Self-Propelled Droplet Transport on Shaped-Liquid Surfaces

TL;DR

This paper introduces a novel shaped-liquid surface design that enables self-propelled droplet transport with minimal resistance, leveraging dual-scale topography and wettability gradients for applications in microfluidics, self-cleaning, and heat transfer.

Contribution

It proposes a simple, effective surface structure that facilitates long-range, self-propelled droplet movement without external energy input, overcoming traditional resistance issues.

Findings

Enables long-range droplet transport on shaped-liquid surfaces.

Captures and transports impacting droplets even on inverted surfaces.

Provides design principles for applications in microfluidics and self-cleaning surfaces.

Abstract

The transport of small quantities of liquid on a solid surface is inhibited by the resistance to motion caused by the contact between the liquid and the solid. To overcome such resistance, motion can be externally driven through gradients in electric fields, but these all inconveniently involve the input of external energy. Alternatively, gradients in physical shape and wettability - the conical shape of cactus spines to create self-propelled motion. However, such self-propelled motion to date has limited success in overcoming the inherent resistance to motion of the liquid contact with the solid. Here we propose a simple solution in the form of shaped-liquid surface, where solid topographic structures at one length scale provides the base for a smaller length-scale liquid conformal layer. This dual-length scale render possible slippery surfaces with superhydrophobic properties. Combined to an heterogeneous topography, it provides a gradient in liquid-on-liquid wettability with minimal resistance to motion and long range directional self-propelled droplet transport. Moreover, the liquid-liquid contact enables impacting droplets to be captured and transported, even when the substrate is inverted. These design principles are highly beneficial for droplet transport in microfluidics, self-cleaning surfaces, fog harvesting and in heat transfer.

Figures

c of the main manuscript shows that the apparent contact angle depends linearly of the solid fraction $f_{s}$ for a droplet on rails with uniform fraction. Fig. [8](#S1.F8) indicates that this is also the case on tails with divergent width, for both pinned and moving droplets. For high solid fractions ($f_{s}>0.7$), the pocket of air expected to be sustained underneath the droplet are sometimes filled with oil. This explains the decrease of the apparent contact angle to $\theta_{e}$, as predicted by Equation 1 of the main manuscript, for some measurements in this region.

c of the main manuscript shows that the apparent contact angle depends linearly of the solid fraction $f_{s}$ for a droplet on rails with uniform fraction. Fig. [8](#S1.F8) indicates that this is also the case on tails with divergent width, for both pinned and moving droplets. For high solid fractions ($f_{s}>0.7$), the pocket of air expected to be sustained underneath the droplet are sometimes filled with oil. This explains the decrease of the apparent contact angle to $\theta_{e}$, as predicted by Equation 1 of the main manuscript, for some measurements in this region.

Equations

$$\cos {\theta }_e=\frac{{\gamma }_{oa}-{\gamma }_{wo}}{{\gamma }_{eff}}$$

$$\cos \theta =f_1\cos {\theta }_e-(1-f_1)$$

$$g=\left({\gamma }_{oa}+{\gamma }_{wo}\right)\cos \theta$$

$$F_d=2kR\left[g(x=-R)-g(x=R)\right]$$

$$F_d=8k{\gamma }_{oa}\alpha R^2$$

$$F_{v,b}={\int }_{S_b}{\mu }_w\frac{\partial v_b}{\partial y}\text{leavevmode}d{S}_b\propto f_s{\mu }_{w}vR$$

$$F_{v,o}={\int }_{S_b}{\mu }_o\frac{\partial v_o}{\partial y}\text{leavevmode}d{S}_b\propto f_s{\mu }_{w}vR$$

$$F_{v,c}\propto f_s{\mu }_{o}v\ln \left(\frac{R}{L_o}\right)R$$

$$F_p=kR\left({\gamma }_{wo}+{\gamma }_{oa}\right)\left(\cos {\theta }_a-\cos {\theta }_r\right)$$

$$\cos {\theta }_a-\cos {\theta }_r=$$

$$=$$

$$F_p=-kR\left({\gamma }_{wo}+{\gamma }_{oa}\right)2\sin {\theta }_{eq}\sin \left(\frac{\Delta \theta }{2}\right)$$

$$8k{\gamma }_{oa}\alpha R^2\propto f_s{\mu }_o\nu ln\left(\frac{R}{L_o}\right)R$$

$$v\propto \frac{8k{\gamma }_{oa}}{{\mu }_o\ln \left(R/L_o\right)}\frac{\alpha R}{f_s}$$

Full text

Self-Propelled Droplet Transport on Shaped-Liquid Surfaces

Gaby Launay

Smart Materials & Surfaces Laboratory, Faculty of Engineering & Environment, Northumbria University, Newcastle upon Tyne, NE1 8ST, UK

Muhammad Subkhi Sadullah

Department of Physics, Durham University, Durham, DH1 3LE, UK

Glen McHale

Smart Materials & Surfaces Laboratory, Faculty of Engineering & Environment, Northumbria University, Newcastle upon Tyne, NE1 8ST, UK

Rodrigo Ledesma-Aguilar

Smart Materials & Surfaces Laboratory, Faculty of Engineering & Environment, Northumbria University, Newcastle upon Tyne, NE1 8ST, UK

Halim Kusumaatmaja

Department of Physics, Durham University, Durham, DH1 3LE, UK

Gary G. Wells

Smart Materials & Surfaces Laboratory, Faculty of Engineering & Environment, Northumbria University, Newcastle upon Tyne, NE1 8ST, UK

email: gary.wells@northumbria.ac.uk

The transport of small quantities of liquid on a solid surface is inhibited by the resistance to motion caused by the contact between the liquid and the solid. To overcome such resistance, motion can be externally driven through gradients in electric fields [1, 2], temperature [3, 4, 5], light [6, 7] and pressure [8], or structural topography combined with vibration or phase change [9, 10], but these all inconveniently involve the input of external energy. Alternatively, gradients in physical shape and wettability – the conical shape of cactus spines [11] and the wettability of butterfly wings [12] – occur naturally and can be engineered into surfaces [7, 13, 14, 15, 16, 17] to create self-propelled motion. However, such self-propelled motion to date has limited success in overcoming the inherent resistance to motion of the liquid contact with the solid. Here we propose a simple solution in the form of shaped-liquid surface, where solid topographic structures at one length scale provides the base for a smaller length-scale liquid conformal layer. This dual-length scale render possible slippery surfaces with superhydrophobic properties [18]. Combined to an heterogeneous topography, it provides a gradient in liquid-on-liquid wettability with minimal resistance to motion and long range directional self-propelled droplet transport. Moreover, the liquid-liquid contact enables impacting droplets to be captured and transported, even when the substrate is inverted. These design principles are highly beneficial for droplet transport in microfluidics, self-cleaning surfaces, fog harvesting and in heat transfer.

An ideal design of a system to move small quantities of fluid should not require complex propulsion mechanisms based on continual input of energy. It should neither present large threshold sticking forces to be overcome nor limit the distance of transport. The ability to provide fine control of speed of motion, to adhere and transport liquids without boundary walls and to do so in multiple orientations, whether uphill or inverted, would provide additional benefits. Motion without energy input, has led to a focus on surfaces with a gradient in physical properties and in the use of topographic features [14]. These features provide a gradient in the wetting properties of the solid surface, whilst retaining a uniform surface chemistry, and create a self-propulsion force on droplets. However, direct contact between the droplet and the solid is needed to drive motion, and this introduces static and dynamic friction forces. The consequent minimum self-propulsion forces required for transport necessitate large gradients in the topographic texture. Large gradients, in turn, lead to limited control of a droplet’s velocity when transported over long distances. To overcome these limitations, we hypothesized that contact with the solid could be completely replaced by another liquid, but in a manner that still allows a gradient in wettability to exist. To do so, we used a dual length scale substrate providing both a liquid surface and a gradient in its surface texture. This enables liquid-on-liquid wetting to drive the motion of droplets, where the driving mechanism is mediated by an asymmetry built within an underlying solid substrate which shapes the liquid surface. To illustrate liquid-on-liquid wetting, first consider placing a water droplet on a thin film of silicone oil that coats a flat hydrophobic solid surface. Because silicone oil completely wets water, the oil will encase the droplet and isolate it from the solid surface [19, 20].

Even though there is no direct contact of the droplet with the solid surface, it still exhibits an apparent contact angle, $\theta_{e}$, due to the interfacial tensions pulling on the outer and inner surfaces of the oil layer at the droplet’s edge (see inset to Fig. 1a). This leads to an effective Young’s Law for liquid-on-liquid wetting [21, 22]

[TABLE]

where $\gamma_{eff}=\gamma_{wo}+\gamma_{oa}$. Here, $\gamma_{oa}=$19.8\text{,}\mathrm{mN}\text{,}{\mathrm{m}}^{-1} and $\gamma_{wo}=$38\text{\,}\mathrm{mN}\text{\,}{\mathrm{m}}^{-1} are the surface tensions of the oil-air and water-oil interfaces and predict $\theta_{e}=$108.4\text{,}\mathrm{\SIUnitSymbolDegree}. If the oil is replaced by another immiscible liquid having a negative spreading power on water, the liquid may still isolate the droplet from the solid surface but does not encase the droplet and so $\gamma_{eff}=\gamma_{wa}$. Since equation [1](#S0.E1) has no explicit dependence on the solid surface, it can be regarded as a definition of wettability for an immiscible liquid surface in the thin film limit. To create liquid-on-liquid wetting with a surface capable of being shaped, we introduce a small length scale solid texture using a hydrophobic nanoparticle-based coating. This allows the silicone oil to be retained and provides a continuous liquid surface [[23](#bib.bibx23), [24](#bib.bibx24)]. The apparent contact angle of a droplet of water on this liquid surface is $\theta_{e}=$109.3\pm 0.7\text{\,}\mathrm{\SIUnitSymbolDegree} consistent with equation 1 (Fig. 1a) and confirms the absence of direct contact with the solid surface. Measurements show that the droplet on the liquid surface exhibits a very low contact angle hysteresis ($\Delta\theta\approx$1\text{,}\mathrm{\SIUnitSymbolDegree}) and sliding angle ($\theta_{s}<$1\text{\,}\mathrm{\SIUnitSymbolDegree} for a $5\text{\,}\mathrm{\SIUnitSymbolMicro L}$ droplet) thus removing the principal constraint on self-propulsion using a gradient surface. This approach allows us to create a conformal liquid coating on a solid substrate which can be textured at a larger length scale and so unlocks the ability to create wetting contrasts and gradient wettability with a liquid surface. We now consider the static wetting of a water droplet on liquid surfaces with textures designed to control wettability. For the larger length scale, we used arrays of rectangular cross-section rails $60\text{\,}\mathrm{\SIUnitSymbolMicro m}$ high and $75\text{\,}\mathrm{\SIUnitSymbolMicro m}$ apart, and with solid surface area fractions $f_{s}$ between $0.1$ and $0.9$. We applied the nanoparticle-based coating as our second, smaller, length scale to enable the silicone oil to create a liquid surface that conforms to the large-scale solid texture. We observed that a water droplet rests on top of the rails leaving pockets of air underneath [25] (Fig. 1b and inset), with low contact angle hysteresis ($\Delta\theta\approx$1\text{,}\mathrm{\SIUnitSymbolDegree}$$) and a contact angle (viewed across the rails) which increases with decreasing rail fraction (Fig. 1c). This is reminiscent of the Cassie-Baxter state for superhydrophobic surfaces [26, 27], but here the contact angle, $\theta$, is determined by the liquid (equation 1) [28]

[TABLE]

and $f_{l}\approx f_{s}$ is the liquid surface area fraction. This prediction agrees with measurements and provides an excellent fit by including a small correction $f_{l}=f_{s}+0.059$ (Fig. 1c). This correction is consistent with the estimated nanoparticle-based coating thickness in our experiments ($\approx$2\text{,}\mathrm{\SIUnitSymbolMicro m}$$, giving a correction of $\approx 0.053$). This suggests the droplet is in a mixed wetting state suspended above a composite liquid and air surface. To further confirm there is no contact with the solid we considered the same substrate without silicone oil. Crucially, because of the high contact angle of $165\text{\,}\mathrm{\SIUnitSymbolDegree}$ on this oil-free surface compared to the lower apparent contact angle of $108.4\text{\,}\mathrm{\SIUnitSymbolDegree}$ on those with oil, the variation of contact angle with rail fraction is significantly weaker (compare dashed and dashed-dotted lines in Fig. 1c). This implies that the gradient in wettability for droplet self-propulsion is significantly stronger for the liquid surface and amplifies the effect of the liquid surface in removing pinning. To verify the droplet recognizes contrasts in wettability on a composite liquid-air surface, we placed a droplet at the boundary between two regions of different rail fraction. The droplet spontaneously moved to the region of higher wettability defined by equation (2) (Supplementary Fig. S5 and movie S1).

We now consider the self-propulsion of a water droplet on a liquid surface induced by a designed gradient in wettability. For the larger length scale, we fabricated an array of $2\text{\,}\mathrm{cm}$ long and $60\text{\,}\mathrm{\SIUnitSymbolMicro m}$ high rails, where the width of each rail diverges along its length. At the low surface fraction end of the substrate, the apex of each rail was $75\text{\,}\mathrm{\SIUnitSymbolMicro m}$ apart (Supplementary Fig. S3). The liquid surface area fractions $f_{l}$ increases linearly from 0 to 1 at a rate $\alpha=$0.05\text{,}{\mathrm{mm}}^{-1}$$ as the structure is traversed from the apex to the base. Without the liquid surface provided by silicone oil, a droplet remains pinned to the solid despite the wettability gradient (Supplementary Fig. S6).

On the shaped liquid surface, however, we observed a sustained self-propulsion over the pattern (Fig. 2a). This propulsion can also be designed into complex pathways, as illustrated by transport along a curved path (Supplementary Movie S3). Returning to our design criteria and the desire for controlled long distance transport and fine control of droplet speed, we discovered an inverse linear relationship between speed of motion and rail fraction $v\propto 1/f_{s}$ (Fig. 2b). The slowing-down dynamics occurs for a range of droplet volumes ($V=2.5-10$\mathrm{\SIUnitSymbolMicro L}) and wettability gradients ($\alpha=0.03-0.1${\mathrm{mm}}^{-1}). By balancing the driving force $F_{d}\propto\alpha\gamma_{oa}R^{2}$ due to the wettability gradient along the droplet’s perimeter (where $R$ is the droplet’s base) with viscous resistance $F_{v}\propto\mu_{o}f_{s}Rv$ from the edge of the drop, we find $v\propto\gamma_{oa}\alpha R/\mu_{o}f_{s}$ (Supplementary Information). This explains the observed slowing-down with increasing rail fraction (Fig. 2b) and predicts an increase of speed with wettability gradient and droplet volume, which we confirmed by measuring the typical droplet speed, $v_{t}$, defined as the speed at the surface point where $f_{s}=0.5$ (Fig. 2c). Our measurements show self-propulsion ceases below a rail fraction gradient $\alpha\approx$0.03\text{,}{\mathrm{mm}}^{-1}$$ implying the presence of a small pinning force (Fig. 2c).

To measure this force, we tested the ability of droplets to move up inclined surfaces (Fig. 3a). We observed upwards self-propulsion below an inclination angle $\beta_{1}$, implying that the driving force outweighs gravity and pinning (bottom region in Fig. 3b), but above an angle $\beta_{2}$ droplets run downhill (upper region in Fig. 3b). From the conditions for mechanical equilibrium, the driving force is $F_{d}=\frac{1}{2}\rho_{d}Vg(\sin\beta_{2}+\sin\beta_{1})$ and the pinning force is $F_{p}=\frac{1}{2}\rho_{d}Vg(\sin\beta_{2}-\sin\beta_{1})$. We deduce $F_{p}\approx$1.11\pm 0.25\text{,}\mathrm{\SIUnitSymbolMicro N} across the range of rail fraction gradients, similar to the small value estimated from contact angle hysteresis measurements, $F_{p}\approx$0.48\text{\,}\mathrm{\SIUnitSymbolMicro N} (Supplementary Information). Fig. 3c shows the measured driving force also agrees well with the simple theoretical model (Supplementary Information). Overall, these measurements confirm that pinning is overcome by the driving force over a broad range of wettability gradients and droplet volumes. With such small gradients (down to $\alpha\approx$0.03\text{,}{\mathrm{mm}}^{-1}) sustained transport is possible over long distances (here $L=1/\alpha\approx$30\text{\,}\mathrm{mm}). Furthermore, because of the small pinning force, droplets can also be moved at low speeds (down to $0.067\text{\,}\mathrm{mm}\text{\,}{\mathrm{s}}^{-1}$ for a $5\text{\,}\mathrm{\SIUnitSymbolMicro L}$ droplet). In contrast, without the shaped liquid surface (Fig. 3c), we find that the pinning force is considerably larger and cannot be overcome by the driving force. In addition to reducing pinning and allowing self-propulsion, the liquid surface significantly increases the ability to capture impacting droplets. The number of bounces of a droplet before it adheres to a solid upon impact depends on the kinetic and surface energy remaining after energy dissipation because of spreading and retraction [29, 30].

On a superhydrophobic surface, the small contact area and low dynamic friction of a droplet leads to a bouncing rate that rapidly increases with increasing droplet impact speed (which we varied by setting the initial release height $h_{r}$ (Fig. 4a,c), making droplet capture and transport difficult. In contrast, our shaped liquid surfaces increase dissipation upon impact through viscous friction [31] and have stronger adhesion due to the wettability of the liquid surface. As such, they prevent bouncing over a broad range of release heights and promote capture and subsequent transport via self-propulsion, even uphill (Fig. 4b,d). Furthermore, Fig. 4e shows that these surfaces are capable of capturing and transporting hanging droplets, impacting the surface from below.

The shaping of liquid surfaces, enabled by a dual length-scale structure, is an advantageous approach to creating wettability contrasts for droplets on liquid surfaces. In this work we used these principles to design liquid surfaces with strong gradients in wettability and low resistance to droplet motion. On a shaped liquid surface, it is possible to gently transport droplets with ease over long distances along defined paths and with fine control of their speed. The design principles presented here provide a foundation for new types of microfluidic devices and, because of the ability to capture and transport impacting droplets in any orientation, may also have wider applications, such as fog harvesting and heat transfer.

1 Supplemental Material

1.1 Surface fabrication process

**We describe here the fabrication process of the super-hydrophobic and shaped liquid surfaces.

**Micro-structured surface are manufactured by photolithography. Glass wafers are cleaned prior to the process, using IPA and acetone rinses. To promote the photoresist adhesion, wafers are etched using a reactive-ion etcher (RIE) and bounded with HDMS (Microchem). Surfaces are then spin-coated $30\text{\,}\mathrm{s}$ at $1750\text{\,}\mathrm{r}\mathrm{p}\mathrm{m}$ with a photoresist (SU8-2025 from Microchem), to obtain a layer of $60\text{\,}\mathrm{\SIUnitSymbolMicro m}$. After a pre-bake ($10\text{\,}\mathrm{min}$ at $95\text{\,}\mathrm{\SIUnitSymbolCelsius}$), patterns are transposed from a chrome mask (JD Photo Data) on the surface using a mask aligner (at an UV power of $300\text{\,}\mathrm{m}\mathrm{J}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$). The extra photoresist is removed using EC solvent (Microchem) after a post-exposure bake ($6\text{\,}\mathrm{min}$ at $95\text{\,}\mathrm{\SIUnitSymbolCelsius}$). Surfaces are then subjected to a last bake ($10\text{\,}\mathrm{min}$ at $170\text{\,}\mathrm{\SIUnitSymbolCelsius}$) to finalize the cross-linking process. The structures obtained are $60\text{\,}\mathrm{\SIUnitSymbolMicro m}$ high and can be as small as $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ wide. The hydrophobic porous layer is deposited by spraying a commercial solution (Glaco™ mirror coat zero®) of silanized nano-particles. SEM images show that this process creates a conformal porous layer on the micro-structures (Fig. 5). At this point, the surface is a super-hydrophobic patterned surface, on which water droplets lie in Cassie-Baxter state (Fig. 6a). $20\text{\,}\mathrm{cSt}$ silicone oil is imbibed into the surface by dip-coating it at $0.1\text{\,}\mathrm{mm}\text{\,}{\mathrm{s}}^{-1}$. This result in a thick oil layer that fills the micro-structures and prevents the formation of air pockets underneath the droplet (Fig. 6b). To obtain a liquid surface, the excess oil is removed using a water jet. The success of this operation is ensured by verifying that deposited droplets are in Cassie-Baxter state (Fig. 6c) and exhibit a very low contact angle hysteresis.

1.2 Micro-structure geometry

**Fig. 7 presents the rail geometry used to create wettability gradients with liquid surfaces.

**

1.3 Contact angle measurements

**We present here the methods used to measure droplet contact angles and contact angle hysteresis.

**Static contact angles are measured using side visualizations of droplets and an in-house drop shape analyzer software (PyDSA). The robustness of the measurements is ensured by performing repeats and by comparing the different methods of contact angle measurements provided by the software.

The contact angle hysteresis is measured using the inflation-deflation method. A droplet of $4\text{\,}\mathrm{\SIUnitSymbolMicro L}$ is first deposited on the surface and put in contact with a small syringe needle. The side-view of the droplet is recorded using a camera while the drop is inflated and then deflated slowly through the needle. The evolution of the base radius and contact angles are extracted from the resulting video using pyDSA. From those measurements, the advancing and receding contact angles are deduced as the contact angles for which the base radius begins to respectively increase and decrease. The contact angle hysteresis is then calculated as the difference between the advancing and receding angles. For a liquid surface, measurements indicate that the CAH remains low ($\Delta\theta=$0.66\pm 0.57\text{,}\mathrm{\SIUnitSymbolDegree}$$ on 20 measurements) and is independent of the solid fraction in the range $f_{s}\in[0.1,0.9]$.

1.4 Apparent contact angle evolution with the solid fraction

**We present here additional measurements of the droplet apparent contact angle as a function of the solid fraction $f_{s}$.

**Fig. 1c of the main manuscript shows that the apparent contact angle depends linearly of the solid fraction $f_{s}$ for a droplet on rails with uniform fraction. Fig. 8 indicates that this is also the case on tails with divergent width, for both pinned and moving droplets. For high solid fractions ($f_{s}>0.7$), the pocket of air expected to be sustained underneath the droplet are sometimes filled with oil. This explains the decrease of the apparent contact angle to $\theta_{e}$, as predicted by Equation 1 of the main manuscript, for some measurements in this region.

1.5 Droplet response to a wettability contrast on a shaped liquid surface

**A droplet deposited between two regions of different rails fraction spontaneously moves towards the region of higher wettability (higher fraction) (Figure 9).

**

See also Movie S1.

1.6 Droplet on patterned super-hydrophobic surfaces

**We compare here the effect of the wettability gradient on a droplet motion for a super-hydrophobic and a liquid surface (Figure 10).

**

See also Movie S2.

1.7 Force balance model

**We derive here the model used to estimate the driving and pinning forces for a droplet on a shaped liquid-surface.

**

1.7.1 Driving force

The elementary force per unit length acting at a specific point of the contact line of a droplet (and towards the droplet center) reads:

[TABLE]

The resulting force acting on the droplet can be obtained by integrating this elementary force on the contact line. In the case of a droplet on a continuous, linear gradient of solid fraction $\alpha=\frac{\partial f_{s}}{\partial x}$, and assuming the droplet to have a rectangle footprint of length $R$ and width $kR$, the capillary force acting on the droplet is simply:

[TABLE]

Using Eq. 2 from the main manuscript and Eq. 3, Eq. 4 simplifies in

[TABLE]

1.7.2 Viscous resistance

As soon as a droplet is put in movement, a force due to viscous friction apposes its motion. Different mechanisms participate to this viscous dissipation: the dissipation in the droplet bulk, in the oil layer underneath the droplet and at the droplet contact line. The force linked to the dissipation in the droplet bulk $F_{v,b}$ is expected to generate a resisting force scaling as

[TABLE]

with $S_{b}$ the surface of contact between the droplet and the surface and $v_{b}$ the velocity in the droplet bulk. Assuming the velocity at the top of the oil layer to be $v_{o}(y=h)=vh\mu_{w}/R\mu_{o}$ (with $h$ the oil thickness), the force linked to the dissipation in the layer of oil scales as

[TABLE]

Using a Taylor expansion of the Cox-Voinov equation, we can show that the force linked to the dissipation at the contact line scales as:

[TABLE]

with $L_{o}$ the slip length. All those forces scale as $f_{s}vR$. However, the force linked to the dissipation at the contact line is expected to be dominant in our configuration because of the viscosity ratio ($\mu_{o}/\mu_{w}=20$) and the high ratio $R/L_{o}$. Being the dominant dissipation force, $F_{v,c}$ will be called $F_{v}$ in the sequel.

1.7.3 Pinning force

Because of chemical and topological heterogeneities, a droplet on a surface always exhibits a contact angle hysteresis. This hysteresis leads to the presence of a pinning force, that need to be overcome before a droplet can be put in motion. Still assuming a droplet with a rectangular footprint, the pinning force can be deduced from the advancing and receding contact angles (respectively $\theta_{a}$ and $\theta_{r}$), as

[TABLE]

The difference in contact angle cosine can be rewritten as:

[TABLE]

with $\Delta\theta=\theta_{a}-\theta_{r}$ the contact angle hysteresis and $\theta_{eq}$ the equilibrium contact angle, estimated to be equal to $(\theta_{a}+\theta_{b})/2$. Introducing Equation 11 into 9 leads to:

[TABLE]

1.7.4 Terminal velocity

A droplet terminal velocity can be estimated by balancing the driving and dissipation forces (respectively $F_{d}$ and $F_{v}$):

[TABLE]

which can be rewritten as

[TABLE]

1.8 Hanging droplet motion

**We present here additional visualisations of hanging droplets moving due to wettability gradients (Figures 11 and 12).

**(See also Movie S4). The droplets are generated using a reservoir of water, whose free-surface is excited at its characteristic frequency using a shaker. Droplets in a given range of size are successfully captured thanks to the high normal adhesion of the liquid surface. Because of the inherent wettability gradient, droplets are then self-propelled towards the high wettability regions (right here). Samples are carefully leveled prior to each experiments, to ensure that gravity is not playing any role in the droplets motion.

1.9 Captions of the movies:

Movie S1.

Droplet rapid motion due to a discontinuity in wettability of a shaped liquid surface. The solid fraction is $f_{s}=0.1$ on the left and $f_{s}=0.9$ on the right. When released, the droplet moves rapidly towards the region of higher wettability (higher solid fraction). The scale bar is $1\text{\,}\mathrm{mm}$.

Movie S2.

Droplet deposited on a shaped liquid surface with a gradient of solid fraction ($\alpha=$0.05\text{,}{\mathrm{mm}}^{-1}$$). The gradient of wettability and the low pinning force allow the droplet to move towards the high wettability regions. The scale bar is $1\text{\,}\mathrm{mm}$.

Movie S3.

Top view of two droplets deposited on a shaped liquid surface with a gradient of solid fraction along a curved path. Droplets move along the curved rails towards the region of higher wettability. For convenience, movies are accelerated by a factor of 4.

Movie S4.

Droplets projected on an upside-down shaped liquid surface. Droplets are captured due to the surface high normal adhesion and then driven towards the high wettability region (right here). Droplets of different sizes move at different velocities.