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Enhancement of coalescence-induced nanodroplet jumping on superhydrophobic surfaces Fang-Fang Xie, Gui Lu, Xiao-Dong Wang, and Dan-Qi Wang Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b02428 • Publication Date (Web): 22 Aug 2018 Downloaded from http://pubs.acs.org on August 26, 2018

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Enhancement of coalescence-induced nanodroplet jumping on superhydrophobic surfaces

Fang-Fang Xie1,2,3, Gui Lu1,2,3*, Xiao-Dong Wang1,2,3*, Dan-Qi Wang1,2,3 1

State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China; 2

Research Center of Engineering Thermophysics, North China Electric Power University, Beijing 102206, China;

3

Key Laboratory of Condition Monitoring and Control for Power Plant Equipment of Ministry of Education, North China Electric Power University, Beijing 102206, China.

ABSTRACT: Coalescence-induced droplet self-jumping on superhydrophobic surfaces has received extensive attentions over the past decade because of its potential applications ranging from anti-icing materials to self-sustained condensers, in which a higher jumping velocity vj is always expected and favorable. However, the previous studies have shown that there is a velocity limit with vj≤0.23uic for microscale droplets and vj≤0.127uic for nanoscale droplets, where uic is referred to as the inertial-capillary velocity. Here we show that the jumping velocity can be significantly increased by patterning a single groove, ridge, or more hydrophobic strip, whose size is comparable with the radius of coalescing droplets, on a superhydrophobic surface. We implement molecular dynamics simulations 1

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to investigate the coalescence of two equally sized nanodroplets (8.0 nm in radius) on these surfaces. We found that a maximum vj=0.23uic is achieved on the surface with a 1.6 nm high and 5.9 nm wide ridge, which is 1.81 times higher than the nanoscale velocity limit. We also demonstrate that the presence of groove, ridge, and strip alters coalescence dynamics of droplets, leading to a significantly shortened coalescence time which remarkably reduces viscous dissipation during coalescence; thus, we believe that the present approach is also effective for microscale droplet jumping.

KEYWORDS: superhydrophobic surfaces; coalescence; jumping velocity; liquid bridge; substrate structure.

*Corresponding Author: Gui Lu, Tel. and Fax: +86-10-62321277, E-mail: [email protected] *Corresponding Author: Xiao-Dong Wang, Tel. and Fax: +86-10-62321277, E-mail: [email protected]

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INTRODUCTION Recently, coalescence-induced jumping of droplets on superhydrophobic surfaces has received great attentions [1-7] due to its promising applications in condensation heat transfer [8-10], self-cleaning [11, 12], anti-icing [13-15], antidew [16], water harvesting [17], and so forth. Among these applications, a larger jumping velocity or height is always expected and favorable. However, previous studies have demonstrated that there is a velocity limit of vj≤0.23uic for microscale droplets [18-20] and vj≤0.127uic for nanoscale droplets [21, 22], implying about 1.6~5.3% excess surface energy converted into effective kinetic energy. Here, uic=(γ/ρR)1/2 is referred to the inertial-capillary velocity, where γ is the surface tension of liquid, ρ is the density of liquid, and R is the initial radius of droplets before coalescence. During coalescence, liquid viscosity and adhesion between the droplet and surface cause energy dissipation. Therefore, a feasible way to raise the jumping velocity is to reduce viscous dissipation and adhesion work as much as possible. Many efforts have been devoted to reducing adhesion work by making surfaces more superhydrophobic [1, 5, 8, 23-25] via various micro-/nanostructures. The upper limit of contact angle is 180 degrees, so that adhesion work is minimized on such surfaces. Furthermore, liquid may penetrate into surface asperities during the coalescence, leading to a wetting transition from the Cassie to the Wenzel state [26, 27]; thus, surface super-hydrophobicity fails and hence the jumping velocity will decrease significantly due to the increased adhesion work [28]. Viscous dissipation depends on liquid viscosity and coalescence time [20]. The coalescence time is defined as 3


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the time elapse beginning when two droplets just contact and ending when the coalesced droplet jumps from the surface. In practical applications, the viscous properties are not tunable when the fluid is selected. Therefore, the only approach for reducing viscous dissipation is to shorten the coalescence time. When two droplets come into contact with each other on a superhydrophobic surface, a liquid bridge is generated between the two droplets and grows rapidly till it impacts the surface. It has been demonstrated that the impact of the liquid bridge on surfaces is responsible for the jumping [1, 18, 22]. Accordingly, the coalescence time is closely associated with the growth and impact of the liquid bridge. In this work, we explore three effective approaches to reduce the coalescence time. The first and second are to put a single groove or ridge on a superhydrophobic surface, and the third is to coat a strip with higher hydrophobicity on a superhydrophobic surface. The groove, ridge, or strip is just located beneath the center of two droplets and its size is comparable with the droplet radius, as shown in Fig. 1. The basic idea is that the presence of groove or ridge alters impact dynamics of liquid bridge of the coalesced droplet and the mixed-wettability reduces the height of mass center of the coalesced droplet, so that a stronger impact or a shorter distance for liquid bridge impacting the surface can be achieved. To verify our idea, coalescence and jumping of two equally sized nanodroplets on the designed surfaces are investigated via molecular dynamics (MD) simulations. We expect that the jumping velocity will increase significantly, and hence the velocity limit would be broken.

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MD MODELING AND SIMULATIONS The initial configurations of the three-dimensional simulated systems are shown in Fig. 1, where a solid Au plate and two spherical Ar droplets are placed in a 51.48×30×29.78 nm3 box. It should be noted that although coalescence-induced jumping of water droplets has been extensively investigated in the previous studies; however, owing to the polar interactions between water molecules, an extremely time-consuming computation is required by MD simulations to model water droplets. Furthermore, our preliminary test shows that on a flat platinum surface with contact angle of 180°, the jumping velocity is vj=0.125uic for two water droplets with equal radius of 3.5 nm, indicating that the dimensionless jumping velocity, vj/uic, also follows the inertial-capillary scaling law. Therefore, the dimensionless jumping velocity is independent of the fluid type and solid materials, but only depends on the Ohnesorge number and the contact angle between fluid and substrate. Thus, to save computational cost, argon droplets are chosen here. The Au plate is modeled by face-centered cubic with lattice constant of 4.08 Å and is placed on the bottom of the box. Figures. 1(a) to 1(c) respectively represent three different plates decorated with a single groove, ridge, or more hydrophobic strip. The two droplets with radii of 8.0 nm and a spacing of 10.0 Å are placed just above the plate, and their center of mass is located on the bisector plane perpendicular to the groove, ridge, or strip. The depth and width are Hg and Wg for the groove, the height and width are Hr and Wr for the ridge, and the width is Ws for the strip. All the five geometric parameters are adjustable to search for a proper design yielding a larger jumping velocity. Because the velocity limit vj=0.127uic for nanoscale droplets is obtained for flat 5


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surfaces with a 180° contact angle [21, 22]. For fair comparison, the contact angle is set to 180° for the yellow region, while 162° for the gray region, as shown in Fig. 1. The interactions between Au-Au are described by the embedded atom model (EAM) [29]. The interactions between Ar-Ar and Au-Ar are based on Lennard-Jones 12-6 potentials, expressed as, 12 6 Vij = 4ε (σ / r ) − cij (σ / r )   

(1)

where σ donates the particle spacing when the potential is zero, and ε represents the minimum value of the potential. In this work, σ=3.41 Å and ε=10.3 meV. The interaction parameter cij between Au and Ar is adjusted to obtain different intrinsic contact angles, and cij=1 denotes the interaction between Ar and Ar. After preparation of the initial configurations and interaction force fields, the energy minimization is performed. The system is firstly equilibrated in the NVT ensemble for 10 ns at T=85 K by using the Nose–Hoover thermostat. Based on our tests, if the contact angle is directly set to 180°, droplets will depart from the Au plate during the energy minimization due to the weak interactions between the droplets and plate. Therefore, the contact angle is gradually increased during this period. The value of cij is firstly set as 0.3 for 4 ns, which can yield a large enough force between the droplets and plate to prevent the droplets from jumping off the surface; however, the contact angle is only 139° with this value of cij, which does not meet the requirement of 180°. Subsequently, the value of cij is reduced to 0.25 for 3 ns, and the contact angle increases to 162°. Finally, the value of cij is reduced to 0.2 for 3 ns, at which the contact angle reaches 180° and vapor-liquid coexistence occurs. The relation between the 6


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contact angle and the value of cij is shown in Fig. S1 in Supporting Information. After that, the thermostat is turned off and the system is run in the NVE ensemble for 6 ns to achieve a final equilibrium. It should be noted that, to prevent the two droplets from coalescence during this period, the interactions between two droplets is set to zero by specifying εAr-Ar=0 if two argon molecules belong to different droplets. After preparation of the equilibrium, the coalescence is performed in the NVE ensemble for 3000 ps. During this period, the interactions between two droplets are recovered. To make two droplets coalesce faster, a constant velocity of vx=+/-3 m s-1 is applied to the left/right droplets; however, once the droplets start to contact, the applied velocity is removed. We also examine other two velocities of vx=+/-1 and +/-2 m s-1, the extracted jumping velocities of the coalesced droplet for the three simulations are almost the same, with the maximum deviation less than 2.02%. The similar treatment was also employed in Ref. [21]. In the whole simulation process, periodic boundary conditions are applied in all three directions with a cut-off radius of 2.5 σ [21, 22] and a time-step of 6 fs. The positions and velocities of the atoms are recorded every 1000 time-steps. Velocity-Verlet algorithm is used to solve the particle motion equation. Simulation results are presented and analyzed by Ovito. The model was validated in our previous work [22] by comparing jumping velocities with Liang et al.’s ones [21] as well as comparing the simulated liquid bridge growing dynamics with Egger’s scaling law [30].

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ENERGETIC ANALYSIS The coalescence-induced droplet jumping results from the release of surface energy of droplets. The released excess surface energy is partly converted into kinetic energy during coalescence, and most of the energy is dissipated by both adhesion work and viscous dissipation. Therefore, energy conversion during coalescence of two droplets is analyzed to understand the influence of different superhydrophobic surfaces. Figure S2 in Supporting Information shows the schematic of two droplet placed on a superhydrophobic surface, just before their coalescence (Fig. S2(a)) and just jumping from the surface (Fig. S2(b)). On the basis of energy conservation law, the kinetic energy for droplet jumping, Ek, can be calculated by, Ek = ∆Es − ∆Ew − ∆Evis

(2)

where ∆Es is the released excess surface energy of the droplets, which can be expressed as [31], 2

 2-3cos θ + cos 3 θ  3 2 ∆Es = 4πγ (1 − cos θ ) R 2 -4πγ   R 2  

(3)

where θ is the contact angle of the droplet on the plate. The adhesion work, ∆Ew, comes from the variation of solid-liquid area and can be expressed as [31, 32], ∆Ew = 2γ (1 + cos θ ) Asl =2πγ sin 2 θ (1 + cos θ ) R 2

where Asl is the solid-liquid interface area.

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(4)

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The viscous dissipation of the droplets, ∆Evis, can be calculated by [33],

∆Evis = 2∫

t

∫

0 Ω

Φd Ωdt ≈2ΦΩτ tot

µ  ∂vi

∂v j  Φ=  +  2  ∂x j ∂xi  Ω=

(5)

2

(6)

π R 3 ( 2 − 3cos θ + cos 3θ )

(7)

3

where Φ represents the dissipation function, t is the time, Ω is the volume of a single droplet, τtot is the coalescence time defined as a time span from the onset of coalescence to the jumping of droplets, µ is the liquid viscosity, vi and vj is the component velocities, and xi and xj is the coordinates. Accurate calculation of viscous dissipation is very difficult, several approximate expressions were proposed [20, 31-33]; however, the viscous dissipation mainly comes from the coalescence direction (x-direction) along which two droplets merge with each other [34]. Hence, a more appropriate estimation should be, Φ =2τ xxτ xx ≈

1 u µ  2 R

2

(8)

where u is the average coalesced velocity of each droplet, which can be calculated by [20], u ≈ τ tot ⋅ ∆p ⋅ π R 2

3 ρπ R ( 2 − 3cos θ + cos 3 θ )

(9)

3

where ∆p=2γ/R is the interfacial tension pressure. Substituting Eqs. (7) to (9) into Eq. (5), the viscous dissipation can be rewritten as, ∆ E vis =

12πµγ 2 τ3 ρ 2 R 3 (2-3cos θ + cos 3 θ ) tot

(10)

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It should be noted that, the coalescence time is taken as the characteristic capillary time scale in Ref. [20], expressed as,

τ tot =

ρ R3 γ

(11)

Eq. (11) shows that the coalescence time is dependent on the initial radius of droplets and the fluid properties, and hence once these parameters remain unchanged, the coalescence time will be constant. For the present decorated surfaces, the size of groove, ridge, or strip is comparable with the initial radius of droplets. Thus, the dynamics of liquid bridge impacting the surfaces is inevitably altered by the presence of groove, ridge, or strip. Therefore, it is expected that the coalescence time will be dependent not only on the droplet radius and fluid properties but also on the size of groove, ridge, or strip. This expectation will be verified in the present molecular dynamics simulations by comparing the snapshots of coalescence on the flat surface with the decorated surfaces. Meanwhile, the coalescence time for these surfaces will also be extracted directly from the simulations to highlight the effects of the groove, ridge, or strip on it. It can be seen from Eqs. (3), (4), and (10) that, when the fluid properties, contact angle, and initial radius of droplets are fixed, the excess surface energy and adhesion work remain constant, while the viscous dissipation is proportional to the cube of the coalescence time. Accordingly, the kinetic energy for droplet jumping and the corresponding jumping velocity will be increased as the coalescence time

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reduces. On the basis of this reason, we expect that the limit of jumping velocity could be broken on the present decorated surfaces.

RESULTS AND DISCUSSION The time-lapse snapshots of the coalescence-induced jumping of two droplets on a flat superhydrophobic surface as well as on a superhydrophobic surface with a single groove, ridge, or strip are illustrated in Figs. 2(a) to 2(d). Similar with the cases on flat surfaces [21, 22, 35-37], the coalescence-induced jumping process can be divided into four stages: (i) the liquid bridge forms and grows; (ii) the bottom of the liquid bridge starts to impact the surface; (iii) the coalesced droplet starts to detach from the surface; (iv) the bottom of the coalesced droplet has detached from the surface and moves up in vapor. The corresponding component velocity of mass center of the coalesced droplet in the y-direction perpendicular to the surface is shown in Fig. S3 in Supporting Information. From this curve, the jumping velocity and the coalescence time can be extracted. The coalesced droplet jumps from the surface at τtot=1176 ps for the flat surface, 948 ps for the grooved surface, 900 ps for the ridged surface, and 924 ps for the stripped surface, and the corresponding jumping velocity is vj=3.39, 5.01, 6.37, and 5.45 m s-1, respectively. Figures 3(a) to 3(c) show the dimensionless jumping velocity, vj/uic, scaled with the inertial-capillary velocity. On the flat surface, the maximum dimensionless jumping velocity is 0.127, which agrees with the previous reports [21, 22]; however, the velocity limit is indeed broken for the three kinds of designed surfaces with specific geometric parameters. 11


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The surface tension, density, and viscosity of liquid Ar are γ=8.16±0.04 mJ m-2, ρ=1.31×103 kg m-3, and µ=190±1 µPa s [21, 22]. Therefore, the Ohnesorge number Oh=µ/(ργR) is 0.65 for the 8.0 nm Ar droplet. Recently, on the basis of experimental and numerical studies, Cha et al. [38] and Vahabi et al. [39] concluded that the Ohnesorge number in the coalescence-induced jumping on a superhydrophobic surface cannot be larger than 0.4. However, this conclusion is debatable due to the following two reasons. First, the experimental observation of dynamics of submicro-scale droplets is currently very challenging, partially because of the limited time and space resolutions of microscopic image capturing systems in the experiment [21]. Some early experiments [20, 34, 40-43] reported that the smallest droplet radius for observable jumping is about 10 µm for water. However, the recent experimental by Enright et al. [19] showed that jumping can still be observed for 0.5 µm water droplets. They presented that no jumping for droplets smaller than 10 µm can be attributed to the influence of surface adhesion in previous experiments. Second, the numerical simulations based on continuum-level models, such as phase-field model [18], showed that when the droplet radius is reduced from microscale to nanoscale, the liquid bridge cannot impact the solid surface and hence jumping no longer occurs. The no-slip boundary condition is commonly applied to the solid surface in continuum-level models; however, it is well-known that slip becomes very significant on superhydrophobic surfaces, for example, the slip length can reach 90 nm for liquid Ar flowing on a superhydrophobic Au plate [21]. Hence, the slip cannot be ignored for nanoscale droplets. Therefore, employing the no-slip condition is responsible for no jumping of nanoscale droplets in continuum-level 12


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models. Recently, Liang et al. [21] investigated the coalescence-induced jumping of nanoscale Ar droplets on an Au plate via MD simulations. Because MD simulations do not require any specific boundary conditions, Liang et al.’s simulations showed that jumping can still take place for nanoscale droplets with Oh ranging from 0.36 to 0.55. Subsequently, Xie et al. [22] and Gao et al. [35] also implemented MD simulations and confirmed that jumping can occur for nanoscale droplets with Oh larger than 0.4.

Effect of A Single Groove As seen in Fig. 3(a), for the surfaces with a groove, vj/uic exceeds the limit only at larger groove depths and smaller groove widths (Hg=1.6 nm and Wg=1.8, 3.9, 5.9, and 8.0 nm; Hg=1.2 nm and Wg=1.8 and 3.9 nm), and the maximum vj/uic=0.181 occurs at Hg=1.6 nm and Wg=1.8 nm. For other groove configurations, the velocity limit is not broken, particularly, for the grooves with Hg=0.8 nm, the jumping velocity is always lower than the limit no matter how the groove width changes. Because the impact of liquid bridge on surfaces is responsible for the droplet jumping [1, 6, 18, 22], the dynamics of liquid bridge impacting groove is analyzed in detail for various groove depths and widths, as shown in Fig. 4. It can be seen that the liquid bridge exhibits two different kinds of impact behaviors. For the grooves with larger depths and smaller widths (Hg=1.6 nm and Wg=1.8, 3.9, 5.9, and 8.0 nm; Hg=1.2 nm and Wg=1.8 and 3.9 nm), the expanding liquid bridge does not penetrate into the grooves throughout the whole coalescence process, while the complete penetration is observed for the 13


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other grooves. Owing to the negative curvature of the liquid bridge, the liquid within the droplets flows towards the liquid bridge driven by capillary pressure, leading to a rapid expansion of the liquid bridge along the direction perpendicular to the surface. The impact of liquid bridge on surfaces is similar to a droplet impacting textured surfaces. The previous studies [44-46] have shown that when the impact velocity, v, (or Weber number, We=ρRv2/γ) is large enough, liquid can penetrate into the surface asperities, causing the wetting transition from the Cassie-Baxter to the Wenzel state. Furthermore, the transition is triggered more easily for liquids with a lower surface tension [47] or grooves with a smaller ratio of depth to width [48, 49] due to their low energy barriers. For liquid argon, its surface tension is only 8.16±0.04 mJ m-2 far lower than that of water, so that the transition takes place more easily than water. On basis of the reasons above, the observed penetration of liquid bridge into the grooves with smaller depths and larger widths can be explained by the wetting transition from the Cassie-Baxter to the Wenzel state. Obviously, once the transition is triggered, the impact of liquid bridge on surfaces will be delayed, which leads to a longer coalescence time (Fig. 5(a)) and a larger viscous dissipation, and hence the jumping velocity reduces as compared with that on the flat surface. Furthermore, it should be noted that when the penetration of liquid bridge takes place, the liquid bridge requires more time to touch and impact the groove bottom for the grooves with a larger depth. Therefore, for the grooves with a large depth of 1.6 nm whose width ranging from 12.0 to 14.0 nm, the jumping velocity reduces more significantly than that of the grooves with a small depth of 1.2 or 0.8 nm. 14


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For the grooves with larger depths and small widths (Hg=1.6 nm and Wg=1.8, 3.9, 5.9, and 8.0 nm; Hg=1.2 nm and Wg=1.8 and 3.9 nm), the penetration of liquid bridge does not occur. Intuitively, the jumping velocity should be identical to that on the flat surface; however, it exceeds the velocity limit. This can be explained by the capillary pressure through the meniscus of liquid bridge when the surface is decorated with a groove. Because liquid cannot penetrate into the grooves, gas is trapped in the grooves. Meanwhile, the grooves have a 180° contact angle, according to the Young-Laplace equation, a large capillary pressure between the trapped gas and the liquid will be generated. A narrow groove width leads to a smaller meniscus radius, thus generating a larger capillary pressure [50-52]. Figure 6 shows the average pressure on the surface with or without groove (Hg=1.6 nm and Wg=1.8 nm) as a function of time. It can be seen that when the expanding liquid bridge starts to touch and impact the surface, an abrupt increase in pressure appears at the bottom of the coalesced droplet, which leads to acceleration of the coalesced droplet. Subsequently, the pressure gradually decreases due to deformability of liquid, and finally returns to the previous value as the coalesced droplet evolves into a nearly-spherical shape. At this moment, the acceleration of the coalesced droplet stops and the component velocity, vy, reaches its maximum value. Because the gas trapped in the groove induces the extra capillary pressure and exhibits strong liquid repellency [40], a higher peak pressure occurs for the surface with the groove of Hg=1.6 nm and Wg=1.8 nm, as shown in Fig. 6(b). Thus, a strong counterforce is exerted on the coalesced droplet, which accelerates the coalescence process and shortens the coalescence time (Fig. 5(a)), leading to the broken velocity limit. 15


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Effect of A Single Ridge When a ridge is placed on the surface, the velocity limit is broken in almost all simulations, only for the cases with Wr≥16.0 nm vj/uic approaches the limit, as shown in Fig. 3(b). This is because when Wr≥2R, the two droplets coalesce actually on the ridge with 180° contact angle. Moreover, the maximum vj/uic=0.230 is observed at Hr=1.6 nm and Wr=5.9 nm, which is 1.81 times higher than the velocity limit. For the surfaces with a ridge, the jumping velocity is always larger than the velocity limit. This is can be explained from the following two aspects. First, on the ridged surfaces, the expanding liquid bridge directly impacts the ridge, as shown in Fig. 7. On the flat surface, the liquid bridge impacting the surface occurs at t=420 ps, while on the ridged surface with Hr=1.6 nm and Wr=5.9 nm the impacting appears at t=204 ps. Thus, the coalescence time is inevitably shortened. Figures 2(a) and 2(c) shows that the coalescence time is 1176 ps on the flat surface, while it reduces to 900 ps on the ridged surface with Hr=1.6 nm and Wr=5.9 nm. Second, when the liquid bridge just impacts the ridge, the coalesced droplet still remains dumbbell-shaped, the negative curvature of the bridge drives liquid mass towards the bridge, causing a stronger impact than that on the flat surface. This conclusion can also be verified from the growth dynamics of liquid ridge. When Re=(ργrb/µ2)>>1, droplet coalescence locates in the inertial regime, and the width of liquid bridge, rb, follows a scaling law expressed as rb/R=Cb(t/τ)1/2 [30], where Cb is a constant whose value is about the order of unity. Liang et al.’ [21] and our previous 16


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studies [22] demonstrated that this scaling law is valid up to rb/R≈1. According to this law, the component velocity of the liquid bridge in the y-direction perpendicular to the surface can be calculated by,

ub =

drb Cb R − 12 = 1 t dt τ2

(12)

Apparently, ub decreases with time. Since the liquid bridge impacts the ridge earlier than the flat surface, the component velocity at the impacting moment is higher on the ridged surface. Accordingly, a stronger impact occurs on the ridged surface, leading to a larger counterforce of the ridge and hence a higher jumping velocity. It should be noted that, for three different ridge heights of Hr=0.6, 1.0, and 1.6 nm, the maximum jumping velocity always occurs at Wr=5.9 nm, as shown in Fig. 3(b). To explain the optimal ridge width, snapshots of droplets coalescing on the ridged surfaces with the same Hr=1.6 nm but different Wr=1.8, 5.9, and 10.0 nm are illustrated in Fig. 8. Because the liquid bridge directly impacts the ridge and the impacting area is the smallest for the narrow ridge with Wr=1.8 nm, only a small amount of momentum is transported to the ridge such that a small counterforce of the ridge is exerted on the coalesced droplet. However, when the ridge width is too large, the two lateral edges of the ridge will hinder the coalescence of droplets. Therefore, there is an optimal ridge width to reach the maximum jumping velocity. We also implement extra simulations with the droplet initial radius R changed from previous 8.0 nm to 7.5 and 9.0 nm. The results show that the optimal ridge width is Wr=4.9 nm for R=7.5 nm and Wr=7.0 nm for R=9.0

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nm, and the maximum vj/uic is 0.264 and 0.227, respectively. Thus, it can be concluded that the optimal ridge width is always smaller than the droplet initial radius and increases with the droplet initial radius. However, when the ridge height is lower than 0.6 nm (Hr=0.2 nm and Hr=0.4 nm), owing to the weakened effect of two lateral edges the jumping velocity continuously increases until Wr=16.0 nm, with the maximum jumping velocity occurring at Wr=14.0 nm. Figure 3(b) also shows that the maximum of the jumping velocity increases with increasing the ridge height. However, this trend is not always true. We also implement extra simulations with Hr=2.2, 3.2, 3.7 and 4.2 nm to search the optimal ridge height. The maximum of jumping velocity is 0.252 for Hr=2.2 nm, 0.293 for Hr=3.2 nm, 0.290 for Hr=3.7 nm, and 0.266 for Hr=4.2 nm, indicating that when Hr exceeds 3.2 nm, the maximum of jumping velocity starts to decrease, and hence the optimal Hr is 3.2 nm for the coalescence of 8.0 nm droplets. It can be reasonably expected that the optimal Hr will increase with the increase in droplet initial radius.

Effect of A More Hydrophobic Strip Intuitively, when a superhydrophobic surface with 162° contact angle is patterned by a strip with 180° contact angle, the jumping velocity on this mixed-wettability surface should be larger than that on the surface with a single 162° contact angle but smaller than that on the surface with a single 180° contact angle. However, astonishingly, the velocity limit is still broken when the strip width ranges from 5.9 to 14.0 nm, with the maximum vj/uic=0.197 occurring at Ws=10.0 nm, as shown in Fig. 3(c). 18


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For a narrow strip, the bottom of the coalesced droplet is mainly in contact with the surface with 162° contact angle before the liquid bridge impacts the strip, as shown in Fig. 9(a); hence, the energy dissipation caused by adhesion work is relatively higher. As the strip width increases, the energy dissipation caused by adhesion work gradually reduces (Fig. 9(b)) and finally is approximately identical to that on the flat surface with 180° contact angle. Therefore, the jumping velocity increases with the strip width. Furthermore, when the strip width exceeds 5.9 nm, it is found that the height of mass center of the coalesced droplet decreases due to the mixed-wettability, as compared with that on the flat surface with 180° contact angle, as shown in Fig. 10. Thus, the time required by liquid bridge impacting the surface reduces, leading to a shorter coalescence time (Fig. 5(c)) and a lower viscous dissipation. For example, the height of mass center of the coalesced droplet is 80.0 Å on the flat surface at t=120 ps, while it reduces to 74.8 Å for the stripped surface with Ws=10.0 nm. Our simulations also show that the height of mass center of the coalesced droplet reaches the minimum at Ws=10.0 nm, above which it starts to increase with increasing Ws. When the strip width exceeds 16 nm, the coalescence actually takes place on the strip; hence, the jumping velocity is identical to that on the flat surface with 180° contact angle. The coalescence time τtot for all the simulations is illustrated in Figs. 5(a) to 5(c), where the dashed line represents the coalescence time required by superhydrophobic surfaces with 180° contact angle. Comparison of τtot and vj/uic curves demonstrates that vj/uic is inversely proportional to τtot. As expected, the coalescence time indeed shortens for the designed surfaces with those specific parameter values as 19


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described before, the viscous dissipation accordingly reduces during coalescence, which is responsible for the break of the velocity limit.

CONCLUSIONS In summary, we proposed a new approach to enhance coalescence-induced droplet jumping and raise the corresponding jumping velocity by patterning a single groove, ridge, or more hydrophobic strip on a superhydrophobic surface. Coalescence and self-jumping of two equally sized nanodroplets on such designed surfaces were investigated via molecular dynamics simulations. We found that the velocity limit vj=0.127uic was broken on all the three kinds of patterned surfaces and the surface with a 1.6 nm high and 5.9 nm wide ridge yields a 1.81 times higher jumping velocity than the limit. We also revealed that the greatly reduced viscous dissipation caused by the reduction in coalescence time is responsible

for

this

jumping

enhancement.

Although

we

have

focused

on

nanoscale

coalescence-induced jumping on the three designed superhydrophobic surfaces, the same mechanisms of the reduction in coalescence time can also occur at microscale. Thus, we expect that this approach could be extended to microscale self-jumping enhancement, and hence offers potential for a wide range of applications.

SUPPORTING INFORMATION

The contact angles for various values of cij. 20


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Schematic of two droplets placed on a superhydrophobic surface.

Transient component velocity in the y-direction on various surfaces.

ACKNOWLEDGMENTS This study was partially supported by the National Science Fund for Distinguished Young Scholars of China (No. 51525602), and the Fundamental Research Funds for the Central Universities (Nos. 2018QN040 and 2017ZZD006).

REFERENCES [1] Boreyko, J.B.; Chen, C.H. Self-propelled dropwise condensate on superhydrophobic surfaces. Phys. Rev. Lett. 2009, 103, 184501. [2] Nam, Y.; Kim, H.; Shin, S. Energy and hydrodynamics analyses of coalescence-induced jumping droplets. Appl. Phys. Lett. 2013, 103, 161601. [3] Wang, K.; Liang, Q.; Jiang, R.; Zheng, Y.; Lan, Z.; Ma, X.H. Numerical simulation of coalescence-induced jumping of multidroplets on superhydrophobic surfaces: Initial droplet arrangement effect. Langmuir 2017, 33, 6258-6268. [4] Qu, X.P.; Boreyko, J.B.; Liu, F.J.; Agapov, R.L.; Lavrik, N.V.; Retterer, S.T.; Feng, J.J.; Collier, C.P.; Chen, C.H. Self-propelled sweeping removal of dropwise condensate. Appl. Phys. Lett. 2015, 106, 221601. 21


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Page 22 of 41

[5] Enright, R.; Miljkovic, N.; Al-Obeidi, A.; Thompson, C.V.; Wang, E.N. Condensation on superhydrophobic surfaces: the role of local energy barriers and structure length scale. Langmuir 2012, 28, 14424-14432. [6] Wang, K.; Liang, Q.; Jiang, R.; Zheng, Y.; Lan, Z.; Ma, X.H. Self-enhancement of droplet jumping velocity: the interaction of liquid bridge and surface texture. RSC Advances, 2016, 6, 99314-99321. [7] Chu, F.Q.; Yuan, Z.P.; Zhang, X.; Wu, X.M. Energy analysis of droplet jumping induced by multi-droplet coalescence: The influences of droplet number and droplet location. Int. J. Heat Mass Tran. 2018, 121, 315-320. [8] Miljkovic, N.; Enright, R.; Nam. Y.; Lopez, K.; Dou, Y.; Sack, J.; Wang, E.N. Jumping-droplet-enhanced condensation on scalable superhydrophobic nanostructured surfaces. Nano Lett. 2013, 13, 179-187. [9] Rykaczewski, K.; Paxson, A.T.; Anand, S.; Chen, X.; Wang, Z.; Varanasi, K.K. Multimode multidrop serial coalescence effects during condensation on hierarchical superhydrophobic surfaces. Langmuir 2013, 29, 881-891. [10] Dietz, C.; Rykaczewski, K.; Fedorov, A.G.; Joshi, Y. Visualization of droplet departure on a superhydrophobic surface and implications to heat transfer enhancement during dropwise condensation. Appl. Phys. Lett. 2010, 97, 033104. [11] Watson, G.S.; Gellender, M.; Watson, J.A. Self-propulsion of dew drops on lotus leaves: a potential mechanism for self-cleaning. Biofouling 2014, 30, 427-434. 22


Page 23 of 41 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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[12] Wisdom, K.M.; Watson, J.A.; Qu, X.; Liu, F.; Watson, G.S.; Chen, C.H. Self-cleaning of superhydrophobic surfaces by self-propelled jumping condensate. Proc. Natl. Acad. Sci. 2013, 110, 7992-7997. [13] Boreyko, J.B.; Collier, C.P. Delayed frost growth on jumping-drop superhydrophobic surfaces. ACS Nano 2013, 7, 1618-1627. [14] Zhang, Q.; He, M.; Chen, J.; Wang, J.; Song, Y.; Jiang, L. Anti-icing surfaces based on enhanced self-propelled jumping of condensed water microdroplets. Chem. Commun. 2013, 49, 4516-4518. [15] Chen, X.M.; Ma, R.Y.; Zhou, H.B.; Zhou, X.F.; Che, L.F.; Yao, S.H.; Wang, Z.K. Activating the microscale edge effect in a hierarchical surface for frosting suppression and defrosting promotion. Sci. Rep. 2013, 3, 2515. [16] Feng, J.; Pang, Y.; Qin, Z.; Ma, R.; Yao, S. Why condensate drops can spontaneously move away on some superhydrophobic surfaces but cannot on others? ACS Appl. Mater. Interfaces 2012, 4, 6618-6625. [17] Zhang, K.; Liu, F.; Williams, A.J.; Qu, X.; Feng, J.J.; Chen, C.H. Self-propelled droplet removal from hydrophobic fiber-based coalescers. Phys. Rev. Lett. 2015, 115, 074502. [18] Liu, F.J.; Ghigliotti, G.; Feng, J.J.; Chen, C.H. Numerical simulations of self-propelled jumping upon drop coalescence on non-wetting surfaces. J. Fluid Mech. 2014, 752, 39-65. [19] Enright, R.; Miljkovic, N.; Sprittles, J.; Nolan, K.; Mitchell, R.; Wang, E.N. How coalescing droplets jump. ACS Nano 2014, 8, 10352-10362. 23


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Page 24 of 41

[20] Wang, F.C.; Yang, F.; Zhao, Y.P. Size effect on the coalescence-induced self-propelled droplet. Appl. Phys. Lett. 2011, 98, 053112. [21] Liang, Z.; Keblinski, P. Coalescence-induced jumping of nanoscale droplets on super-hydrophobic surfaces. Appl. Phys. Lett. 2015, 107, 143105. [22] Xie, F.F.; Lu, G.; Wang, X.D.; Wang, B.B. Coalescence-induced jumping of two unequal sized nanodroplets. Langmuir 2018, 34, 2734-2740. [23] Mulroe, M.D.; Srijanto, B.R.; Ahmadi, S.F.; Collier, C.P.; Boreyko, J.B. Tuning superhydrophobic nanostructures to enhance jumping-droplet condensation. ACS Nano 2017, 11, 8499-8510. [24] Kim, A.; Lee, C.; Kim, H. Simple approach to superhydrophobic nanostructured Al for practical antifrosting application based on enhanced self-propelled jumping droplets. ACS Appl. Mater. Interf. 2015, 7, 7206-7213. [25] Hamed, V.; Wang, W.; Davies, S.; Mabry, J.M.; Kota, A.K. Coalescence induced self-propulsion of droplets on superomniphobic surfaces. ACS Appl. Mater. Interf. 2017, 9, 29328-29336. [26] Lu, G.; Wang, X.D.; Duan, Y.Y. A critical review of dynamic wetting by complex fluids: from Newtonian fluids to non-Newtonian fluids and nanofluids. Adv. Colloid Interface Sci. 2016, 236, 43-62. [27] Chu, D.F.; Dong, X.Y.; Zhao, Q.; Gu, J.K.; Wang, Z.J. Photosensitization priming of tumor microenvironments improves delivery of nanotherapeutics via neutrophil infiltration. Adv. Mater. 2017, 29, 1701721.

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[28] Attarzadeh, R.; Dolatabadi, A. Coalescence-induced jumping of micro-droplets on heterogeneous superhydrophobic surfaces. Phys. Fluids 2017, 29, 012104. [29] Foiles, S.M.; Baskes, M.I.; Daw, M.S. Embedded-atom-method functions for the fcc metals Cu, Ag, Au, Ni, Pd, Pt, and their alloys. Phys. Rev. B 1986, 33, 7983. [30] Eggers, J.; Lister, J.R.; Stone, H.A. Coalescence of liquid drops. J. Fluid Mech. 1999, 401, 293-310. [31] Li D.; Qian C.L.; Gao S.W.; Zhao X.B.; Zhou Y.M. Self-propelled drop jumping during defrosting and drainage characteristic of frost melt water from inclined superhydrophobic surface. Int. J. Refrig. 2017, 79, 25-38. [32] Li G.Q.; Alhosani M.H.; Yuan S.J.; Liu H.R.; Ghaferi A.A.; Zhang T.J. Microscopic droplet formation and energy transport analysis of condensation on scalable superhydrophobic nanostructured copper oxide surfaces. Langmuir 2014, 30, 14498-14511. [33] Chandra S.; Avedisian C.T. On the collision of a droplet with a solid surface. Proc. R. Sot. London A 1991, 432, 13-41. [34] Lv, C.J.; Hao, P.F.; Yao, Z.H.; Song, Y.; Zhang, X.W.; He, F. Condensation and jumping relay of droplets on lotus leaf. Appl. Phys. Lett. 2013, 103, 021601. [35] Gao, S.; Liao, Q.; Liu W.; Liu. Z. Coalescence-induced jumping of nanodroplets on textured surfaces. J. Phys. Chem. Lett. 2018, 9, 13-18.

25


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Page 26 of 41

[36] Li, H.X.; Yang, W.L.; Aili, A.; Zhang, T.J. Insights into the impact of surface hydrophobicity on droplet coalescence and jumping dynamics. Langmuir 2017, 33, 8574-8581. [37] Liu, X.L.; Cheng, P. 3D multiphase lattice Boltzmann simulations for morphological effects on self-propelled jumping of droplets on textured superhydrophobic surfaces. Int. Commun. Heat Mass 2015, 64, 7-13. [38] Cha, H.; Xu, C.; Sotelo, J.; Chun, J.M.; Yokoyama, Y.; Enright, R.; Miljkovic, N. Coalescence-induced nanodroplet jumping. Phys. Rev. Fluids 2016, 1, 064102. [39] Vahabi, H.; Wang, W.; Davies, S.; Mabyr, J.M.; Kota, A.K. Coalescence-induced self-propulsion of droplets on superomniphobic surfaces. ACS Appl. Mat. Interfaces 2017, 9, 29328. [40] Hou, Y.M.; Yu, M.; Chen, X.M.; Wang, Z.K.; Yao, S.H. Recurrent filmwise and dropwise condensation on a beetle mimetic surface. ACS Nano 2015, 9, 71-81. [41] Liu, T.Q.; Sun, W.; Sun, X.Y.; Ai, H.R. Mechanism study of condensed drops jumping on superhydrophobic surfaces. Colloid Surf. A 2012, 414, 366-374. [42] Lv, C.; Hao, P.; Yao, Z.; Niu, F. Departure of condensation droplets on superhydrophobic surfaces. Langmuir 2015, 31, 2414-2420. [43] Nam, Y.; Kim, H.; Shin S. Energy and hydrodynamic analyses of coalescence-induced jumping droplets. Appl. Phys. Lett. 2013, 103, 161601.

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[44] Bartolo, D.; Bouamrirene, F.; Verneuil, E.; Buguin, A.; Silberzan, P.; Moulinet, S. Bouncing or sticky droplets: Impalement transitions on superhydrophobic micropatterned surfaces. Europhys. Lett. 2006, 74, 299-305. [45] Reyssat, M.; Pepin, A.; Marty, F.; Chen, Y.; Quere, D. Bouncing transitions on microtextured materials. Europhys. Lett. 2006, 74, 306-312. [46] Gao, S.; Liao, Q.; Liu, W.; Liu, Z. Nanodroplets impact on rough surfaces: A simulation and theoretical study. Langmuir 2018, 34, 5910-5917. [47] Vahabi, H.; Wang W.; Movafaghi S.; Kota A.K. Free-standing, flexible superomniphobic films. ACS Appl. Mat. Interfaces 2016, 8, 21962-21967. [48] Feng, J.; Qin, Z.Q.; Yao, S.H. Factors affecting the spontaneous motion of condensate drops on superhydrophobic copper surfaces. Langmuir 2012, 28, 6067-6075. [49] Jeong W.J.; Ha, M.Y.; Yoon, H.S.; Ambrosia, M. Dynamics behavior of water droplets on solid surfaces with pillar-type nanostructures. Langmuir 2012, 28, 5360-5371. [50] Shi, Y.; Tang, G.H.; Xia, H.H. Investigation of coalescence-induced droplet jumping on superhydrophobic surfaces and liquid condensate adhesion on slit and plain fins. Int. J. Heat Mass Tran. 2015, 88, 445-455. [51] Zheng, Y.M.; Bai, H.; Huang, Z.B.; Tian, X.L.; Nie, F.Q.; Zhao, Y.; Zhai, J.; Jiang, L. Directional water collection on wetted spider silk. Nature 2010, 463, 640-643.

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[52] Xiu, Y.; Zhu, L.; Hess, D.W.; Wong, C.P. Hierarchical silicon etched structures for controlled hydrophobicity/superhydrophobicity. Nano Lett. 2007, 7, 3388-3393.

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Figure Captions Figure 1. Initial configurations of two Ar droplets coalescing on a superhydrophobic Au surface decorated with (a) a groove, (b) a ridge, (c) or a more hydrophobic strip. The contact angle is set to be 180° on the yellow region, while 162° on the gray region. Figure 2. Snapshots of the coalesced droplets (a) on a flat superhydrophobic surface as well as a superhydrophobic surface decorated with (b) a groove of Hg=1.6 nm and Wg=1.8 nm, (c) a ridge of Hr=1.6 nm and Wr=5.9 nm, or (d) a more hydrophobic strip of Ws=10.0 nm. The last snapshot for every series shows the droplet just jumping from the surface. Figure 3. Dimensionless jumping velocity of the coalesced droplets on a superhydrophobic surface decorated with (a) a groove; (b) a ridge; or (c) a more hydrophobic strip. Here the dashed line denotes the limit value 0.127 of dimensionless jumping velocity on the flat superhydrophobic surface, and the solid symbols denote that the liquid bridge cannot penetrate into the grooves. Figure 4. Snapshots for impact of the liquid bridge on the superhydrophobic surfaces with a groove of (a) Wg=1.8 nm: Hg=1.6 nm (left) and Hg=1.2 nm (right); (b) Hg=0.8 nm: Wg=1.8 nm (left), Wg=5.9 nm (middle), and Wg=12.0 nm (right); and (c) Wg=12.0 nm: Hg=1.6 nm (left) and Hg=1.2 nm (middle) as well as Wg=8.0 nm and Hg=0.8 nm (right). Figure 5. The coalescence time on the superhydrophobic surface decorated with (a) a groove; (b) a ridge; or (c) a more hydrophobic strip. Here the dashed line denotes the coalescence time on the flat superhydrophobic surface. 29


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Figure 6. Temporal evolution of the velocity of coalesced droplet in the y-direction and the average pressure on the bottom of the coalesced droplet: (a) on the flat surface and (b) on the surface with a groove. Figure 7. Snapshots for impact of the liquid bridge on the superhydrophobic surface without (left, the impact occurs at t=420 ps) or with (right, the impact occurs at t=204 ps) a ridge. Figure 8. Snapshots for impact of the liquid bridge on the superhydrophobic surfaces with a ridge of Hr=1.6 nm: (a) Wr=1.8 nm; (b) Wr=5.9 nm; and (c) Wr=10.0 nm. Figure 9. Snapshots for coalescence on the superhydrophobic surface decorated with a strip of (a) Ws=1.8 nm and (b) Ws=10.0 nm. Figure 10. Snapshots for coalescence on the superhydrophobic surface without or with a strip. The strip width is Ws=10.0 nm. The height of mass center for the surface with a strip reduces by ∆h due to the mixed-wettability, as compared with that for the flat surface with 180° contact angle.

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(a)

(b)

(c)

Figure 1. Initial configurations of two Ar droplets coalescing on a superhydrophobic Au surface decorated with (a) a groove, (b) a ridge, (c) or a more hydrophobic strip. The contact angle is set to be 180° on the yellow region, while 162° on the gray region.

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(ps)

0

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588

768

1176 (a)

948 (b)

900 (c)

924 (d)

Figure 2. Snapshots of the coalesced droplets (a) on a flat superhydrophobic surface as well as a superhydrophobic surface decorated with (b) a groove of Hg=1.6 nm and Wg=1.8 nm, (c) a ridge of Hr=1.6 nm and Wr=5.9 nm, or (d) a more hydrophobic strip of Ws=10.0 nm. The last snapshot for every series shows the droplet just jumping from the surface.

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0.24

(a)

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0.12 0.09

θstrip=180°, θsurface=162°

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Ws (nm) Figure 3. Dimensionless jumping velocity of the coalesced droplets on a superhydrophobic surface decorated with (a) a groove; (b) a ridge; or (c) a more hydrophobic strip. Here the dashed line denotes the limit value 0.127 of dimensionless jumping velocity on the flat superhydrophobic surface, and the solid symbols denote that the liquid bridge cannot penetrate into the grooves. 33


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(a)

(b)

(c) Figure 4. Snapshots for impact of the liquid bridge on the superhydrophobic surfaces with a groove of (a) Wg=1.8 nm: Hg=1.6 nm (left) and Hg=1.2 nm (right); (b) Hg=0.8 nm: Wg=1.8 nm (left), Wg=5.9 nm (middle), and Wg=12.0 nm (right); and (c) Wg=12.0 nm: Hg=1.6 nm (left) and Hg=1.2 nm (middle) as well as Wg=8.0 nm and Hg=0.8 nm (right).

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(a)

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Hg=1.6 nm 1500 1200 900 600

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1500 1200 900 600

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Ws (nm) Figure 5. The coalescence time on the superhydrophobic surface decorated with (a) a groove; (b) a ridge; or (c) a more hydrophobic strip. Here the dashed line denotes the coalescence time on the flat superhydrophobic surface.

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6.0

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Figure 6. Temporal evolution of the velocity of coalesced droplet in the y-direction and the average pressure on the bottom of the coalesced droplet: (a) on the flat surface and (b) on the surface with a groove.

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Page 37 of 41 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Figure 7. Snapshots for impact of the liquid bridge on the superhydrophobic surface without (left, the impact occurs at t=420 ps) or with (right, the impact occurs at t=204 ps) a ridge.

37


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

396 ps

624 ps

Page 38 of 41

744 ps

(a) 264 ps

504 ps

624 ps

(b) 192 ps

348 ps

552ps

(c) Figure 8. Snapshots for impact of the liquid bridge on the superhydrophobic surfaces with a ridge of Hr=1.6 nm: (a) Wr=1.8 nm; (b) Wr=5.9 nm; and (c) Wr=10.0 nm.

38


Page 39 of 41 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

72 ps

174 ps

420 ps

(a) 60 ps

186 ps

420 ps

(b) Figure 9. Snapshots for coalescence on the superhydrophobic surface decorated with a strip of (a) Ws=1.8 nm and (b) Ws=10.0 nm.

39


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

t=120 ps

Page 40 of 41

t=120 ps

∆h

Figure 10. Snapshots for coalescence on the superhydrophobic surface without or with a strip. The strip width is Ws=10.0 nm. The height of mass center for the surface with a strip reduces by ∆h due to the mixed-wettability, as compared with that for the flat surface with 180° contact angle.

40


Page 41 of 41

Table of Contents Graphic

0.24 0.21 0.18

vj/uic

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

0.15 0.12 0.09

Hg=1.6 nm Hr=1.6 nm

0.06 0.03 0.00

θstrip=180°, θsurface=162° 0

2

4

6

8

10 12 14 16 18 20 22 24

Wg/Wr/Ws (nm)

41


Enhancement of Coalescence-Induced ... - ACS Publications

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