Droplet Self-Propulsion on Slippery Liquid-Infused Surfaces with Dual-Lubricant Wedge-Shaped Wettability Patterns

Young’s equation is fundamental to the concept of the wettability of a solid surface. It defines the contact angle for a droplet on a solid surface through a local equilibrium at the three-phase contact line. Recently, the concept of a liquid Young’s law contact angle has been developed to describe the wettability of slippery liquid-infused porous surfaces (SLIPS) by droplets of an immiscible liquid. In this work, we present a new method to fabricate biphilic SLIP surfaces and show how the wettability of the composite SLIPS can be exploited with a macroscopic wedge-shaped pattern of two distinct lubricant liquids. In particular, we report the development of composite liquid surfaces on silicon substrates based on lithographically patterning a Teflon AF1600 coating and a superhydrophobic coating (Glaco Mirror Coat Zero), where the latter selectively dewets from the former. This creates a patterned base surface with preferential wetting to matched liquids: the fluoropolymer PTFE with a perfluorinated oil Krytox and the hydrophobic silica-based GLACO with olive oil (or other mineral oils or silicone oil). This allows us to successively imbibe our patterned solid substrates with two distinct oils and produce a composite liquid lubricant surface with the oils segregated as thin films into separate domains defined by the patterning. We illustrate that macroscopic wedge-shaped patterned SLIP surfaces enable low-friction droplet self-propulsion. Finally, we formulate an analytical model that captures the dependence of the droplet motion as a function of the wettability of the two liquid lubricant domains and the opening angle of the wedge. This allows us to derive scaling relationships between various physical and geometrical parameters. This work introduces a new approach to creating patterned liquid lubricant surfaces, demonstrates long-distance droplet self-propulsion on such surfaces, and sheds light on the interactions between liquid droplets and liquid surfaces.


Dewetting of Glaco from Teflon AF1600
The dewetting of the Glaco superhydrophobic solution from Teflon AF1600 coated areas of substrates was confirmed using Scanning Electron Microscopy (SEM) images of substrates after lithographic patterning.Figure S1 shows a surface patterned with one side coated with Teflon AF1600 and one side with Glaco (i.e. a binary surface).

Contact Angles for Droplets on Different Surfaces
Example side profile view images of droplets on various surfaces are given in Fig. S2.

Derivation of Capillary Forces on a Wedge-Shaped Wettability Region
Circular Geometry and Coordinate System.The coordinate system is presented in Figure S4.
We consider an x-y plane reference system, with a wedge pattern defined by the half-angle at the wedge ξ.We consider the tip of the wedge being at the origin of the reference system, with the wedge symmetrical to the x-axis.The droplet footprint is a circumference defined by a constant radius r and by the centre x 0 .We assume that the droplet centre x 0 moves alongside the x-axis, with the droplet always remaining on the wedge main axis.The intersection points between the circumference and the wedge are defined by the angles at the centre and .Capillary Forces.The net force from the droplet on the wedge area is given by integrating from 0 to the wedge-circle intersection, (S1) where  i is the contact angle on the inside of the wedge.This gives (S2) Similarly, for the forces outside the wedge integrate from the wedge-circle intersection to 90 o , (S3) where  o is the contact angle on the outside of the wedge The total force on the droplet is then, (S4)

Figure S1 .
Figure S1.Scanning electron microscope (SEM) images of a binary GLACO-Teflon AF1600 surface.The right half of the sample is first coated with Teflon.Subsequently, the all sample is dip coated in Glaco five times (U in =1.00 mm/s, U w =1.00 mm/s).After the five coatings, GLACO develops in a thick coating layer on the left side of the sample (white region), where no Teflon is present.On the right side of the sample, the Teflon prevents the silica nanoparticles attaching to the surface, with only a small amount depositing which does not change the superficial chemistry of the region.

Figure S2 .
Figure S2.Images of 4 μl droplets on different types of surfaces.(a) DI water on Krytox Teflon AF1600-based SLIPS, (b) DI water on Olive oil Glaco-based SLIPS, (c) Olive oil droplet on a Krytox Teflon AF1600-based SLIPS, (d) Krytox on Olive oil Glaco-based SLIPS.

Figure
FigureS3shows a sequence of side profile images for a droplet of water deposited on the boundary between an Olive oil-based SLIPS region and a Krytox-based SLIPS region.The droplet rapidly moves across so it rests entirely on the Olive oil-based SLIPS which has the lower value of (liquid) Young's law contact angle.

Figure S3 .
Figure S3.Side view of a 4 μl DI water droplet deposited on the boundary region between Olive oil and Krytox on a OOG-KT Composite SLIPS.(a) The droplet is still attached to the needle.(b) The droplet touches the surface and detaches from the needle with an immediate movement.(c) The droplet sits on the Olive oil region due to the lesser contact angle. S4

Figure S4 .
Figure S4.Schematic representation of the wedge geometry considering a circular droplet footprint.x 0 is the geometrical centre of the droplet.ξ is the half-angle at the wedge,  b and  f are the angles on the x-y plane defined by the intersections between the droplet footprint and the wedge geometry; the subscripts b and f refer to the back and front of the droplet, under the assumption that the droplet moves in the positive direction on the x-axis. DV cos is the projection of the capillary force on the x-y plane.

Figure S5 .
Figure S5.Motion of 4 l water droplets self-propelling on composite wedge-shaped SLIP surfaces with wedge opening half-angles of =4 o , 5 o , 6 o , 7 o , 8 o , 9 o , 10 o , 11 o , 12 o , 13 o and 14 o for panels (a)-(k), respectively.In each case, the dotted line is the measured data series of droplet centre position as a function of time with the open circle symbols selected dated points from the series.The solid lines are fits to eq 20, x o (t)=x f tanh(t+t i )/), using three parameters t i ,  and x f , and the horizontal dashed lines are the measured length and the maximum transverse radius of the droplet (from top view images).