Flow‐Induced Long‐Term Stable Slippery Surfaces

Abstract Slippery lubricant‐infused surfaces allow easy removal of liquid droplets on surfaces. They consist of textured or porous substrates infiltrated with a chemically compatible lubricant. Capillary forces help to keep the lubricant in place. Slippery surfaces hold promising prospects in applications including drag reduction in pipes or food packages, anticorrosion, anti‐biofouling, or anti‐icing. However, a critical drawback is that shear forces induced by flow lead to depletion of the lubricant. In this work, a way to overcome the shear‐induced lubricant depletion by replenishing the lubricant from the flow of emulsions is presented. The addition of small amounts of positively charged surfactant reduces the charge repulsion between the negatively charged oil droplets contained in the emulsion. Attachment and coalescence of oil droplets from the oil‐in‐water emulsion at the substrate surface fills the structure with the lubricant. Flow‐induced lubrication of textured surfaces can be generalized to a broad range of lubricant–solid combinations using minimal amounts of oil.

. A photograph of the experimental setup (top) and a schematic of the filling experiment (bottom).

S2. Supplementary Videos
This section contains a list of videos (.avi).
Video S1: Silicone oil droplets in aqueous environment refusing to coalesce even if the contact line gets deformed considerably.

S3.1 Charging of Oil Droplets
The conceptual idea behind the addition of a positively charged surfactant is illustrated in Figure S3. The emulsions remain sufficiently stable over the time scales of hours to days, also upon the addition of positively charged surfactant. In Figure S3a we present photographs of the emulsions taken immediately after preparation, 1h after preparation, 5h after preparation, and 24 after preparation. In both cases (without CTAB and with CTAB), the emulsions remain turbid. Visual inspection hardly reveals any differences. In Figure 3b and 3c, it is shown that the polydisperse character of the emulsion not containing CTAB ( Figure S3b) is preserved upon the addition of CTAB ( Figure S3c). Both, negative as well as positive charges on the oil droplets do stabilize an emulsion ( Figure S4) [11,25] . Addition of the cationic surfactant reduces the effective charge of the droplet, resulting in a decrease of electrostatic repulsion. As long as the charge repulsion between the charges on the oil droplet does not become too pronounced, the filling of the structure can progress. If no surfactant is added, the emulsions are very stable, but no attachment and coalescence on the solid is observed ( Figure   S5).  Figure S4 and Video S1 show that without the addition of positively charged surfactant (CTAB) the negative charges on the oil droplets are sufficiently strong to prevent coalescence in an aqueous environment even when the droplets are pushed into one another. Figure S4. Negatively charged oil droplets in aqueous environment preferring deformation over coalescence.
At the later stages of the filling experiment; the filling process does slow down ( Figure S14).
We expect that this is due to progressive accumulation of charges at the surface of the deposited oil film, which eventually leads to charge repulsion ending the filling process. The addition of a negatively charged surfactant (SDS, sodium dodecyl sulfate) and neutral surfactant Pluronic  F-127 is not effective since it does not decrease the electrostatic repulsion between the emulsion drops.
Since a suitable content of a cationic surfactant has been added to reduce the electrostatic repulsion between the drops, the coalescence of droplets at the micropillar surface and thus formation of droplet bridges from pillar tops over the pillar walls to the bottom of the substrate depends on the flux of newly arriving drops. The flow of the emulsion facilitates the transport of the oil droplets.
The primary size or the size distribution of the arriving droplets is not relevant since the success of the filling process does not depend on either of them. The key element to the successful lubricant replenishment is the amount of cationic surfactant that is used. Also, surfaces in aqueous environments are typically negatively charged [11] . The interaction between the oil drops and the micropillar is the first event in any filling experiment. However, negatively charged oil droplets and a negatively charged surface lead to charge repulsion and droplet attachment to the micropillar array is not observed. After addition of 0.14% CMC of CATB, the emulsions are still stable for hours, i.e. enough for the timespan of the filling experiments which is typically ~1 h. If the concentration of CTAB is too high (>1000 g L -1 ), there is only a thin oil film formed on the surface with very little subsequent coalescence of oil droplets since the charge density of positive charges becomes too high. In this case, the charge repulsion between the positive charges in the oil droplets dominates and hence prevents the filling.

S3.2 Influence of Surfactant Concentration
The standard CTAB concentration used in this work (500 g L -1 ), an amount of 2 wt.-% oil and assuming an average radius of R = 2 m for the oil drops, each drop is covered by less than 1310 3 surfactant molecules. Full coverage of the drop surface by CTAB would imply 510 7 molecules per drop assuming each CTAB molecule covers [26]  1 nm 2 . Hence, the surfactant molecules cover less than 0.03% of the surface area of the drops. hardly any droplets still attach to the bottom substrate. No droplet attachment at all is observed above the CMC. This is expected to happen due to the surfactant accumulation at the solid substrate, thus preventing attachement of any surfactant covered oil drops. Figure S6 summarizes our observations. A concentration below 500 g L -1 does not overcome the charge repulsion between the negatively charged oil droplets. Hence, no coalescence and consequently no droplet attachment and droplet growth is possible. Above concentrations of 500 g L -1 which remain well below the CMC of CTAB (1 mM) droplet attachment predominantly takes place on the bottom substrate, and the structure gets partially filled starting from the bottom substrate, but droplet attachment at pillars and droplet growth on pillars does no longer take place. As soon as an oil layer is formed on the bottom substrate, which may also embrace the pillars, the filling stops due to charge repulsion and increased emulsion stability. The addition of more CTAB above a concentration of 500 g L -1 has led to the increased repulsion of positive charges, which dominates upon further increasing the CTAB-concentration. For CTAB concentrations approaching and exceeding the CMC, no droplet attachment is observed anymore since in these cases the strong presence of CTAB leads to strong charge repulsion between the individual oil droplets as well as between the oil droplets and the bottom substrate. Since surfactant is always present on the bottom substrate, the strong presence of CTAB leads to a positively charged surface which repels the now positively charged oil droplets.

S3.3 Attachment of Oil Droplets
As can be seen in Figure 3a multiple oil droplets can attach to the same pillar and repeatedly accommodate droplets even after some portions of the underlying bottom substrate around a pillar have already been filled with oil. Figure S7 and S8 show magnified versions of the area marked by green squares and a white circle in Figure 3a.

S3.4 Orientation of Droplets with Respect to the Micropillar and Size of Descending Droplets
At the early stages of the filling procedure, droplets are oriented towards the flow direction, as shown in Figure S9, S10, and S11. Lifetime of drops and their growth is presented in Figure   S12 and S13.     Towards the end of the filling procedure droplets primarily attach at the pillars' top faces. Now, the center of the droplets can be to the left or right of the center of the pillars, reflected in an orientation angle larger than 90°.

S3.5 Lateral Adhesion versus Shear-Induced Depinning
After attachment of a single oil drop to a pillar wall, continuity of shear stresses across the interface might remove the droplet from the wall. However, after attachment, the droplets do not leave the pillar nor do not change their position. This implies that the lateral adhesion force needs to overcome the depinning force and the shear force.
The lateral adhesion force of attached oil drops on pillar walls [27] can be estimated as The shear stress required to depin a sessile droplet from a flat surface surrounded by an immiscible fluid is estimated by [9,28] : With R = 2.5 µm for the small and R = 50 µm for the large droplets, the required yield stress would amount to 7.6 10 4 Pa and 0.4 10 4 Pa, respectively, more than three orders of magnitude larger than the flow induced shear stress at the surface. Thus, consistently with the experiments, the calculations suggest that the droplets remain in place after attaching the pillar walls.

S3.6 Slowing Down of Filling
Without the addition of a surfactant, the underlying structure will not be filled with oil.
Likely, accumulation of the surfactant molecules in the oil film causes the slowing down of the filling process with time ( Figure S14).

S3.7 Quadratic Pillars
As a next step, the droplet distribution (number of droplets) around big rectangular pillars (40 m edge length) has been elucidated to compare it to the droplet orientation found for the situation of cylindrical pillars (see main text, Figure 4a, b). The definition of an orientation angle is not appropriate for the case of rectangular pillars owed to the presence of sharp edges. In Figure S15 the time evolution of the droplet distribution is presented for a filling experiment conducted with a micropillar array consisting of large rectangular pillars. The standard experimental conditions were established (see main text). Owed to geometry three cases need to be distinguished. There is no radial symmetry. The side of the rectangle facing the flow direction (black curve, in Figure S15), the side turned away from the flow direction (blue curve, in Figure S15) and the two equivalent sides parallel to the flow direction (red curve, in Figure S15)

S3.8 Filling of Porous Substrates of Varying Geometry and with Different Oils
The approach used in this work can potentially be extended to various other porous substrates different from micropillar substrates. To find out whether the mechanism also applies to nanoscopic, porous structures and whether lubricant can be replenished, we investigated glass coated with silicone nanofilaments [13] , Figure S16. Silicone nanofilaments consist of a fibrous structure of filaments having a diameter of approximately 70 nm. Figure S16. SEM image of a glass substrate coated with silicone nanofilaments (Scale bar = 3 m).
In Figure    So far only silicone oil was used. This poses the question whether the filling mechanism works analogously for different lubricating oils. Figure S18a shows the filling of a micropillar structure with the lubricant poly--olefin (PAO,  = 0.78-0.82 g mL -1 ). The key strength of PAO lies in its compatibility with many essential lubricant additives, such as antioxidants, anti-wear additives, anti-corrosion agents, friction modifiers or viscosity modifiers [29] . Again, 500 g L -1 CTAB was added to the water phase before emulsification (2 wt.-% of PAO). The top row shows the fluorescence channel (water in blue, oil and pillars in black) and the bottom row the corresponding image in the transmission channel. The filling kinetics resembles those observed for silicone oil. Also, with a fluorinated lubricant Krytox  103 [7,15,30] , which is a commonly used lubricant for slippery surfaces, the same mechanism of droplet attachment to the pillars walls, growth, coalescence and descending by droplet-bridging was observed ( Figure S19). Analogously to the case of using silicone oil as lubricating oil, filling of the structure with PAO and Krytox  103 did not occur without added surfactant, hinting that PAO and Krytox emulsions are negatively charged.
To test the generic nature of flow-induced filling, we investigated the influence of the surface structure. Figure S18b shows the top view of the time evolution of a pillar array consisting of square pillars (edge length: 20 m, center-to-center spacing: 40 m, pillar height: 10 m).
Again, the top row depicts the fluorescence channel, and the bottom row the transmission channel. The transmission channel image sequence demonstrates that the droplet growth and coalescence take place analogously to the case of cylindrical pillars and the pillar substrate gets filled with oil in an approximately square-like pattern [31] . Different from the cylindrical pillars ( Figure 2 and Figure 3); here each pillar accommodates several droplets due to the large pillar size. Moreover, droplets also attach to the side walls and the pillars backsides, Figure S15. The side facing the flow direction is not preferred over the side turned away from the flow direction regarding more frequent drop attachment. Figure   In the main text successful filling of the micropillar structure with silicone oil and PAO is demonstrated. The structure can also be successfully filled with Krytox  103 and Fluorinert TM FC-70 as is shown in Figure S19 showing droplet attachment and growth to the micropillars.
Successful filling is also observed in the case of customary food-grade olive oil (not shown). Below, in Figure S20, an experiment supporting this expectation is presented. In the standard filling experiments reported in this manuscript based on an oil-in-water emulsion containing 2 wt.-% 50 cSt silicone oil as well the positively charged surfactant CTAB at a concentration of 500 gL -1 , the viscosity ratio of the lubricant to aqueous phase is  oil / water  50. To test whether the viscosity ratio matters, we conducted the same experiment using a ratio  oil / water  50, by exchanging pure water as the aqueous phase by a water-glycerol mixture having a viscosity of  100 mPas; a change by a factor of 100. Still, the filling mechanism works analogously, i.e. oil droplets (dyed yellow) attach to pillars, growth through coalescence, and descend onto the substrate. Water is replaced by oil at the pillars. It has to be noted that in the case of a lubricating oil less dense than water, buoyancy will keep the oil from contacting the bottom substrate as well as pillars. Secondly, without the addition of a surfactant, the underlying structure will not be filled with oil. For hydrophilic substrates, filling does not take place. The hydrophobic (oleophilic) nature of the underlying solid substrate needs to be preserved, as is shown in Figure S22, in order to facilitate filling. the micropillar arrays. [32] Flow field considerations suggest that a minimum flow velocity is necessary to facilitate transport of oil droplets in-between the pillars and to the bottom substrate. [33] Figure S23. Ratio between the upper limit of the filling rate ̇ to the amount of depleted lubricant ( ̇ /q d ) as a function of flow rate Q.

S3.10 Hydrodynamic Drag Force
Theoretically the hydrodynamic drag force of water (viscosity η w = 1 mPa s) F hyd should prevent coalescing and filling [11,17] . For spheres diverges if the distance x between the particle of radius R and the surface approaches zero. Because drops are deformable, no analytical expression exists. Still, only at sufficiently high velocity or long contact times; the water film separating the drop and the pillar is sufficiently thinned during the impact that a defect can induce rupturing of the water film. [18]