamination so that it can be used repeatedly.
In a drop fluidic device, it is virtually impossible to avoid contamination if the drop makes contact with the substrate. In certain cases, it might be possible to avoid contamination if the components present in a drop do not preferentially adsorb on the supporting surface, as we observed previously 33 with a drop containing silicone glycol surfactant. In some cases, it has been possible to immerse the droplet in oil 23 , which avoids contamination. However, the mechanisms underlying such a drop propulsion require elaborate micro-fabrication of the substrate. Leidenfrost effect induced transport of drop 35 , in which a vapor layer separates the drop from the substrate, may also be sought to solve the contamination issue; however, such a method is usually suitable for high temperature applications, which has been exemplified in the synthesis of nanoparticles for which the thermal condition in the drop is exactly what is needed 36 .
Precise methods, nevertheless, need to be developed to stop and actuate a drop and to do so at low temperature 37 before such a method finds wider applications. Recently, some promising developments [38][39][40] in actuating drops via magnetic field have also taken place, by impregnating magnetic particles in an oil with which water drops exhibit very low hysteresis, or coating the drop with magnetic particles yielding similar effect.
The platform used in all the drop fluidic devices is, however, static, the efficacy of which is ultimately limited by the spatial density of the complex array of electronic circuitries. In order to circumvent such a limitation, we were inspired by a question: what if the platform of the drop fluidic device is not static but it is one in which the landscape can dynamically evolve? To illustrate this point, let us consider a platform made of elastically deformable film embedded with magnetic particles arranged in a square grid, where each or few of the particle(s) can be independently addressed by an external magnetic field. As with such a square matrix array of particles, the local deformations of the film are amenable to a Boolean on/off sequence, an extremely large array of curvature induced energy landscapes can be produced on the film. If, now, a mechanism is designed with which a liquid drop would sense such a landscape and be disposed to movement, vast possibilities of the direction of motion of a single, binary and multiple drops could be designed on the surface.
What we describe here is our first set of results demonstrating the feasibility of the above concept by using a magnetically addressable elasto-capillary effect that results from the asymmetric deformation of the water droplet on the surface of an oil laden elastomeric film. The interfacial thermodynamic conditions of all the interfaces here are such that the water drop does not adhere to the elastomer. The oil also spreads completely on the surface of water, thereby ensuring that the water drop is softly confined between the air-oil and solid-oil interfaces. As soon as the elastomeric film is deformed to a circular wedge, the drop becomes subjected to an unbalanced Laplace force prompting it to move towards the center of the deformed region where the unbalanced capillary force vanishes. Now by changing the landscape of the film, the direction of the motion of the drop can be diverted, thereby creating enumerable gaits of droplet motion.

Mechanism of Droplet Propulsion
As indicated above, the fundamental method of drop propulsion is to create a differential curvature on two opposite sides of a drop that propels it towards the direction where the curvature differential would be minimal. Such a curvature gradient can be produced with a surface displaying a gradient of wettability 1 , placing the drop on a conical wire 41,42 or in a geometrical wedge 12 . The physics underlying all these droplet propulsion mechanisms has been extensively reviewed in recent literature 1,12 . Our starting point is also to create a curvature differential in a drop placed in a wedge 12 , but in which one of the surfaces is a deformable air-oil interface.
Let us consider a water drop interacting with a solid surface, through a third phase, which in our case is an immiscible oil. The stability of the thin oil film intervening the water drop and the solid support is determined by the long range electrodynamic fluctuation forces. While the electrodynamic fluctuations at all possible spectral ranges contribute to interactions in condensed phases, two significant contributors to the net interaction arise from the excitations of the media in the ultraviolet and microwave frequencies. If the dielectric susceptibilities of the three phases are either in ascending or descending order (i.e. frequencies, a repulsion is ensured between water drop and the substrate, so the oil film remains stable against all perturbations. Although, it is quite possible to select a solid substrate and an oil based on their electrodynamic spectra to ensure such a stability of the oil film, the following simple thermodynamic method suffices for that purpose 43  ensuring that a thin oil film remains stable between a drop of water and the substrate, which in our case is either a Trimethylsiloxy Terminated Polydimethylsiloxane (DMS-T22) grafted silicon wafer or a Polydimethylsiloxane (PDMS) elastomer. This non-adherence of the droplet to the substrate is also beneficial when it is desired that the droplet is removed from the oil laden substrate at any stage of a specific drop fluidic operation. As shown in Figure 1, the entire drop can be removed into a capillary tube without any trace of it remaining on the substrate, which would be difficult to perform if the drop adheres to the substrate. In this latter case, a smaller droplet would remain behind the substrate as a result of the pinch off instability of the parent droplet during the detachment process. In addition to the fact that the drop does not adhere to the substrate, another fortunate situation with all the oils used in our experiments is that they spread readily on the surface of water. Taken together, we have a situation in which a water droplet remains sandwiched between the air-oil and the solid-oil interfaces. The Laplace pressure exerted by the film, the tension of which is composed of the interfacial tensions of the air/oil and oil/water interfaces, press the water drop downward, which is counteracted by an equivalent Laplace force generated within the drop due to its own deformation.

A drop under soft confinement
A droplet sandwiched between air and the solid (Figure 1) deforms against the solid support with a contact width b, the value of which is determined by the balance between the total Laplace pressure exerted by the thin film above the drop and the spring action of the drop itself. In order to make an estimate of b, we draw inspiration from the Hertzian contact mechanics 44 , according to which, the width of the contact deformation of a sphere (radius R and modulus E) on a flat surface due to a normal Figure 1. A) Schematic of a water droplet on a DMS-T22 grafted silicon wafer through a film of oil in which the water drop does not contact with the support. The oil spreads on water drop as well, causing the drop to be deformed by the Laplace pressure generated by the curved film above the droplet B) when the support is tilted, the left-right symmetry of the Laplace pressure (as in figure A) is broken. A gradient of this pressure propels the drop from the thinner towards the thicker part of the oil film C) A water drop sandwiched between the air-oil interface (a poly phenyl methyl functional silicone oil ) and a solid substrate can be removed by capillary action in just 0.3 seconds. The region in the oil, from which the water drop was recovered, shows an indentation upon the removal of the drop as it has not fully relaxed due to viscous slowing down.
force F N is . For our problem to be dimensionally similar to that of Hertz, the normal force is to be expressed as the total vertical Laplace force: h Rh a   that deforms the water drop having an equivalent modulus as . / R E ow    F is the film tension that is contributed by the air/oil and oil/water interfaces. All the geometrical variables of this problem are shown in Figure 1. We thus obtain the following expression for b: At this juncture, a question might be asked regarding the importance of the role of gravity in the deformation of the water drop. The explicit condition for neglecting gravity is that the drop must be smaller than the capillary length and that this length becomes bigger as the density difference gets smaller. The effect of gravity can thus be reduced by using oils of specific gravities (0.97 to 1.05) close to that of water. As most part of the water drop is immersed in oil, it experiences negligible effect of gravity because of very small density contrast. However, the part of the drop above the base of the air/oil interfaces may, conceivably, be deformed by gravity. Thankfully, however, the film tension that prevents this gravity induced deformation is composed of two interfaces (oil/water and air/oil). In any event, we measured the radius of curvature of the spherical cap of the drop above the base of the oil, and found it to be very close (see Supporting Information, Figure S1) to that of the deposited droplet (11L to 55 L) in spherical shape approximation. These observations thus strengthen the assumption underlying the derivation of eq 2, in that the deformation of the drop against the rigid support is mainly due to Laplace force exerted by the spherical cap.  air-silicone oil interface show that the width of the base (b) contact diameter increases with the size of the drop (A), but it decreases with the thickness h of the oil above the drop (B). The quantitative data are summarized in figure C. D. The data presented in figure C are processed according to eq 2 by plotting the width (b) of contact as a function of In order to verify the validity of eq 2, we used a polydimethylsiloxane grafted (5 nm thick) silicon wafer as the substrate that was covered with a silicone oil of density 0.95. Water drops of volume ranging from 3 L to 70 µL were deposited on the flat silicon wafer by varying the thickness (h) of the oil between air and the substrate. The results summarized in Figure 2 show that the width of the contact deformation of the drop and the substrate decreases with h, but increases with R as is predicted by eq 2 qualitatively. When all the data are rescaled by defining an equivalent drop radius as cluster around each other that is in compliance with eq 2. Even though the thinnest film deviated somewhat from the rest, we point out that the slope of the b-R* plot differs from the rest by only 10%.

Motion of a Squeezed Drop in a Wedge Geometry.
Let us now turn our attention to the case when the silicon wafer is placed at an angle with respect to the air-oil interface. In this case, the drop is deformed asymmetrically that, in turn, exerts an unbalanced Laplace force to the drop. The Laplace pressure difference between the left (P 1 ) and right (P 2 ) sides of the droplet is approximately given by the following equation: here, h 1 and h 2 are the depths of the oil layer on either side of the drop as shown in Figure 1. As the thickness h varies linearly with distance x, eq 3 can also be written as follows: here, n is the unit vector. This pressure acts on the area projected by the drop normal to the direction of motion, thus the total force experienced by the drop is ~ (P 1 -P 2 )R 2 , or Substitution of the expression for b (eq 2) in eq 5 leads to an expression for the total capillary motor force, which can be equated to a Stokes like drag force ~

RV
 to obtain the velocity (V) of the drop. We ignore any contact line drag 45 Eq 6 predicts that the droplet velocity increases with R and decreases with h with exponents greater than unity; both these predictions have been verified experimentally (see below).

Figure 3. A).
A 10 l size water drop sandwiched between a PDMS grafted silicon wafer and an air-oil (DMS-T12) interface moves right (see up and down panels, e.g. A to A*, etc). B). By contrast, even a larger (20 l) water drop moves very slowly when it is completely submerged in the oil. C). The importance of the Laplace pressure induced compression of a water drop is demonstrated here using two oil layers (DMS-T12 on top of a high temperature silicone oil), which are immiscible with each other (Note: high temperature silicone oil is the commercial name of the oil meaning it is stable at high temperature). Here, even though the drop is in between the silicon wafer and the interface produced by two different oils, the interfacial tension of the oil is negligible compared to the surface tension of an air-oil interface. Because of very low Laplace pressure, a 20 l size water drop moves much more slowly compared to that shown in Figure 3A. The wedge angle is 4°in all the figures. Figure 3 summarizes the basic observations of the motion of a water droplet placed at the wedge formed by the air-oil and solid-oil interfaces. The solid support used here is a silicon wafer, which is modified by grafting an ultra-thin (5 nm) film of polydimethylsiloxane, the surface energy of which is ~ 24 mN/m. A rectangular quartz cell was partially filled with a silicone oil, following which the PDMS grafted silicon wafer was placed inside it on the top of a solid support. The quartz cell containing the silicon wafer was rotated about two orthogonal axes in such a way that it was as flat as possible in one direction, but was slightly tilted in the orthogonal direction. In these experiments, the air-oil interface remains always flat, while the wedge angle is modified by tilting the silicon wafer relative to the air-oil interface. As a drop of water is placed atop the silicon wafer through oil, it moves towards the wider gap of the wedge (Movie1).
Gravity has negligible effect in this type of droplet motion that can be easily shown by an experiment in which it moves very slowly when it is completely immersed in the oil (Movie2).
By contrast, the importance of the surface tension of the oil, and thus the role of the resulting Laplace pressure, becomes evident in an experiment in which part of the water drop protrudes out of a thin oil film, the surface tension of which is reduced by placing another immiscible oil above it (Figure 3). The plan view of the motion of a water droplet over the silicon wafer covered with a thin oil film is shown in Figure 4 (Movie3). It is interesting to note that when more than one droplet are deposited on the silicon wafer, they attract each other due to the ensuing capillary interaction 48,49 between them, often fusing before they reach the end of the wedge (Movie4). This fusion of droplets in the thin oil film is of significant importance in designing a suitable drop fluidic device to carry out specific reactions inside the droplets. with both R and h as can be gleaned from eq 6. When these velocities are plotted against the dimensionless radius RR * /h 2 , all the data ( Figure 5) reasonably cluster around a straight line (correlation coefficient 98%) passing through origin (0, 0), which vindicates the geometrical factor of eq 6. Since the velocity depends so strongly on both R and h, extreme care needs to be taken in quantifying the motion of the droplet in a wedge, where h changes along its length, which was also observed by Reyssat 12 for bubbles moving in a wedge of two rigid walls.
Thus, when the wedge angle is altered, we found it convenient to measure the velocity in the middle of the wedge in order to provide a uniformity of results obtained from different experiments. Thus by keeping both R and h as constants, the velocity of water drop was measured at different wedge angles and with liquids of various viscosities that, in turn, alters V*.

Drop manipulation using dynamically evolving topography
The experiments (Figures 3 to 5) performed with a silicon wafer helped us to identify the range of possibilities that can be exploited to control the motion of a water droplet in a wedge. Based on these results, we are poised to address the next objective of this work that is to modify the landscape of a fluidic platform using a deformable film 50 ( figure 6). To this end, we used a Figure 6. Schematic of the modulation of the shape of an oil laden elastomeric membrane of PDMS dispersed with magnetic particles by external electro-magnetic fields that can be simultaneously or sequentially turned on or off. This magnetic field induced deformation of the film alters its landscape with variable curvature that creates wedges thus causing motion of water drops as shown.
polydimethylsiloxane elastomer of modulus ~1 MPa, the surface energy of which is comparable to that of the PDMS grafted silicon wafer. While various silicone oils (Table 1 and Figure 5) could be applied on the surface of such a PDMS film to prevent the adhesion of droplet with the substrate, care had to be taken to ensure that the oil does not swell the PDMS. This requirement eliminates the commonly found silicone and a hydrocarbon based oils. However, a silicone oil in which some of the methyl groups are substituted with phenyl functionality induced little or no swelling of PDMS elastomer. Two such oils, commercially known as AR 20 and a high temperature silicone oil (designated here as HTS), were satisfactory for this purpose, HTS is superior to AR 20 as it induced no swelling of the elastomer. The densities (1.01 to 1.05 gm/cc) of these oils (Table 1) are also not much different from water thereby minimizing the effects of gravity. Thin films of a mixture of two commercial elastomers (Sylgard 184 and Sylgard 186) were cast on a large (diameter 14 cm) polystyrene dish via a spin coating process. The thickness of the resultant film varies somewhat, but it is typically close to 1 mm. While the film is still liquid, small magnetic steel particles (1.56 mm diameter) were placed on the film forming a square matrix array following which the film was cured in an oven to carry out the crosslinking reaction. Figure 7. Panel A) Small metal spheres are embedded in an elastomeric silicone film that is then coated with a thin film of oil. When a magnetic field is produced close to the metal spheres, the film deforms locally perpendicular to its surface. This creates a capillary wedge or circular symmetry around the particles. A water drop (colored red here) senses this changed landscape and moves towards the minimum of the bell-shaped energy potential. Moving the magnetic field parallel to the substrate, both short and long range motions of the drop, including its rotation and translation can be achieved. Panel B) By controlling the movement of the magnetic field, multiple droplets can be manipulated on the surface. Here two droplets (colored red and blue) are brought close to each other till they fuse. The fused drop can then be transported to another location of the surface. Thickness of oil film 0.4 mm Once the crosslinking is complete, the metal particle impregnated elastomeric film could be easily lifted off the polystyrene dish and fastened in between two concentric rings (8.5 cm diameter) that generated a certain amount of pretension. Such a film could be vibrated with a gentle tap exhibiting a low frequency (7 Hz We feel that this effect should be rather small in our pre-stressed film having an effective elastic tension of 1N/m, which is much larger than any of the surface and interfacial tensions ~ 25 mN/m in our system. Furthermore, as the water drop does not contact the elastomeric film, there is virtually no exchange of mechanical action from one to the other. After fastening the film in between the concentric rings with the metal particles facing upward, it was leveled and then coated with a thin (0.5 mm) layer of an oil film (AR20 or HTS). The surface of the PDMS elastomeric film was slightly undulated periodically in 2d due to the presence of metal particles, which has the advantage of creating a gently varying energy potential with various minima, where deposited water droplets could be trapped only to be actuated later when a magnetic field is applied beneath the film. The elastomer then deforms locally with a maximum deflection of 3 mm forming a centrosymmetric wedge that creates a Laplace pressure gradient in the drop, causing it to move. By moving the magnetic field spatially, various types of motion can be induced to the droplet. For example, the drop can be made to rotate (Figure 7A; Movie5); it can be translated up to a certain distance on the surface and then altering its direction, or the drop can be made to move sinusoidally mimicking somewhat a reptilian gait (Movie6). When two or multiple droplets are released on the surface, they can be brought into contact with each other till they fuse to become a single droplet ( Figure   7B; Movie7).
Having achieved the elementary objectives of the drop motion, we ventured to explore if such a device could be heated locally to carry out a specific temperature dependent chemical transformation. As an illustrative example, we considered a water drop containing a surfactant (Triton X 100) that forms a transparent micellar solution at room temperature. But, when the solution is heated to about 60 o C, it becomes milky by undergoing a cloud point transition. In order to achieve this objective a laminated heater is placed below the film, where such a transformation is expected to occur. By deforming the film with an external magnetic field in such a way that the deformed region contacts the electrically preheated resistive heater, the drop of the aqueous surfactant solution does indeed exhibit the expected cloud point transition ( Figure   8). Such a drop can also be moved out of the heated zone so that it becomes transparent again (Movie8). Such a process can be repeated countlessly without the fear of the underneath elastomeric film being contaminated by the surfactant. Figure 8. The method of drop manipulation as shown in Figure 7 can also be used to carry out temperature sensitive reactions in the droplet. Here a droplet of a micellar solution of a surfactant (Triton X-100) is manipulated on the elastomeric film, by bringing it in contact with the part of the surface that contacts a laminated resistive heater so that its temperature rises to about 60 o C. The micellar solution undergoes a cloud point transition at this temperature, causing the drop to be translucent white. When this drop is moved away from the hot zone, it becomes transparent instantaneously as the heat transfer is relatively fast in small drops. There is no evidence of surface contamination as this process of moving the droplet in and out of the hot zone can be repeated countless times.

Drop
The operations reported above can be carried out with a simple magnet that can be moved underneath the film. No spatial movement of a magnet is, however, needed in this operation, as the film can be deformed locally by turning electromagnetic fields on and off at different parts underneath the elastomeric film, thereby dynamically altering its landscape. While, metal particles placed in a regular array has the advantage of locally tapping the water droplet that is to be further deformed by external field, it is also possible to carry out this operation by impregnating the elastomeric film with a uniform dispersion of small   m  10 80  iron particles. movements of a single water droplet (colored red) on this film can be achieved simply by switching the magnetic fields on (+) and off (-) that dynamically modulates the energy landscape of the film. The dynamically evolving energy landscape can also be used to manipulate two water droplets (colored red and blue), eventually fusing them and moving the fused drop to a different location on the surface. This system should be amenable to logic gates, so that the entire drop dynamics is controlled by a computer.
Here, to begin with, the water drops can be trapped at specific locations of the film by predeforming it with spatially inhomogeneous electromagnetic field. By then turning on and off the electromagnetic fields at different locations following a predetermined logic, the motion of a single or multiple drops can be easily manipulated. An example of such an operation is shown in Figure 9, in which the positions of two droplets have been translocated orthogonal to its previous positions and then they were brought into contact to induce fusion (Movie9). The fused drop can afterwards be moved to a different location on the surface, if needed. An image of the iron powder impregnated membrane as seen under a low power microscope (40X) has also been provided in Figure 9.

CLOSING REMARKS AND FUTURE OUTLOOK
What we achieved in this work is a controlled actuation of a single or multiple droplets on a surface using a magnetically deformable elasto-capillary device that is simple to fabricate. The rapidly. All these thermal cycling processes should be accomplished with the fluidic concept described here that can be controlled by a pre-programmed computer. An additional advantage of carrying out the reactions with a water drop surrounded by oil film, as described here, is that the evaporation of water would be minimized, thereby making it suitable for carrying out many other biochemical reactions engineering with the aforementioned device. One limiting factor with the reactions carried out with drops is the delay of the time needed to fuse them in a more viscous medium. However, the preliminary studies carried out by us show that the fusion time can be drastically reduced by gently vibrating the elastomeric film that can be carried out with an alternating electromagnetic field superimposed upon a direct field. Future works in our laboratory would target the above objectives.
Before closing the paper, we make a few comments about the theoretical aspects of the work carried out in this paper and what could be done to improve the current analysis. The assumptions that led to the derivation of eq 6 are somewhat simplified. For example, we assumed that the oil film forms a spherical cap above the water droplet. While this assumption is valid when the difference of density between the water drop and the surrounding oil is very small, gravity should play some role when the difference of the density is not negligible. The complete analysis of the drop shape can be carried out by functional minimization of the Lagrangian of the system comprising the gravitational and the interfacial free energies while taking into corrections needed for the thickness dependent van der Waals free energy of the oil film above the drop as it could be thin enough to be influenced by the disjoining pressure acting upon it. While the driving force of the drop propulsion could be obtained following the above analysis, derivation of the viscous resistance that such a drop would experience is more complex that might require a numerical analysis under appropriate level set conditions. All these analyses would also need amendments from the curvature of the elastomeric film that varies in the centrosymmetric wedge following a power law, unlike the analysis carried out in this work. Tools exist to carry out the rigorous analyses as outlined above, but at the cost of losing the simplicity presented in this paper.
Perhaps, a more worthwhile analysis would be to consider the role of the curved wedge 12  Finally, being inspired by the suggestion of one of the referees, we make some comments regarding the scaling of the drop fluidic technology reported here and its relation to the role of gravity. With drops of volumes ranging from 3 L to 90 µL, gravity did not appear to play any significant role as long as the density of the surrounding liquid was close to that of water, which ensures a very large capillary length. As far as very small drops are concerned, we do not see any theoretical limitation of using this technique, as long as the oil film is thin enough and stable.
For nanoscopic film, it will be subjected to the so called "disjoining pressure", and of course if there is a gradient of the disjoining pressure, it could be used to induce motion of nano-droplets.

Elasto-capillarity induced motion of water droplet sandwiched between an inclined
substrate and an air/oil interface. The set-up used for this experiment was same as that described above. Here, the metallic stage supporting the wafer was inclined to a pre-determined angle, following which a drop of water was gently injected either using a syringe (Gilmont R ) or a pipetor (eppendorf) on the surface of the oil towards the narrower part of the wedge. As the drop gradually moved towards the thicker part of the wedge, both top and side views of it were videorecorded and subsequently analyzed. This same procedure was iterated for different wedge angles starting from 0.5 to 3.5° for each of the liquids listed in Table1. The viscosities of all the test liquids were measured using traditional Ostwald viscometer (Cannon) using water (viscosity 1cP at 20°c) as the reference liquid. The surface and interfacial tensions of various liquids were measured using a home built drop weight tensiometer.

Magnetic particle embedded polymeric membrane preparation and its use for drop manipulation.
For all the studies, the elastomeric film was prepared using a mixture of polydimethylsiloxanes (Dow Corning) Sylgard 184 and Sylgard 186 in the ratio of 1:1 w/w. Our previous 54 study showed that such a film is robust against stretching and bending. They could also support significant normal force. Mixtures of these liquid polymers were spun cast on a clean polystyrene petridish (diameter 14cm, obtained from Fischer Chemicals) at 1000 r.p.m for 15 sec in a Photo-Resist Spinner (Headway Research, Inc.). Steel balls (1.56 mm diameter) were placed on the PDMS at a uniform separation of ~ 10 mm from each other in the form of a square grid. The particles penetrated the PDMS film touching the underneath support, while part of it protruded outward above the base of air-film interface. This was then cross-linked in hot air oven at 75°C for 2 hours. After peeling the crosslinked film from the petridish, it was fastened in between two concentric rings (8.5 cm diameter) such that it neither sagged down nor was too stretched to be deformed by magnetic field. This was then placed on a glass plate clamped to an aluminum frame that could be moved translated laterally and tilted. Similar membranes impregnated with iron powder were prepared following the same procedure by sprinkling the iron particles evenly on the surface of the spun cast PDMS film and cured in an oven as