The limit of droplet rebound angle

Abstract

Regulating the motion state of droplets after impacting on solid surfaces is crucial in many fields including self-cleaning, energy harvesting, and microfluidics. The rebound angle of the droplet is a key factor in determining its motion state. However, up until now, the limit of droplet rebound angle remains unidentified. Here, we reveal a previously undiscovered droplet rebound behavior that the droplet rolls rapidly along the surface with a rebound angle close to 0 degrees, the limit of the droplet rebound angle. Such unexpected behavior originates from the droplet behaving like two mutually perpendicular springs enabled by continuous asymmetric adhesion provided by the heterogeneous modified nanostructure. This boundary-rolling behavior of droplets contributes to scientific and technical advances in various fields that involve droplet-impact, as illustrated through examples of enhanced cleaning efficiency (improved by 349%) and well-controlled droplet transport in tortuous passages which can hardly be achieved before without external fields coupling.

Introduction

The research on droplet impact on solid surfaces has extensively explored the motion state of droplets during and after impact^{1,2,3,4,5,6,7}. Various phenomena have been observed, including vertical rebounding^8,9, lateral bouncing^{10,11,12,13,14}, and pinning^15,16,17. Diverse surface designs endow the droplets with different rebound angles $\theta$, which is defined as the angle between the droplet trajectory and the tangent to the surface at the moment of the droplets detaching from the impact center. The regulation of $\theta$ is essential for a wide range of applications such as self-cleaning^{18,19,20,21,22}, energy harvesting^{23,24,25,26,27}, and microfluidics²⁸.

Droplet passive transport, the movement of droplets driven solely by the kinetic energy of the droplet itself^18,22,29, contrasts with droplet active transport that is contingent on the external energy input, such as electric field³⁰, thermal field^11,31, magnetic field³², light field³³, and sound field³⁴, etc. Deciphering the code for $\theta$ regulation of droplet passive transport is of great significance both to apprehend the droplet intrinsic movement as well as optimizing the droplet-impact-related processes, such as droplet active transport on diverse and demanding conditions. Previous studies have shown that droplets typically rebound within a $\theta$ range of 26.7°–90° on condition of passive transport, as shown by statistics in Supplementary Note 1. Droplet rebound angles below 26.7° are rarely reported. Up until now, the limit of droplet rebound angle has remained elusive and challenging to reach.

Here, we address this issue by constructing heterogeneously modified nanostructures on the superhydrophobic surface, and thus report a previously unknown droplet rebounding behavior: after impacting on the solid surface, the water droplet rolls rapidly along the surface with $\theta$ close to 0 degrees, the limit of the droplet rebound angle. The heterogeneous nanostructures endow the droplet with a nanoscale continuous three-phase contact line as well as punctilious allocation, thus allowing us to reveal the underlying mechanism by introducing the concept of two mutually perpendicular springs system. The two adhesive forces in mutually perpendicular directions reduce the vertical velocity to zero while simultaneously accumulating the large lateral velocity, which can be analogized to the asymmetrical fixed hinge exerting forces on the spring system. The force measurement indicated that the reaction force of the asymmetric adhesion force can effectively offset the second pressure peak exerted by the droplet on the substrate. This counterintuitive finding differs significantly from previous studies^35,36. By investigating both the droplet and substrate characteristics, we identified the necessary conditions for achieving the rolling behavior of droplets with a rebound angle θ of 0° along the surface. This rolling behavior along surfaces can be extended to inclined surfaces, concave surfaces, and convex surfaces, which offer potential optimization and inspire new designs for various droplet-impact-related applications, as illustrated through examples of significantly enhanced cleaning efficiency and well-controlled droplet transport in tortuous passages, which can hardly be achieved before without external assistance.

Results

Horizontal rebound

By constructing a superhydrophilic (SHL) arc (contact angle <3°, see contact angle characterization in Supplementary Fig. 2) on the superhydrophobic (SHB) surface (contact angle = 158°), we obtain a patterned wettability surface (PW surface). The line width L of the SHL arc in Fig. 1 is 200 μm, and the opening radian angle α is π/5. The radius of the SHL arc R is 2.5 mm, which is close to the maximum spreading radius of the droplets. The droplet maximum spreading radius is related to droplet diameter D and impact velocity v₀. The arc radius in the experiment needs to be less than or equal to the maximum spreading radius, to ensure that each position of the droplet can be pinned by the superhydrophilic arc when it retracts and thus be affected by the adhesion force (see details in Supplementary Note 2). The boundary between the SHL and SHB regions is clearly shown in the inset on the upper right side of Fig. 1a. The left SHL region is covered with water. The insets on the lower right side show the nanostructures both on the SHL and SHB regions, indicating the nanoscale continuous three-phase contact line. Weber number is ${{{\rm{We}}}}=\rho {v}_{0}^{2}{R}_{0}/\gamma$, where $\rho$, ${R}_{0}$ and $\gamma$ are the droplet density, radius, and surface tension, respectively. When a water droplet (We = 32.8) is liberated and impacts on the PW surface, the droplet spreads in 2.2 ms and then the liquid film is peeled off along the arc from left to right under the action of the SHL arc, as shown in Fig. 1b (t = 3.4 ms and t = 6.4 ms) and Supplementary Video 1. At t = 14.4 ms, the liquid film moves to the right and is about to break away from the surface.

Fig. 1: Horizontal droplet rebound.

a Chronophotography of the side view of the droplet rebounding from the PW surface. The droplet rebound angle $\theta$ is 0°. The moments for each picture are (from 0 − 6): −11.2 ms, − 7 ms, − 2.2 ms, 0 ms, 18.2 ms, 26.8 ms and 36.4 ms, respectively. Scale bar: 2 mm. The inset shows the schematic diagram of the parameters of the PW surface (patterned wettability surface). Dark blue-gray areas represent SHL (superhydrophilic) areas, and light gray areas represent SHB (superhydrophobic) areas. L, R, and α are the SHL arc width, the radius of the SHL arc, and the opening angle of the arc, respectively. The inset on the upper right shows the ESEM characterization. The inset on the lower right shows the SEM characterization. b Top and side views of the droplet rebound from the PW surface. Scale bars: 2 mm. c Schematics showing the mechanical analogies of the droplet rebounding on the PW surface. The fixed hinge and its binding effect on the spring system are represented by blue and green dashed lines, respectively. m, ${v}_{0}$, $v$, ${F}_{{{{\rm{spr}}}}-{{{\rm{z}}}}}$, and ${F}_{{{{\rm{spr}}}}-{{{\rm{x}}}}}$ represent the droplet mass, the droplet initial velocity, the droplet velocity after detaching from the fixed hinge, the z-direction force of the vertical spring, and the x-direction force of the horizontal spring, respectively. The overall stretchability of the droplet due to surface tension is represented by two mutually perpendicular springs.

Immediately afterwards, the droplet rolls, surprisingly, horizontally along the surface with a rebound angle $\theta$ of 0 degrees, as shown in the side-view chronophotography in Fig. 1a (Supplementary Video 2). Supplementary Fig. 4 shows such distinct rebound behavior by comparing the droplet rebounding trajectories on the PW surface with that on the SHB surface. Previous studies have proved that the droplet rebound process can be analogized with the vertically oriented spring^{16,29,37,38,39}. Here we demonstrate that the horizontal rolling behavior of droplets can be analogized with two mutually perpendicular springs (vertically and horizontally) under the action of the adhesion force exerted by the nanoscale continuous structure, as shown in Fig. 1c. During the movement of the two springs, the fixed hinge will exert force at the appropriate time. Consequently, the two springs experience two forces during contraction and expansion:

$${F}_{{{{\rm{spr}}}}-{{{\rm{z}}}}}={k}_{{{{\rm{z}}}}}\Delta z$$

(1)

and

$${F}_{{{{\rm{spr}}}}-{{{\rm{x}}}}}={k}_{{{{\rm{x}}}}}\Delta x$$

(2)

where ${F}_{{{{\rm{spr}}}}-{{{\rm{z}}}}}$, ${k}_{{{{\rm{z}}}}}$, $\Delta z$, ${F}_{{{{\rm{spr}}}}-{{{\rm{x}}}}}$, ${k}_{{{{\rm{x}}}}}$, and $\Delta x$ are the z-direction force, the stiffness, the distance pushed or pulled of the vertical spring, the x-direction force, the stiffness, and the distance pushed or pulled of the horizontal spring, respectively. The two forces ${F}_{{{{\rm{spr}}}}-{{{\rm{z}}}}}$ and ${F}_{{{{\rm{spr}}}}-{{{\rm{x}}}}}$ and the springs velocities meet the following relationships: $\int {F}_{{{{\rm{spr}}}}-{{{\rm{z}}}}}{dt}=m{v}_{0}$, and $\int {F}_{{{{\rm{spr}}}}-{{{\rm{x}}}}}{dt}={mv}$, where ${v}_{0}$, $v$, and $t$ are the initial velocity of the system, the velocity after breaking away from the fixed hinge support, and the time when the fixed hinge takes effect, respectively. By elegant control of the fixed hinge support, the two springs can effectively convert vertical velocity ${v}_{0}$ into horizontal velocity $v$.

Vertical adhesion of the droplet

This interesting horizontal rolling behavior of water droplets, which reaches the limit of $\theta$, drives us to explore the mechanism behind it. Just as the fixed hinge of the two-spring system exerts force on the springs shown in Fig. 1c, the force regulation of the droplet is obtained by the SHL pattern on the SHB surface, as shown in Fig. 2a. After the droplet reaches the maximum spreading, the droplet edge is pinned by the SHL arc during the droplet retraction. The SHL arc exerts two effects on the droplet, and the droplet is subjected to two forces, just like the ${F}_{{{{\rm{spr}}}}-{{{\rm{z}}}}}$ and ${F}_{{{{\rm{spr}}}}-{{{\rm{x}}}}}$ of the spring system. On the one hand, in the z-direction, the droplet is subjected to a dominant vertical adhesion force ${F}_{{{{\rm{z}}}}}={\gamma L}_{{{{\rm{z}}}}}\sin \beta$, by virtue of the continuous nanoscale structure, where $\gamma$, $\beta$, and ${L}_{{{{\rm{z}}}}}$ are the surface tension, dynamic contact angle, and the action length of the z-direction force (see details in Supplementary Note 3), respectively⁴⁰. And the accumulation of ${F}_{{{{\rm{z}}}}}$ over time t reduces the vertical rebound velocity of the droplet. When $L$ = 200 μm, α = π/10, the vertical velocity at the moment of droplet separation can be reduced to 0 m/s. Different vertical velocities of the arc for different α are shown in Supplementary Fig. 5i. Measurements of the impact force F in the vertical direction, as shown in Fig. 2b, demonstrates that the presence of the SHL arc reduces the droplet-to-substrate force to nearly 0 mN during retraction, which is quite different from that on SHB surface as well as the previous studies^35,36 (The second peak of the force is significantly reduced, see the inset in Fig. 2b and Supplementary Note 4). According to Newton’s third law, we can speculate that it is the reaction force of the vertical adhesion force ${F}_{{{{\rm{z}}}}}$ of the substrate that offsets the droplet’s pressure on the substrate, which can be expressed as ${F}_{{{{\rm{z}}}}}={F}_{{{{\rm{SHB}}}}}-{F}_{{{{\rm{PW}}}}}$, where ${F}_{{{{\rm{SHB}}}}}$ and ${F}_{{{{\rm{PW}}}}}$ are the measured impact and retracting force of z-direction on the SHB surface and PW surface, respectively. According to the momentum theorem, the disappearance of the second droplet impact force peak denotes that the droplet is unable to accumulate vertical upward velocity. The vertical upward velocity from force measurement integration meets well with that from high-speed photography, as shown in Supplementary Note 5. Since it is hard to observe the real-time contact angle $\beta$ at each contact line position in the experiment, the change of ${F}_{{{{\rm{z}}}}}/\sin \bar{\beta }={\gamma L}_{{{{\rm{z}}}}}$ with time t is plotted, where $\bar{\beta }$ is the weighted arithmetic mean of $\beta$, as shown in Fig. 2c. Due to the existence of the SHL arc, the droplet is subjected to the persistent adhesion force ${F}_{{{{\rm{z}}}}}$ of the substrate compared to the SHB surface during the whole rebounding process. The same trend of ${F}_{{{{\rm{z}}}}}/\sin \bar{\beta }$ and ${F}_{{{{\rm{SHB}}}}}-{F}_{{{{\rm{PW}}}}}$ shown in Supplementary Fig. 5j supports the above analysis.

Fig. 2: Mechanism of droplet horizontal rebound.

a Schematic illustration of droplet retraction on the PW surface. b Measured impact and retraction force of z-direction F on the PW surface F_PW and SHB surface F_SHB. t = 0 is set when the droplet reaches its maximum spreading radius. c The force F_z in the vertical direction during the retraction process of the droplet on the PW surface and the SHB surface. Here, the force is expressed as ${F}_{{{{\rm{z}}}}}/\sin \bar{\beta }={\gamma L}_{{{{\rm{z}}}}}$ and plotted by data of ${\gamma L}_{{{{\rm{z}}}}}$. d The force F_x in the vertical direction during the retraction process of the droplet on the PW surface and the SHB surface. The “+” and “−” denote the force is to the right and left, respectively, as distinguished by different colors. Here, the force is expressed as ${F}_{{{{\rm{x}}}}}/\overline{(\cos {\beta }_{1}-\cos {\beta }_{2})}=\gamma {L}_{{{{\rm{x}}}}}$ and plotted by data of $\gamma {L}_{{{{\rm{x}}}}}$.

Horizontal adhesion of the droplet

On the other hand, in the x-direction, the contact angles of the left (${\beta }_{1}$) and right sides (${\beta }_{2}$) are different during the retraction process of the droplet. Therefore, like the ${F}_{{{{\rm{spr}}}}-{{{\rm{x}}}}}$ of the spring system, the droplet will experience an asymmetric x-direction adhesion force ${F}_{{{{\rm{x}}}}}=\gamma {L}_{{{{\rm{x}}}}}(\cos {\beta }_{1}-\cos {\beta }_{2})$^41,42, where ${L}_{{{{\rm{x}}}}}$ is the action length of the x-direction force (see details in Supplementary Note 6). The accumulation of ${F}_{{{{\rm{x}}}}}$ over time t converts to the lateral rebound velocity of the droplet. Similar to Fig. 2c, since it is hard to accurately obtain the real-time contact angle of the left ends ${\beta }_{1}$ and right ends ${\beta }_{2}$ of the droplet, ${F}_{{{{\rm{x}}}}}/\overline{(\cos {\beta }_{1}-\cos {\beta }_{2})}=\gamma {L}_{{{{\rm{x}}}}}$ is used to characterize the adhesion force received by the droplet in the horizontal direction, where$\overline{\,(\cos {\beta }_{1}-\cos {\beta }_{2})}$ is the weighted arithmetic mean of $\left(\cos {\beta }_{1}-\cos {\beta }_{2}\right)$, as shown in Fig. 2d. According to the relative size of the left (${\beta }_{1}$) and right contact angles (${\beta }_{2}$), the variation process of ${F}_{{{{\rm{x}}}}}/\overline{(\cos {\beta }_{1}-\cos {\beta }_{2})}$ can be divided into three stages according to the side view of the droplet rebounding process shown in Supplementary Fig. 6c–f, namely ${\beta }_{1}$ > ${\beta }_{2}$, ${\beta }_{1}={\beta }_{2}$, and ${\beta }_{1}$ < ${\beta }_{2}$. When ${\beta }_{1}$ > ${\beta }_{2}$, the x-direction force on the droplet is to the right, which is denoted by “+” in Fig. 2d; and when ${\beta }_{1}$ < ${\beta }_{2}$, the x-direction force on the droplet is to the left, which is denoted by “−”. Here, the value of the lateral adhesion force can be calculated if we assume that the left and right contact angles observed from the side view are equivalent to the actual left and right contact angles, respectively (see Supplementary Fig. 6g–i). The calculated droplet lateral velocity is close to that measured by high-speed photography, which indicates that the lateral force analysis above is reliable.

General principles on the pattern design of the substrate

Next, the general principles that enable water droplets’ horizontal rebound behaviors from aspects of substrate design and droplet conditions are investigated. First, the pattern design of the substrate. The pattern design regulating ${F}_{{{{\rm{z}}}}}$ and ${F}_{{{{\rm{x}}}}}$ can be analogized to the fixed hinge of the spring system regulating ${F}_{{{{\rm{spr}}}}-{{{\rm{z}}}}}$ and ${F}_{{{{\rm{spr}}}}-{{{\rm{x}}}}}$ by means of adjusting $\Delta z$ and $\Delta x$ in (1) and (2), respectively. From the above analysis, it can be seen that the line width $L$ and the radian angle $\alpha$ of the arc vacancy are key factors in regulating the adhesion force during the droplet rebound process. Here, two dimensionless parameters are defined, namely the liquid film peripheral coverage vacancy $\alpha ^{\prime}$:

$$\alpha ^{\prime}=\alpha /(2\pi )$$

(3)

and the liquid film gripping degree ${L}^{{\prime} }:$

$$L^{\prime}=L/D$$

(4)

to describe the regulation of the SHL pattern on the droplet, where D is the droplet diameter. ${\alpha }^{{\prime} }$ and ${L}^{{\prime} }$ need to work together to ensure that the vertical adhesion can just offset the vertical upward velocity during the droplet retraction process. At the same time, the lateral adhesion force provided by the pattern needs to enable the droplet to accumulate sufficient lateral velocity and to detach from the pattern. As shown in Fig. 3a, when the gripping degree of the liquid film ${L}^{{\prime} }$ is small (${L}^{{\prime} }$ = 0.047) and the peripheral coverage vacancy $\alpha ^{\prime}$ is small (${\alpha }^{{\prime} }$ = 0.029), the droplet can rebound laterally (the blue area). As $L^{\prime}$ becomes larger, a wider region of ${\alpha }^{{\prime} }$ enables the horizontal rebound behavior of the droplet. When the liquid film gripping degree ${L}^{{\prime} }$ exceeds 0.38, it is no longer possible to achieve horizontal rebounding behavior by increasing $\alpha ^{\prime}$. It is indicated that the adhesion forces of the lateral and vertical directions are uncoordinated, and the droplet will remain pinned on the substrate as a result of the high adhesion force (the yellow area) or rebound upward laterally as a result of inadequate vertical adhesion (the pink area). It should be noted that the large ${L}^{{\prime} }$ (i. e., 0.38) is accompanied by large amounts of water residue remaining on the SHL patterns (Maximum ${{V}}_{{{{\rm{residual}}}}}/{V}_{{{{\rm{droplet}}}}}$ is around 26.9 %, see details in Supplementary Note 7), meaning that only a small part of the droplet can achieve horizontal rebounding behaviors.

Fig. 3: Phase diagram.

a Rebound behavior of droplets impacting on different patterned surfaces characterized by ${L^{\prime}}=L/D$ and $\alpha ^{\prime}=\alpha /(2\pi )$. R = 2.5 mm, D = 2.1 mm. We = 32.8. Solid squares represent droplet rebound laterally, and open squares represent droplets pinned, rebound upward laterally, and rebound vertically, with each behavior distinguished by background color. b Rebound behavior of droplets impacting on different patterned surfaces characterized by $Q={v}_{{{{\rm{z}}}}}/{v}_{{{{\rm{x}}}}}$ at different ${{{\rm{We}}}}$. ${{{{\rm{We}}}}}_{{{{\rm{c}}}}}$ is the droplet critical splashing Weber number on a superhydrophobic surface, and ${{{{\rm{We}}}}}_{{{{\rm{c}}}}-{{{\rm{PW}}}}}$ is the droplet critical splashing Weber number on the PW surface. Different Weber numbers here are obtained by regulating the droplet impact velocity v₀, with the droplet size unchanged.

General principles on the droplet condition

For the droplet conditions, the droplets need to meet the following two conditions to achieve horizontal rebounding behavior: (i) the liquid film needs to be intact during the spreading process, where the intact liquid film ensures the interaction between the SHL patterns and the liquid film periphery, and (ii) the droplet has enough kinetic energy to detach from the SHL patterns on the surface. For the springs, (i) corresponds to the large external force and large deformation, which doesn’t exceed the yield strength and the springs wouldn’t fail or break into pieces. The critical spring fall velocity ${v}_{0-\max }$ represents the maximum external force ${F}_{{{{\rm{spr}}}}-{{{\rm{z}}}}-\max }$ that the spring can withstand: ${v}_{0-\max }=\int {F}_{{{{\rm{spr}}}}-{{{\rm{z}}}}-\max }{dt}/m$. Hence, the spring fall velocity ${v}_{0}$ needs to satisfy: ${v}_{0}/{v}_{0-\max } < 1$ to achieve horizontal rebounding. (ii) corresponds to the small external force and small deformation where the spring is able to get rid of the shackles of the fixed hinge. The critical velocity of spring fall ${v}_{0-\min }$ represents the minimal external force ${F}_{{{{\rm{spr}}}}-{{{\rm{z}}}}-\min }$ that the springs need to achieve the horizontal rebounding behavior: ${v}_{0-\min }=\int {F}_{{{{\rm{spr}}}}-{{{\rm{z}}}}-\min }{dt}/m$. The spring fall velocity ${v}_{0}$ needs to satisfy: ${v}_{0}/{v}_{0-\min } > 1$. To simplify the description, a dimensionless parameter $Q={v}_{{{{\rm{z}}}}}/{v}_{{{{\rm{x}}}}}$ is defined to describe the horizontal rolling behavior of the droplet, where ${v}_{{{{\rm{z}}}}}$ and ${v}_{{{{\rm{x}}}}}$ are the droplet vertical and lateral velocities when the droplet detaches from the surface, respectively. When ${v}_{{{{\rm{z}}}}}\approx 0$ and ${v}_{{{{\rm{x}}}}}\, \ne \, 0\,$($Q=0$), it corresponds to the droplet’s horizontal rebounding behavior along the surface. When ${v}_{{{{\rm{z}}}}}\ne 0$ and ${v}_{{{{\rm{x}}}}}\, \ne \, 0$ ($Q\, \ne \, 0$), it corresponds to the droplet rebounding upward laterally. When ${{v}}_{{{{\rm{z}}}}}\, \ne \, 0$ and ${v}_{{{{\rm{x}}}}}\approx 0\,$($Q\to \infty$), it corresponds to the droplet rebounding vertically. When ${{v}}_{{{{\rm{z}}}}}\approx 0$ and ${v}_{{{{\rm{x}}}}}\approx 0$ ($Q\to \infty$), it corresponds to the droplet pinning on the surface. To meet the droplet condition (i), splashing is a situation that ought to be avoided where the liquid film breaks into smaller pieces of droplets during the droplet spreading/retraction process. Inspired by the critical velocities of the springs system, the droplet critical Weber number ${{{{\rm{We}}}}}_{{{{\rm{c}}}}}$, below which the liquid film is intact and above which the splashing occurs^{43,44,45,46,47,48,49,50}, is introduced to identify the droplet horizontal rebounding behavior limit⁴³:

$${{{{\rm{We}}}}}_{{{{\rm{c}}}}}=\frac{\rho {v}_{{{{\rm{c}}}}}^{2}{R}_{0}}{\gamma }$$

(5)

where ${v}_{{{{\rm{c}}}}}$ is the critical velocity for splashing. We set up three kinds of line width ${L}^{{\prime} }$ arrays: 0.024, 0.095, and 0.238. Three values of ${\alpha}^{{\prime} }$ were set: 0.166, 0.500 and 0.833. It can be seen from Fig. 3b that the droplet critical Weber number ${{{{\rm{We}}}}}_{{{{\rm{c}}}}}$ on superhydrophobic surface is: ${{{{\rm{We}}}}}_{{{{\rm{c}}}}}$ = 47.3, where the liquid film remains intact during retracting process at ${{{\rm{We}}}}$ = 32.8 while small droplets separated from the film edge at$\,{{{{\rm{We}}}}}_{{{{\rm{c}}}}}=\,47.3$. This result is consistent with Riboux and Gordillo’s droplet splashing model on a superhydrophobic surface⁴³. Interestingly, the horizontal rolling behavior of droplets ($Q=0$) can be achieved through pattern design at ${{{\rm{We}}}}$ = 47.3, which indicates that the droplet critical Weber number of the patterned wettability surface ${{{{\rm{We}}}}}_{{{{\rm{c}}}}-{{{\rm{pw}}}}}$ (${L}^{{\prime} }$- ${\alpha}^{{\prime} }$:0.095 – 0.166 and 0.238 – 0.166) is inconsistent with the ${{{{\rm{We}}}}}_{{{{\rm{c}}}}}$ of the superhydrophobic surface. This is because when a small number of satellite droplets appear at the edge of the largest spreading edge of the droplet, the existence of the superhydrophilic arc can merge the satellite droplets with the droplet matrix into a complete droplet again (see details in Supplementary Fig. 8). At larger ${{{\rm{We}}}}$ (i.e., ${{{\rm{We}}}}$ = 64.3), the horizontal rolling behavior cannot be achieved where the droplets have large kinetic energy and cannot remain as a continuous liquid film after impacting the surface, as shown in Supplementary Fig. 9a. Hence, ${{{{\rm{We}}}}}_{{{{\rm{c}}}}-{{{\rm{pw}}}}}$ is 64.3 for the patterned wettability surface where droplet can achieve lateral rebounding in Fig. 3b. At smaller ${{{\rm{We}}}}$ (i.e., ${{{\rm{We}}}}=3.6$), the kinetic energy and the rebound velocity are small so that the droplet cannot overcome the adhesion force to the superhydrophilic arc. Consequently, the droplets are hard to detach from the substrate in time and cannot achieve horizontal rebounding behavior (Supplementary Fig. 9b). Meanwhile, the effect of Weber number on droplet rebound angle when L’ and α’ are constant is discussed in Supplementary Note 8.

The boundary-rolling rebound behavior

The horizontal rebounding behavior can be extended to the boundary-rolling rebound behavior, which is defined as the rebound behaviors rolling along the substrate boundary, as shown in Fig. 4. The droplet rolling upward along the surface against gravity after impacting on the solid surface is considered to be impossible without the coupling of other external fields^11,30,51. Here, by introducing an asymmetric interface, the droplet rolling upward along the surface can be realized by exploiting the droplet’s kinetic energy. As shown in Fig. 4a and Supplementary Video 3, a side view of a droplet rolling upward along an inclined surface (inclination angle = 17.4°) after impacting is presented. Similar to the horizontal surface, the velocity component of the droplet perpendicular to the inclined surface ${v}_{\perp }$ is reduced to 0 under the continued action of the adhesion force, while the velocity component along the inclined surface ${v}_{//}$ gradually accumulates until the droplet detaches from the substrate (see details in Supplementary Note 9). The chronophotography of the side view of the droplet rolling downwards an inclined surface (inclination angle = 17.4°) is shown in Fig. 4b and Supplementary Video 4. Moreover, the droplet can roll upwards along the arc surface without external assistance, as shown in Fig. 4c and Supplementary Video 5. The droplet can also roll downwards along the arc surface without bouncing or detaching from the arc surface, as shown in Fig. 4d and Supplementary Video 6.

Fig. 4: Boundary-rolling rebound behavior of droplets on surfaces with different topography.

a Chronophotographic side view of the droplet boundary rolling upwards along an inclined surface (inclination angle = 17.4°). The moments for each picture are (in the order of the direction of the arrow): − 2.2 ms, 0 ms, 19.0 ms, 28.0 ms, and 37.0 ms, respectively. b Downwards along an inclined surface (inclination angle = 17.4°). The moments for each picture are (in the order of the direction of the arrow): − 2.2 ms, 0 ms, 9.8 ms, 21.8 ms, and 32.2 ms, respectively. c Upwards along the arc surface. The moments for each picture are (in the order of the direction of the arrow): − 2.2 ms, 0 ms, 28.4 ms, 53.4 ms, and 100.8 ms, respectively. d Downwards along the arc surface. The moments for each picture are (in the order of the direction of the arrow): − 2.0 ms, 0 ms, 30.8 ms, 63.2 ms, and 82.0 ms, respectively. Scale bars of (a–d): 2 mm.

Applicable examples

Such droplet boundary-rolling rebound behavior has wide applications in many fields, such as cleaning pollutants, microfluidics, and droplet transport. For instance, the cleaning efficiency can be greatly improved by incorporating the droplet boundary-rolling rebound behavior. As shown in Fig. 5a and Supplementary Video 7, after the droplet contacts the PW surface, the droplet rolls along the inclined surface (inclination angle = 17.4°) covered with contaminants and carries away a large amount of pollutants, leaving a clean pathway indicated by the white dashed line. While the same droplet impacts on the SHB surface, the droplet bounces away (indicated by white dotted circle) and carries away few pollutants (Fig. 5b). After counting the contact area between the droplet and the contaminant, the cleaning efficiency of the PW surface is 349% higher (see details in Supplementary Fig. 11).

Fig. 5: Applicable examples of the boundary-rolling rebound behavior of droplets in pollutant removal and droplet fluidics.

a, b Cleaning performance characterization on PW surface (a) and SHB surface (b). Inclination angle = 17.4°. Scale bars: 10 mm. c Droplets fusion in the L-shaped channel. d Droplet transportation in a complex-shaped channel. Scale bars of (c, d): 10 mm.

The boundary-rolling rebound behavior of droplets expands the ability of droplet transportation and fusion in confined channels, as shown in Figs. 5c, d. For the L-shaped channel shown in Fig. 5c, it is considered impossible to fuse the droplets at the entrance and the outlet through the channel without external field coupling. Here, by exploiting the PW surface, droplet 1 at the entrance of the L-shaped channel converts its vertical velocity into horizontal velocity, moves toward and fuses with droplet 2, during which the energy input is only the kinetic energy of droplet 1 (see Supplementary Video 8). Meanwhile, droplet transportation in the complex-shaped channel can also be realized. As shown in Fig. 5d and Supplementary Video 9, by applying the PW surface, the droplet can climb to the peak of the slope and travel down the slope to the destination, which provides new approaches for droplet transport and droplet fluidics.

Discussion

The interaction force between the droplet and the solid surface controls the droplet motion state after the droplet rebounds. The heterogeneously modified nanostructure provides a compelling way to elegantly regulate the adhesion force. In this work, by designing the heterogeneously modified nanostructure, we introduced a previously unreported droplet rebound behavior with a rebound angle of 0° and determined the limit of the droplet rebound angle. The mechanism lies in two mutually perpendicular springs that are enabled by continuous asymmetric adhesions provided by the heterogeneous nanostructure, wherein a vertical spring receives a continuous adhesion force to minimize the vertical velocity of the droplet, and a horizontal spring simultaneously converts the accumulated momentum into high-speed horizontal droplet motion. The droplet rebound behavior with a rebound angle of 0° can be extended to inclined surfaces, concave surfaces, and convex surfaces.

It’s accepted that a droplet impacting on the SHB surface is accompanied by two pressure peaks due to the impact force at drop touchdown and the Worthington jet formation, respectively. This study proves that the pressure peak originates from Worthington jet formation can be obliterated by heterogeneously modified surface, which provides a method to suppress the Worthington jet formation and may be utilized in the mechanical sensing equipment, etc.

From a broader prospect, the determination of the limit of droplet rebound angle enriches the toolbox for apprehending droplets impacting on solid surfaces, which will enable the optimization of droplet impacting process as illustrated by the example of improving cleaning efficiency and well-controlled droplet transport ability in tortuous passages which can hardly be achieved before without external fields coupling, thus being of considerable significance for multi-dimensional droplet-related manipulation for various applications, water collection, spray cooling, fertilizer and insecticide spraying, to name a few.

Methods

Fabrication of samples

• 1) Superhydrophobic samples.

  The aluminum plates were cut into the desired size and were corroded by mixed acid solution (40 mL HCl (37 wt.%), 12.5 mL H₂O, and 2.5 mL HF (40 wt.%)) for 20 s. After being blown dry and plasma treatment, the plates were grafted with 1H, 1H, 2H, and 2H-Perfluorodecyltriethoxysilane for 3 h at 90 °C. After fluorination, the plates became superhydrophobic.

• 2) Patterned wettability samples.

  Two methods are provided to make the surface superhydrophilic. One is the mask-UV lithography method. The superhydrophilic area is achieved by photomask under UV radiation, removing the grafted 1H, 1H, 2H, and 2H-Perfluorodecyltriethoxysilane molecules. The other is the laser etching method. The superhydrophilic area is achieved by laser etching, where the laser removes the grafted molecules on the surface.

Droplet impacting process

The droplet-impacting processes are conducted at room temperature. Water droplets were generated from fine needles from pre-determined heights. The size of droplets is regulated by needles with different inner diameters.

High-speed photography

The droplet impacting processes are recorded by the Phantom M110 camera and IX speed 510.

Force measurement

The impact force is measured by synchronizing high-speed photography with fast-force sensing. The aluminum plates were placed above a high-precision piezoelectric force transducer (Kistler 9215 A) with a resolution of 0.5 mN. The sampling rate is 100 kHz.