Nanodroplets Impact on Rough Surfaces: A Simulation and

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Nanodroplets Impact on Rough Surfaces: A Simulation and Theoretical Study Shan Gao, Quanwen Liao, wei Liu, and Zhichun Liu Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b00480 • Publication Date (Web): 30 Apr 2018 Downloaded from http://pubs.acs.org on May 1, 2018

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Langmuir

Nanodroplets Impact on Rough Surfaces: A Simulation and Theoretical Study Shan Gao, Quanwen Liao, Wei Liu,* Zhichun Liu*

School of Energy and Power Engineering, Huazhong University of Science and Technology (HUST), Wuhan 430074, China

Corresponding Authors

* E-mail: [email protected] (WL); [email protected] (ZCL)

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ABSTRACT

Droplets impact is widespread in life, and modulating the dynamics of impinging droplets is a significant problem in production. However, on textured surfaces, the micromorphologic change and mechanism of impinging nanodroplets is not well understood, furthermore, the accuracy of theoretical model for nanodroplets need to be improved. Here, considering the great challenge of conducting experiments on nanodroplets, a molecular dynamics simulation is performed to visualize the impact process of nanodroplets on nanopillar surfaces. Compared with macroscale droplets, apart from the similar relation of restitution coefficient with Weber number, we found some distinctive results: is described as a power law of impact velocity and the

relation of with impact velocity or the Reynolds number is exponential.

Moreover, the roughness of substrates plays a prominent role in impact nanodroplets dynamics, on surfaces with lower solid fraction, the lower attraction force induces an easier rebound of impact nanodroplets. At last, based on the energy balance, through modifying the estimation of viscous dissipation and surface energy terms, we proposed an improved model for maximum spreading factor, which shows greater accuracy for nanodroplets, especially in the low to moderate velocity range. The outcomes of this study demonstrate the distinctive dynamical behavior of impinging nanodroplets, the fundamental insight and more accurate prediction is very useful in the improvements of nanodroplets’ hydrodynamic behavior.


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TOC GRAPHICS


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INTRODUCTION

Impact of water droplets is a pervasive phenomenon in the natural world, and it has a widespread range of applications including pesticides deposition1, inkjet printing2, spray cooling3 and micro-fabrication.4 Various physical outcomes may be exhibited after a droplet impacting on a solid substrate such as sticking, spreading, bouncing or splashing. In particular, because of the prominent role in self-cleaning5-11, anti-icing,12-14 anti-fogging15-17 and water harvesting,18,19 the rebound of droplets from solid substrates has attracted increasing attention of surface engineers. The dynamical behavior of an impacting droplet is manipulated by its diameter, physical properties20 and impact velocity,21 among these influence factors, the solid substrate morphology and its physicochemical property22 significantly play a central role in the droplets performance. For a solid substrate with low surface energy and composite micro-nano structures, namely superhydrophobic surface, impacting droplet may bounce back along with some oscillation. The dynamic behavior of droplets during falling and impinging has been intensively investigated through experiments,20,22-30 theoretical investigations31-34 and numerical simulations.35-39 A number of parameters in literature were proposed to analyze this process, including the maximum spreading time (the time needed

to reach the maximum spreading state), the maximum spreading factor

( = ⁄ , where and are the maximum spreading diameter and initial diameter of droplet, respectively), the restitution coefficient ε, ( = ⁄ ,

where and are the velocity before and after bouncing), the Weber number ACS Paragon Plus Environment

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( = / , where , , and are the density, initial diameter, impinging velocity and surface tension of droplet, respectively) and the Reynolds number ( = /, where is the viscosity of droplet). Simultaneously, based on the conservation of energy, there have been appeared different theoretical models, 31-34,40,41

which mainly focus on the exact description of impinging droplet at the

macroscale level. Despite the considerable advantages, the accuracy of previous theoretical models need to be improved for nanoscale droplet. Recent developments in the field of nanopringting and nanocoating have led to a renewed interest in nanodroplet impingement behavior. However, the nanodroplets micromorphologic change remain unclear, the mechanism of nanodroplets impact behavior is not well understood and lacks a more appropriate theoretical model. It is difficult to conduct experimental researches on the dynamic behavior of nanoscale droplets, additionally, the previous models for macroscale impinging droplets, traditional macroscopic and mesoscopic simulation methods are not appropriate at the nanoscale. Conversely, in a time span of picoseconds, molecular dynamics (MD) simulations can provide a powerful alternative to examine the detailed impinging process of nanodroplets. In 2017, Li et al.42 investigated the impinging of nanoscale water droplets on the smooth surface through molecular dynamics simulation, a wide regime of impinging from spreading to breakup was studied. Despite these significant results, the impact behavior of nanodroplets on nanostructure surfaces and the effects of surfaces physicochemical property remain unclear. In this work, unlike the previous simulation studies, we provided a systematic


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quantitative description and analysis for the dynamic behavior of impact nanodroplets on nanopillar surfaces featured various physicochemical properties, through molecular dynamics simulation. During the impinging process, several important parameters, such as the centroid height, the maximum spreading time , the

maximum spreading factor and the restitution coefficient ε are calculated and recorded. For the same nanopillar surfaces, by varying the nanodroplets impact velocity from 1 Å/ps to 8 Å/ps, we found that the is described as a power law of

impact velocity, the Reynolds number or the Weber number ( = 2 ⁄5 . ,

~ . ! , ~ ." ), the relation of with impact velocity or the

Reynolds number is exponential ( = · 1.047() ) , both the findings are

different from the results of previous studies, moreover, the relation of ε with the Weber number is also exponential approximatively ( ~ ." ). For the

nanodroplets with same impact velocity, by varying the sold fraction of nanopillar surfaces, the findings show that remains almost invariable and decreases with decreasing solid fraction, which is ascribed to the decreasing atraction force between substartes and water molecules. At last, based on energy conservation, we proposed a modified model for impinging nanodroplets to explain these findings, which is more accurate for nanodroplets impact process.

MODEL AND METHODOLOGY

All simulations were performed using molecular dynamics simulation to study the evolution of impinging nanodroplets with different initial velocities, on a series of


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nanopillar surfaces with different solid fractions. A large-scale atomic/molecular massively parallel simulator (LAMMPS) package is used to conduct all simulations, where the droplet is constituted by water molecules, and more simple copperlike surfaces are used to construct the substrates. Both water molecules and cooperlike atoms are generated in the face-center cubic lattice initially, and the lattice constant of the unit cell are determined by each density at different temperatures. The properties of the droplet are taken to be the same as those of water at 20 °C, such that the mass density ρ = 998.23 kg/m3, the surface tension σ = 72.75 mN/m, and the viscosity coefficient µ = 1.0087 mPa·s. The Reynolds number, the Weber number and the Ohnesorge number are respectively defined by = /µ, = / ,

*ℎ = /, , where D0 is the initial diameter of a droplet, denotes the impinging velocity. Gravity is not taken into consideration since the droplets are much smaller than the capillary length.

1. Physical Model

The simulation domain is schematically shown in Figure 1, a spherical nanodroplet constructed by 56435 water molecules, with a diameter of 155.4 Å, is initially placed above the solid substrate. 145800 cooperlike atoms constitute the textured surfaces, whose horizontal area correspond to 324.0 Å×324.0 Å. The nanopillared surfaces consists of a square pillar array with height H=18.1 Å, different width W and interpillar spacing S, as detailed in the Table 1.


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Figure 1. Schematic diagram for the simulation domain. Cooper-like atoms are in red, oxygen atoms and hydrogen atoms are in blue and white, respectively. (a) Front view. (b) Side view, and the inset is a component unit of textured surfaces.

Table 1. Physical dimensions of pillars: width W, distance between pillars S, and height H. Structural properties of rough surfaces: solid fraction -. = ⁄/ 0 12 . S(Å) — 9.04 12.65 16.27 19.88 23.49

Structure 1 Structure 2 Structure 3 Structure 4 Structure 5 Structure 6

W(Å) — 27.11 23.49 19.88 16.27 12.65

-. /%2 100 56.24 42.23 30.24 20.26 12.25

H(Å) — 18.1 18.1 18.1 18.1 18.1

2. Simulation Details A four interaction points model of water, the TIP4P model is employed for nanodroplets modeling, whose potential is represented by the following expression, including an intermolecular interaction term and a Coulomb term. 4 = 455 67

899

:;< :=>

"

?

@7

899

:;< :=>

A

? B

CD",FD"

0 ∑CD" × ∑FD"

H;< H=>

"

πIJ :;< :=>

(1)

Where the van der Waals interaction corresponds to the 12-6 type Lennard-Jones potential, i and j denote different water molecules, a and b denote different interaction ACS Paragon Plus Environment

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sites in a water molecule, 55 represent the interaction energy well-depth and

55 represent the finite distance where the value of LJ potential is equal to zero, and denote the vacuum permittivity. The detailed properties of the TIP4P model are

presented in Table 2. The setting of intermolecular forces in this paper are almost identical to that of our previous study.9,14 There is only the van der Waals force in the interaction between cooperlike atoms and oxygen atoms in water molecules, and the interaction parameter K5 is set to equal to K5,14 corresponding to the intrinsic contact angle L = 118.2°. The interactions between cooperlike atoms and hydrogen

atoms are relatively negligible. For the solid substrate, the interactions between cooperlike atoms are also implemented by using the 12-6 LJ potential, and the interaction parameters are taken from the LJ potential of copper.43

Table 2. Properties of the TIP4P model: qH—charge on the hydrogen atom; qO—charge on the oxygen atom; qM—charge on the dummy atom; , —Lennard-Jones parameters; rOH—OH bond length; rOM—distance of dummy atom from oxygen atom; θHOH—HOH bond angle.

TIP4P

qH(e) +0.52

qO(e) 0.0

qM(e) -1.04

(kJmol-1)

0.6487

(Å) 3.1656

rOH(Å) 0.9572

rOM(Å) 0.15

θHOH(˚) 104.52

In all simulations, the periodic boundary conditions are only applied in the horizontal direction while the fixed boundary condition and bounce-back boundary condition are applied in the lower and upper boundaries, respectively. The popular velocity Verlet algorithm with a time step of 1.0 fs is used to integrate the newton’s equation of motion, the long-range columbic force is computed by PPPM ACS Paragon Plus Environment

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(particle-particle particle-mesh) approach within the cutoff distance of 10 Å, the SHAKE algorithm is used to fix the bond distance and angle of water molecule.

Considering the simulation of this huge system is time consuming, and in order to improve the calculating efficiency and shorten the relaxation time, the nanodroplets are initially generated as spherical shape, which corresponding to the minimum energy state. Additionally, nanodroplets are placed close to surfaces to reduce the impinging velocities change during falling. Simulations of impinging nandroplets are performed in the following two stages: (1) the entire water zones are equilibrated with Nose-Hoover thermostat in an NVT ensemble until the temperature of water system achieves a stable value of 298K, and the substrate is simultaneously heated to 298 K with a Langevin thermostat in an NVE ensemble, and in this stage, the intermolecular force between droplet and substrate was set to 0 to eliminate the substrate’s influence, (2) after those preparations steps are done, the Nose-Hoover thermostat applied on the water molecules is removed, and the entire system is integrated with NVE ensemble. Meanwhile all water molecules are imposed an additional momentum in the minus Z direction which marks the beginning of the droplet impingement.

RESULTS AND DISCUSSIONS 1. Nanodroplets with different velocities impact nanopillared surfaces Keeping the surface solid fraction -. = 30.24% and other parameters fixed, the variation of nanodroplets’ morphology is investigated at increasing impact velocities,


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eight cases with different velocities are carried out, the details are listed in Table 3, and the sequential snapshots of each impact process are shown in Figure 2. According to these time-lapse images the impact process can be divided into two regimes: Ⅰ spreading stage, due to the rapid pressure increase caused by inertia force, droplet deforms first, spreads radially with impingement proceeding and extend to its maximum spreading state gradually. Moreover, the bottom of droplet is observed to penetrate structure gap, Table 3. Impact velocity, the Reynolds number and the Weber number for each simulation case. case (Å/ps) Re We

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1.48 2.97 4.45 5.94 7.42 8.91 10.39 11.88 2.06 8.24 18.54 32.96 51.5 74.16 100.94 131.84

and there is a deeper and larger surface penetration for droplets with higher impact velocity. In the spreading stage, with the increase of interfacial area, a portion of droplet’s kinetic energy converts into interfacial energy, and another part is consumed by internal-flow-induced viscous dissipation and adhesion work from substrate, which will be transferred to heat energy. At last, the droplet reaches its maximum spreading and deepest penetration sate, particularly presents a pancake-like shape at high impinging velocity. Ⅰ retraction stage, the droplet starts to recoil under the action of surface tension. During the upward movement of three-phase contact line, the stored interfacial energy converts into kinetic energy again and also partly consumed by viscous dissipation and adhesion work. If the kinetic energy is high enough to compensate the dissipative energy, the extra energy will result in the subsequent


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rebound. In our simulation cases, only the droplet with impact velocity of 1 Å/ps sticks to the surface after impingement.


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Figure 2. Comparison of the dynamical behavior of the nanodroplet impacting on the same nanopillar surfaces at different velocities. Sequencing from (a) to (h) are droplets with impinging velocity of 1 Å/ps, 2 Å/ps, 3 Å/ps, 4 Å/ps, 5 Å/ps, 6 Å/ps, 7 Å/ps and 8 Å/ps.

To investigate the effects of impinging velocity on the droplet impact process, several important parameters are calculated and recorded. Figure 3a shows the evolution of droplets centroid heights, which decrease in spreading stage and increase in retraction stage, and the variation tendency is consistent with the observed result: only at impact velocity of 1 Å/ps, there is a small change in centroid height before and after the impingement, indicating that the droplet always adheres on surface. The maximum spreading time is obtained from the crossover point of this two stages, whose result is demonstrated in Figure 3b. In the range of low to moderate velocity ( ≤ 6Å/ps), we found that the is approximatively described as a

power law of impact velocity, = 2 ⁄5 ., which is different from previous researches as = 8 ⁄3 , and can estimate the maximum spreading time of nanodroplet more appropriately. However, in the high velocity range ( > 6Å/ps), the value of remains constant roughly.


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Figure 3. (a) Temporal evolution of the centroid height of droplet at different impact velocities. (b) Dependency of the maximum spreading time with respect to the impact velocity. Moreover, the dependencies of to Re and We are correspondingly shown in

Figure 4a and 4b. The function equations between and Re and We are obtained

by linear fitting in log-log coordinate, and the maximum spreading time follow the scaling laws ~ . ! , ~ ." . In Figure 4c, we plot the variation of

the restitution coefficient ε with impact velocity. Here, ε is defined as the ratio of the droplet velocity after and before impingement. There is a marked reduction in ε under the increasing impact velocity, which is attributed to the mounting viscous dissipation and adhesion work during impact. Impact with a higher velocity results in a more dramatic deformation and leads to a more dissipation energy further. Simultaneously, there is a higher adhesion work between droplet and substrate, due to the deeper and larger surface penetration. We record the variation of the restitution coefficient ε as a function of Weber number, as illustrated in Figure 4d, ε decreases with increasing Weber number. The power function equation is obtained by linear fitting in a log-log


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coordinate and described as ~ ." , which is closed to the previous experiment results.

Figure 4. (a) and (b) are maximum spreading time as a function of Reynolds number and Weber number, respectively, in which black points denote the value of and the red line is a fit to the black points. (c) Dependency of the restitution coefficient ε to impact velocity. (d) The variation of the restitution coefficient ε as a function of We: black triangles denote the value of ε and the red line is a fit to the black triangles.

To evaluate how the impinging velocity affects the droplet impact, we further calculate the spreading radii for all cases, whose variation with time are shown in Figure 5a. All the spreading radii have an upward and then downward trend, and tend to a constant value eventually. These maximums are extracted to acquire the


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maximum spreading factor . Figure 5b shows the calculated result of versus impinging velocity, there is a power function relationship between them in the range of our simulation velocity. Furthermore, we plot the variation of with

Weber number in Figure 5c. In the previous literature, it is shown that scales

with V or V . However, our calculated result of versus Weber number suggest that the data do not scale with V or V . It’s more likely to be an

exponential function relation between and We, and the function equation is

= 1.106 W 1.047() = · 1.047() in the range of our simulation velocity,

where the is the equilibrium spreading factor at zero impact velocity.

Figure 5. (a) Temporal evolution of the spreading radius of droplet at different impact velocities. (b) and (c) are the variation of maximum spreading factor versus impact velocity and Weber number, respectively, in which black points denote the value of and the red line is a fit to the black points. 2. Nanodroplets impact nanopillar surfaces with different solid fractions Keeping the impact velocity = 5 Å/ps and other parameters fixed, effects of surface solid fraction on the droplet impact process are investigated, we carried out six cases herein, whose details are listed in Table 1. Figure 6 shows selected snapshots


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of identical nanodroplets impacting on surfaces with different solid fractions under We=51.46. A tendency is observed from these time-lapse images: the impinging droplets can more easily bounce from substrates with decreasing solid fraction. Similarly, to analyze the effects of solid fraction, parameters such as and are calculated and recorded. Figure 7a shows the evolution of droplet’s centroid height, where the slope of curve represents the vertical velocity of droplet. on those surfaces with high solid fraction (-. = 100%, -. = 56.24%) , the small change in centroid height before and after the impingement indicates that the droplet always adheres to surface. Besides, for surfaces with lower solid fraction, the greater jump height and larger slope of curve in the retraction stage reveal that impinging droplet can more easily bounce from substrates at a higher takeoff velocity. Figure 7c shows the evolution of droplet’s spreading radius, where the slope of curve represents the contact line velocity, the contact line dynamics is nearly the same in spreading stage, while it becomes different and presents a substrate-dependent retraction behavior. In Figure 7b and 7d, it is observed that depends less on the roughness of substrates

than the dose, and its value remains constant roughly. Taken together,

aforementioned results suggest that impact droplet dynamics is dramatically modulated by the roughness of substrate: with decreasing of solid fraction, the maximum spreading factor decreases and the rebound velocity increases. The explanation for this observation is that surface with low solid fraction exerts a low attraction force on water molecules, leading to a high contact angle and a low


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adhesion work. As a result, there is a small spreading radius and high rebound velocity.

Figure 6. Comparison of the dynamical behavior of identical nanodroplet impacting on nanopillar surfaces with different solid fractions. Sequencing from (a) to (h) are substrates with solid fraction of 100%, 56.24%, 42.23%, 30.24%, 20.26% and 12.25%.


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Figure 7. (a) Temporal evolution of the droplet’s centroid height on different surfaces during impact. (b) The variation of maximum spreading time versus solid fraction. (c) Temporal evolution of the droplet’s spreading radius on different surfaces. (d) The variation of maximum spreading factor versus solid fraction. 3. Theoretical model for maximum spreading factor To understand the impact dynamics of nanodroplets on textured surfaces, we conduct an energy analysis. For nanostructured surface, all energy terms involved before impact and at the maximum spreading state are modified and taken into consideration. The initial kinetic energy is XY = π /12, the surface energy of

droplet before impingement is X[ = π \], where \] denotes the liquid-vapor

surface tension, the interfacial energy of substrate in vapor is X^ = π -. .] /4,

where -. denotes the solid fraction and .] is solid-vapor surface tension. At the


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maximum spreading state, the droplet is treated as a cylinder with a height of H, which can be obtained from the volume conservation π_ /4 = π /6. The dissipation energy during spreading is estimated to be X` = a ghi ae bdΩd = Ω a ghi bd f

f

where b is the viscous dissipation function b = 7

jkl

jmn

0

jkn

?

jkl

jml jmn

≈ p Js q

r

(2)

(3)

where Ω is the characteristic volume for viscous dissipation, is the maximum spreading time and h is the characteristic height during spreading. In the most of previous models, the characteristic height h is simply replaced by the value at maximum spreading state, namely H. In 2015, Li et al.33 proposed a refined model, where the characteristic height is assumed to be ℎ = @ . Here, in our modified

model, we assume that ℎ = @

qJ f

, which is more reasonable because the droplet

velocity change from to 0 during spreading, and boundary conditions are ℎ =

in = 0 and ℎ = _ in = , thus

X` = t / @ 2 "

(4)

At maximum spreading state, the droplet kinetic energy is XY u = 0, the surface

energy of droplet is X[ u = v/2 @ -. 2π /4 0 π _w\], and the interfacial

energy of substrate in liquid is X^ u = π -. .\ /4, where .\ is solid-liquid interface tension. The value of liquid-vapor surface tension \] was taken from the


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property of water in 298K, Young’s equation and Cassie-Baxter’s equation are introduced in the simplification and calculation:cosL = /.] @ .\ 2/\] ,cosLK =

-. cosL 0 -. @ 1, where L is the intrinsic contact angle on smooth surface, LK is

the apparent contact angle on rough surface, L and LK are obtained from our simulation results and computed in the same way as our previous studies.9,14 Substituting the refined estimation of viscous dissipation and other modified energy terms into the energy balance equation: XY 0 X[ 0 X^ = XY u 0 X[u 0 X^ u 0 X`. After simplification, we obtain 3 p2

z) ()

0 1 @ cosL{ s @ p 0 4

z) ()

0 12s 0 8 = 0

(5)

Solving this equation, we find = |

~ }"

z)}

~ }"[

cos arccos36| "

~ }"[ ~ pz)} }"s

(6)

This model is referred to as the modified model, where we introduce refined energy dissipation and modified surface energy terms for structure substrate. Figure 8 illustrates the variation of maximum spreading factor versus impact velocity in different models. Compared to previous models, our refined model is consistent with our simulation results both in trend and in order of magnitude, and provides a more accurate prediction of for nanodroplets, especially in the low to moderate velocity range.


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Figure 8. Comparison between previous models, modified model and simulation results for maximum spreading factor as a function of impinging velocity. CONCLUSIONS

In this work, the impact process of nanodroplets was studied quantitatively using the molecular dynamics simulation. The important parameters, including centroid height, spreading radius, maximum spreading factor, maximum spreading time and restitution coefficient were calculated, recorded and discussed in detail. Apart from some similar rebound phenomena and variation rules, we found there are other distinctive results for nanodroplets compared with macroscale droplets: is

described as a power law of impact velocity and the relation of with impact velocity or the Reynolds number is exponential. And then, we investigated the identical nanodroplet impingement on nanopillar surfaces with various solid fractions.


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Owing to the diverse attraction force between the surfaces and water molecules, impact droplet dynamics is dramatically modulated by the roughness of substrate: with decreasing of solid fraction, the maximum spreading factor decreases and the rebound velocity increases. At last, we modified the estimation of viscous dissipation and surface energy terms for nanodroplets, and an improved model for maximum spreading factor was proposed, which describe nanodroplets’ impact behavior more precisely. We believe these findings provide a fundamental insight into the mechanism of impact nanodroplets, and have implications for predicting and modulating nanodroplets dynamical behavior.

ACKNOWLEDGEMENTS

This project was supported by the National Natural Science Foundation of China (No.51736004 and No.51776079). The study was performed at the National Supercomputer Center in Tianjin, and the calculations were performed on TianHe-1(A).

AUTHOR INFORMATION

Notes The authors declare no competing financial interests.

REFERENCES


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1

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